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Research Article

An identification problem related to mud filtrate invasion phenomenon during drilling operations

Tudor Boacaa Faculty of Letters and Sciences, Petroleum-Gas University of Ploiesti, Ploiesti, Romania;b Institute of Mathematics of the Romanian Academy, Bucuresti, RomaniaCorrespondence[email protected]

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Abstract

In this paper, we study two identification problems related to the mud filtrate invasion phenomenon. We want to determine a parameter (the invasion rate) in the coefficients of the parabolic equation that describes the mud filtrate invasion phenomenon. In the first problem, we determine this parameter starting from the observed values of the mud filtrate dispersion. We reduce the problem to an optimal control problem and prove the existence of the optimal control. In the second problem, we determine the invasion rate imposing the minimum condition of the quantity of mud filtrate that diffuses into the oil reservoir. We also reduce the identification problem to an optimal control problem. We prove the existence of the optimal control and we obtain a simple explicit form of this optimal control. A numerical example is presented for the second problem.

Keywords:

2010 Mathematics Subject Classifications:

1. Introduction

In this paper, we study two identification problems related to the invasion phenomenon arising in the borehole drilling process. During this process the drilling mud is pumped inside of the drill bore (towards the drill bit) and then circulated back to the surface through the annular space between the borehole and the drill bore wall. The drilling mud has sufficiently high density in order to avoid eruptive phenomena so that the hydrostatic pressure of the mud column is higher than the fluid formation pressure. This difference in pressure generates an undesirable phenomenon: the water from the drilling mud (hereinafter referred to as the mud filtrate) invades the formation and radially displaces the hydrocarbons and water from the formation. This invasion alters the reservoir properties in the vicinity of the well bore. The permeability, the porosity and the electrical resistivity of the reservoir are affected. For this reason, the invaded zone is also called the damaged zone.

During the mud filtrate invasion process, a part of the solid particles contained in the drilling mud penetrates the porous medium and another part is deposited on the well wall forming the mudcake. The most used model of the invasion phenomena is the so-called linear single phase mud filtrate invasion model. This model contains two parts:

the mudcake thickness model;
the mud filtrate invasion through the porous media model [Citation1].

a. The mudcake thickness model

Let $ξ_{c}$ be the mudcake thickness, $x_{w}$ the well radius, h the reservoir (formation) thickness, $Φ_{c},$ $ρ_{c}$ the porosity and the mudcake density, respectively, Φ the formation porosity, $c_{s}$ the solid particle dispersion in the drilling mud, k and $k_{c}$ the reservoir permeability and the mudcake permeability, respectively, $u (t)$ the volumetric flow rate (or invasion rate) (see Figure ). Then, the mudcake growth is described by the following Cauchy problem [Citation1]: (1) $\begin{aligned} \frac{d ξ_{c}}{d t} & = c \frac{u (t)}{x_{w} - ξ_{c} (t)}, \end{aligned}$ (1) (2) $\begin{aligned} ξ_{c} (0) & = 0, \end{aligned}$ (2) where (3) $\begin{aligned} u (t) = u_{M} \frac{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}}}{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}} - \ln (1 - \frac{ξ_{c} (t)}{r_{w}})}, \end{aligned}$ (3) (4) $\begin{aligned} u_{M} = \frac{2 π k h Δ p}{μ_{f} \ln \frac{r_{e}}{r_{w}}}, \end{aligned}$ (4) and $c = \frac{(1 - Φ) c_{s}}{2 π h (1 - Φ_{c}) ρ_{c}}$ .

Figure 1. The geometry of the mud-invasion phenomenon. A, wellbore; B, mudcake; C, the invaded zone; D, the drainage zone; $x_{w}$ , the wellbore radius; $x_{e}$ , the drainage radius; $ξ_{c}$ , the mudcake thickness; $d_{p}$ , the penetration depth.

From (Equation2(2) $\begin{aligned} ξ_{c} (0) & = 0, \end{aligned}$ (2) ), (Equation3(3) $\begin{aligned} u (t) = u_{M} \frac{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}}}{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}} - \ln (1 - \frac{ξ_{c} (t)}{r_{w}})}, \end{aligned}$ (3) ) and (Equation4(4) $\begin{aligned} u_{M} = \frac{2 π k h Δ p}{μ_{f} \ln \frac{r_{e}}{r_{w}}}, \end{aligned}$ (4) ) we obtain that $u_{M} = u (0)$ is the maximum value of filtration rate.

b. The mud filtrate dispersion through the porous media model.

In [Citation1], the authors propose the following model in order to determine the mud filtrate dispersion C in the invaded zone: (5) $\frac{\partial C}{\partial t} = \frac{1}{x} \frac{\partial}{\partial x} (x D \frac{\partial C}{\partial x}) - \frac{u (t)}{2 π x h Φ (1 - S_{w i r} - S_{o r})} \frac{\partial C}{\partial x},$ (5) for $(t, x) \in (0, T) \times (x_{w}, x_{e}),$ (6) $\begin{aligned} C (0, x) = C_{0} (x), x \in (x_{w}, x_{e}), \end{aligned}$ (6) (7) $\begin{aligned} C (t, x_{w}) = C_{w}, C (t, x_{e}) = 0, t \in (0, T) . \end{aligned}$ (7) The mixed problem (Equation5(5) $\frac{\partial C}{\partial t} = \frac{1}{x} \frac{\partial}{\partial x} (x D \frac{\partial C}{\partial x}) - \frac{u (t)}{2 π x h Φ (1 - S_{w i r} - S_{o r})} \frac{\partial C}{\partial x},$ (5) )–(Equation7(7) $\begin{aligned} C (t, x_{w}) = C_{w}, C (t, x_{e}) = 0, t \in (0, T) . \end{aligned}$ (7) ) describes the mud filtrate invasion through the porous media. The initial condition (Equation6(6) $\begin{aligned} C (0, x) = C_{0} (x), x \in (x_{w}, x_{e}), \end{aligned}$ (6) ) specifies that at t = 0 the mud filtrate dispersion is $C_{0} .$ In the Dirichlet boundary conditions (Equation7(7) $\begin{aligned} C (t, x_{w}) = C_{w}, C (t, x_{e}) = 0, t \in (0, T) . \end{aligned}$ (7) ), $C_{w}$ represents the mud filtrate dispersion at the wall well. Instead of conditions (Equation7(7) $\begin{aligned} C (t, x_{w}) = C_{w}, C (t, x_{e}) = 0, t \in (0, T) . \end{aligned}$ (7) ), the Dirichlet–Neumann boundary conditions can also be used [Citation2, p.747]: (8) $C (t, x_{w}) = C_{w}, \frac{\partial C}{\partial x} (t, x_{e}) = 0, t \in (0, T) .$ (8) The first term in the right member of Equation (Equation5(5) $\frac{\partial C}{\partial t} = \frac{1}{x} \frac{\partial}{\partial x} (x D \frac{\partial C}{\partial x}) - \frac{u (t)}{2 π x h Φ (1 - S_{w i r} - S_{o r})} \frac{\partial C}{\partial x},$ (5) ) is the conductive term. Here D is the diffusion coefficient. Based on experimental results, several authors proposed for D the following formula [Citation1,Citation3,Citation4]: (9) $D = α {(\frac{u (t)}{2 π x h})}^{β},$ (9) where $α > 0$ and $β > 1$ are empirical parameters. For example, in the sandstone formation $α = 51.7$ and $β = 1.25$ [Citation1]. The second term of the right member of Equation (Equation5(5) $\frac{\partial C}{\partial t} = \frac{1}{x} \frac{\partial}{\partial x} (x D \frac{\partial C}{\partial x}) - \frac{u (t)}{2 π x h Φ (1 - S_{w i r} - S_{o r})} \frac{\partial C}{\partial x},$ (5) ) is the convective term, where $S_{w i r}$ and $S_{o r}$ are the irreducible water saturation and the residual oil saturation, respectively [Citation1].

If we use the following dimensionless variables, denoted by tilde superscript, (10) $\begin{aligned} \tilde{x} = \frac{x}{x_{w}}, L = \frac{x_{e}}{x_{w}}, {\tilde{ξ}}_{c} = \frac{ξ_{c}}{x_{w}}, \tilde{t} = \frac{t}{t_{0}}, \\ \tilde{C} = \frac{C}{C_{w}}, {\tilde{C}}_{0} = \frac{C_{0}}{C_{w}}, \tilde{u} = \frac{u}{u_{M}}, \tilde{D} = \frac{D}{α {(\frac{u_{M}}{2 π x_{w} h})}^{β}}, \end{aligned}$ (10) where $t_{0} = \frac{2 π x_{w}^{2} h Φ (1 - S_{w i r} - S_{o r})}{u_{M}} .$ The model (Equation1(1) $\begin{aligned} \frac{d ξ_{c}}{d t} & = c \frac{u (t)}{x_{w} - ξ_{c} (t)}, \end{aligned}$ (1) )–(Equation2(2) $\begin{aligned} ξ_{c} (0) & = 0, \end{aligned}$ (2) ), (Equation5(5) $\frac{\partial C}{\partial t} = \frac{1}{x} \frac{\partial}{\partial x} (x D \frac{\partial C}{\partial x}) - \frac{u (t)}{2 π x h Φ (1 - S_{w i r} - S_{o r})} \frac{\partial C}{\partial x},$ (5) )–(Equation7(7) $\begin{aligned} C (t, x_{w}) = C_{w}, C (t, x_{e}) = 0, t \in (0, T) . \end{aligned}$ (7) ) becomes (11) $\begin{aligned} \frac{d ξ_{c}}{d t} = c_{1} \frac{u (t)}{1 - ξ_{c} (t)}, \end{aligned}$ (11) (12) $\begin{aligned} ξ_{c} (0) = 0, \end{aligned}$ (12) (13) $\begin{aligned} \frac{\partial C}{\partial t} = \frac{1}{P e} \frac{1}{x} \frac{\partial}{\partial x} (x {(\frac{u (t)}{x})}^{β} \frac{\partial C}{\partial x}) - \frac{u (t)}{x} \frac{\partial C}{\partial x}, \end{aligned}$ (13) (14) $\begin{aligned} C (0, x) = C_{0}, x \in (1, L), \end{aligned}$ (14) (15) $\begin{aligned} C (t, 1) = 1, C (t, L) = 0, t \in (0, T), \end{aligned}$ (15) and the Dirichlet–Neumann boundary conditions (Equation8(8) $C (t, x_{w}) = C_{w}, \frac{\partial C}{\partial x} (t, x_{e}) = 0, t \in (0, T) .$ (8) ) become (16) $C (t, 1) = 1, \frac{\partial C}{\partial x} (t, L) = 0, t \in (0, T),$ (16) where $c_{1} = \frac{Φ (1 - Φ) (1 - S_{w i r} - S_{o r}) c_{s}}{(1 - Φ_{c}) ρ_{c}}$ and (17) $P e = \frac{r_{w}^{2}}{α t_{0} {(\frac{u_{M}}{2 π x_{w} h})}^{β}}$ (17) is the Péclet number. In (Equation11(11) $\begin{aligned} \frac{d ξ_{c}}{d t} = c_{1} \frac{u (t)}{1 - ξ_{c} (t)}, \end{aligned}$ (11) )–(Equation16(16) $C (t, 1) = 1, \frac{\partial C}{\partial x} (t, L) = 0, t \in (0, T),$ (16) ), the tilde notation was omitted.

For convenience we write Equation (Equation13(13) $\begin{aligned} \frac{\partial C}{\partial t} = \frac{1}{P e} \frac{1}{x} \frac{\partial}{\partial x} (x {(\frac{u (t)}{x})}^{β} \frac{\partial C}{\partial x}) - \frac{u (t)}{x} \frac{\partial C}{\partial x}, \end{aligned}$ (13) ) under the form (18) $\frac{\partial C}{\partial t} = \frac{\partial}{\partial x} (P e^{- 1} x^{- β} u^{β} (t) \frac{\partial C}{\partial x}) + (P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t)) \frac{\partial C}{\partial x},$ (18) for $(t, x) \in Q,$ where $Q := (0, T) \times (1, L) .$ Hereinafter we will call the problems (Equation11(11) $\begin{aligned} \frac{d ξ_{c}}{d t} = c_{1} \frac{u (t)}{1 - ξ_{c} (t)}, \end{aligned}$ (11) )–(Equation12(12) $\begin{aligned} ξ_{c} (0) = 0, \end{aligned}$ (12) ) and (Equation18(18) $\frac{\partial C}{\partial t} = \frac{\partial}{\partial x} (P e^{- 1} x^{- β} u^{β} (t) \frac{\partial C}{\partial x}) + (P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t)) \frac{\partial C}{\partial x},$ (18) ), (Equation14(14) $\begin{aligned} C (0, x) = C_{0}, x \in (1, L), \end{aligned}$ (14) ), (Equation15(15) $\begin{aligned} C (t, 1) = 1, C (t, L) = 0, t \in (0, T), \end{aligned}$ (15) ) the linear single phase mud filtrate invasion model.

If all the physical properties of the oil reservoir are known, then we can determine the mudcake thickness by solving the Cauchy problem (Equation11(11) $\begin{aligned} \frac{d ξ_{c}}{d t} = c_{1} \frac{u (t)}{1 - ξ_{c} (t)}, \end{aligned}$ (11) )–(Equation12(12) $\begin{aligned} ξ_{c} (0) = 0, \end{aligned}$ (12) ). Once the mud filtrate thickness is determined we obtain the mud filtrate dispersion by solving the parabolic mixed problem (Equation18(18) $\frac{\partial C}{\partial t} = \frac{\partial}{\partial x} (P e^{- 1} x^{- β} u^{β} (t) \frac{\partial C}{\partial x}) + (P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t)) \frac{\partial C}{\partial x},$ (18) ), (Equation14(14) $\begin{aligned} C (0, x) = C_{0}, x \in (1, L), \end{aligned}$ (14) ), (Equation15(15) $\begin{aligned} C (t, 1) = 1, C (t, L) = 0, t \in (0, T), \end{aligned}$ (15) ) (or (Equation16(16) $C (t, 1) = 1, \frac{\partial C}{\partial x} (t, L) = 0, t \in (0, T),$ (16) )). If we do not know all the physical properties of the oil reservoir and mudcake then we can put the problem of determining these unknown properties using other observed physical properties. Thus we obtain an identification problem which we reduce to an optimal control problem in the coefficients of a parabolic equation.

In the first identification problem, we are dealing with in this paper we aim at determining the volumetric flow rate (or invasion rate) u starting from the observed values, available at a time T, of the mud filtrate dispersion. This objective will be achieved by solving the minimization problem $Minimize \frac{1}{2} {‖ C (T, \cdot) - C_{r e f} (\cdot) ‖}_{L^{2} (1, L)}^{2},$ subject to (Equation18(18) $\frac{\partial C}{\partial t} = \frac{\partial}{\partial x} (P e^{- 1} x^{- β} u^{β} (t) \frac{\partial C}{\partial x}) + (P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t)) \frac{\partial C}{\partial x},$ (18) ), (Equation14(14) $\begin{aligned} C (0, x) = C_{0}, x \in (1, L), \end{aligned}$ (14) ), (Equation15(15) $\begin{aligned} C (t, 1) = 1, C (t, L) = 0, t \in (0, T), \end{aligned}$ (15) ), for all u in an admissible set U, where $C_{r e f}$ is a reference (observed) concentration. Once the invasion rate is obtained we can also determine the mudcake thickness $ξ_{c}$ by solving (Equation1(1) $\begin{aligned} \frac{d ξ_{c}}{d t} & = c \frac{u (t)}{x_{w} - ξ_{c} (t)}, \end{aligned}$ (1) )–(Equation2(2) $\begin{aligned} ξ_{c} (0) & = 0, \end{aligned}$ (2) ).

In the second identification problem, we aim at determining the volumetric flow rate u that produces a minimum amount of mud filtrate that invades the reservoir. Thus we obtain the following minimization problem $Minimize \int_{1}^{L} C (T, x) d x,$ subject to (Equation18(18) $\frac{\partial C}{\partial t} = \frac{\partial}{\partial x} (P e^{- 1} x^{- β} u^{β} (t) \frac{\partial C}{\partial x}) + (P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t)) \frac{\partial C}{\partial x},$ (18) ), (Equation14(14) $\begin{aligned} C (0, x) = C_{0}, x \in (1, L), \end{aligned}$ (14) ), (Equation16(16) $C (t, 1) = 1, \frac{\partial C}{\partial x} (t, L) = 0, t \in (0, T),$ (16) ), for all u in a some set U.

The structure of the paper is: in Section 2 we deal with first identification problem. Here we prove the existence for the state system and the existence for the optimal control. We obtain the first-order condition of optimality using the system of first order variations and the dual system. Section 3 contains the second identification problem. In this case, we obtain a simple explicit form of optimal control $u^{*}$ . In Section 4, we give some numerical results and the last section contains the conclusion of the paper.

2. Dirichlet boundary conditions

In this section, we study an identification problem related to the state systems (Equation18(18) $\frac{\partial C}{\partial t} = \frac{\partial}{\partial x} (P e^{- 1} x^{- β} u^{β} (t) \frac{\partial C}{\partial x}) + (P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t)) \frac{\partial C}{\partial x},$ (18) ), (Equation14(14) $\begin{aligned} C (0, x) = C_{0}, x \in (1, L), \end{aligned}$ (14) ), (Equation15(15) $\begin{aligned} C (t, 1) = 1, C (t, L) = 0, t \in (0, T), \end{aligned}$ (15) ). The purpose of this problem is to determine the filtration rate u starting from the observed values of the mud filtrate dispersion C available at a time T. We consider that the filtration rate $u (t)$ is the control and that the solution $C (t, x)$ of the mixed problem (Equation18(18) $\frac{\partial C}{\partial t} = \frac{\partial}{\partial x} (P e^{- 1} x^{- β} u^{β} (t) \frac{\partial C}{\partial x}) + (P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t)) \frac{\partial C}{\partial x},$ (18) ), (Equation14(14) $\begin{aligned} C (0, x) = C_{0}, x \in (1, L), \end{aligned}$ (14) ), (Equation15(15) $\begin{aligned} C (t, 1) = 1, C (t, L) = 0, t \in (0, T), \end{aligned}$ (15) ) is the state variable. The analysis of the invasion phenomenon presented in the introduction leads us to the conclusion that the filtration rate $u (t)$ varies between a maximum value and a minimum value. These values are determined by the physical characteristics of the reservoir (formation) and of the drilling mud, and by the pressure difference value $Δ p$ . We can also consider that the rate of variation of filtration rate is bounded. Starting from these observations, we will consider the set U under the form of (19) $\begin{aligned} U = {u \in W^{1, \infty} (0, T); u_{m} \leq u (t) \leq 1, \\ u (0) = 1, u (T) = u_{m}, | u^{'} (t) | \leq u_{\infty}, a . e . t \in (0, T)}, \end{aligned}$ (19) where $u_{m} \in R$ , $0 < u_{m} < 1,$ $u_{\infty} > 0$ are given constants. In what follows, we call the set U as the control set.

In what follows we use the standard notations for the Sobolev spaces $W^{m, p},$ $H^{m} = W^{m, 2},$ $H_{0}^{1}$ and the Lebesgue spaces $L^{p},$ $1 \leq p \leq \infty,$ $m \in N^{*} .$ We denote by $C ([0, T]; X)$ the space of X-valued continuous functions on $[0, T]$ and by $L^{p} (0, T; X)$ the space of X-valued $L^{p}$ -Bochner integrable functions on $(0, T) .$ We denote by $H^{1} (0, T; X)$ the space of functions $u \in L^{2} (0, T; X)$ with $\frac{d u}{d t} \in L^{2} (0, T; X) .$ X above is a Banach space.

We define the functional cost (20) $J_{1} (u) = \frac{1}{2} {‖ C (T, \cdot) - C_{r e f} (\cdot) ‖}_{L^{2} (1, L)}^{2},$ (20) where function $C_{r e f} (x)$ represents a known reference (observed) mud filtrate dispersion. This dispersion can be experimentally determined by using reservoir resistivity measurements and resistivity-water saturation correlations [Citation5]. Our goal is to determine $u = u^{*}$ that minimizes J on the set U. We introduce the following minimization problem: (21) $inf_{u \in U} J_{1} (u),$ (21) where the control set U is given in (Equation19(19) $\begin{aligned} U = {u \in W^{1, \infty} (0, T); u_{m} \leq u (t) \leq 1, \\ u (0) = 1, u (T) = u_{m}, | u^{'} (t) | \leq u_{\infty}, a . e . t \in (0, T)}, \end{aligned}$ (19) ).

Let us consider the change of variable (22) $C = y + \frac{L - x}{L - 1} .$ (22) The state system (Equation18(18) $\frac{\partial C}{\partial t} = \frac{\partial}{\partial x} (P e^{- 1} x^{- β} u^{β} (t) \frac{\partial C}{\partial x}) + (P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t)) \frac{\partial C}{\partial x},$ (18) ), (Equation14(14) $\begin{aligned} C (0, x) = C_{0}, x \in (1, L), \end{aligned}$ (14) ), (Equation15(15) $\begin{aligned} C (t, 1) = 1, C (t, L) = 0, t \in (0, T), \end{aligned}$ (15) ) becomes (23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) (24) $\begin{aligned} y (0, x) = y_{0} (x), x \in (1, L), \end{aligned}$ (24) (25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) The coefficients of Equation (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) ) depend on x and t (the dependence on t is achieved by the control u). The key to us is the dependence of the coefficients of Equation (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) ) on control u. Therefore, in order not to complicate the notations, we write (26) $\begin{aligned} a (u) = P e^{- 1} x^{- β} u^{β} (t), \end{aligned}$ (26) (27) $\begin{aligned} b (u) = P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t), \end{aligned}$ (27) (28) $\begin{aligned} f (u) = P e^{- 1} (β - 1) {(L - 1)}^{- 1} x^{- β - 1} u^{β} (t) - {(L - 1)}^{- 1} x^{- 1} u (t), \end{aligned}$ (28) (29) $\begin{aligned} y_{0} (x) = C_{0} (x) + {(L - 1)}^{- 1} (x - L) . \end{aligned}$ (29) The functional cost (Equation20(20) $J_{1} (u) = \frac{1}{2} {‖ C (T, \cdot) - C_{r e f} (\cdot) ‖}_{L^{2} (1, L)}^{2},$ (20) ) takes the following form: (30) $J (u) = \frac{1}{2} {‖ y^{u} (T, \cdot) - y_{r e f} (\cdot) ‖}_{L^{2} (1, L)}^{2},$ (30) where (31) $y_{r e f} (x) = C_{r e f} (x) - \frac{L - x}{L - 1}$ (31) and $y^{u}$ is the solution of (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ) corresponding to $u \in U$ . Problem (Equation21(21) $inf_{u \in U} J_{1} (u),$ (21) ) becomes (P) $inf_{u \in U} J (u),$ (P) where the control set is given by (Equation19(19) $\begin{aligned} U = {u \in W^{1, \infty} (0, T); u_{m} \leq u (t) \leq 1, \\ u (0) = 1, u (T) = u_{m}, | u^{'} (t) | \leq u_{\infty}, a . e . t \in (0, T)}, \end{aligned}$ (19) ). We call this minimization problem problem (P).

In the following, we will use the notations $a^{'} (u^{*}),$ $b^{'} (u^{*}),$ $f^{'} (u^{*})$ for the derivatives of the functions a, b, f with respect to u, calculated for $u = u^{*},$ namely: $\begin{aligned} a^{'} (u^{*}) := β \cdot P e^{- 1} \cdot x^{- β} \cdot (u^{*} (t))^{β - 1}, \\ b^{'} (u^{*}) := β \cdot P e^{- 1} \cdot x^{- β - 1} \cdot (u^{*} (t))^{β - 1} - x^{- 1} \end{aligned}$ and $f^{'} (u^{*}) := \frac{β (β - 1)}{P e \cdot (L - 1)} \cdot \frac{(u^{*} (t))^{β - 1}}{x^{β + 1}} - \frac{1}{(L - 1) \cdot x} .$

Remark 2.1

From (Equation26(26) $\begin{aligned} a (u) = P e^{- 1} x^{- β} u^{β} (t), \end{aligned}$ (26) ), (Equation27(27) $\begin{aligned} b (u) = P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t), \end{aligned}$ (27) ), (Equation28(28) $\begin{aligned} f (u) = P e^{- 1} (β - 1) {(L - 1)}^{- 1} x^{- β - 1} u^{β} (t) - {(L - 1)}^{- 1} x^{- 1} u (t), \end{aligned}$ (28) ) we obtain that the functions $a (u),$ $b (u),$ $f (u)$ have the following properties:

(i) $a (u),$ $b (u),$ $f (u), a^{'} (u), b^{'} (u), f^{'} (u), a_{x} (u), b_{x} (u), f_{x} (u) \in L^{\infty} (Q)$ and the norms in $L^{\infty} (Q)$ of these functions are bounded by positive constants that do not depend on $u \in U$ . In what follows we use several times this property;

(ii) $a_{m} > 0$ exists so that $a_{m} \leq a (u)$ a. e. $(t, x) \in Q,$ $(\forall) u \in U$ ;

(iii) the positive constants $L_{a}, L_{b}$ and $L_{f}$ exist so that for all $u_{1},$ $u_{2} \in U$ we have (32) $\begin{aligned} | a (u_{1}) - a (u_{2}) | \leq L_{a} | u_{1} (t) - u_{2} (t) |, \end{aligned}$ (32) (33) $\begin{aligned} | b (u_{1}) - b (u_{2}) | \leq L_{b} | u_{1} (t) - u_{2} (t) |, \end{aligned}$ (33) (34) $\begin{aligned} | f (u_{1}) - f (u_{2}) | \leq L_{f} | u_{1} (t) - u_{2} (t) |, \end{aligned}$ (34) for all $x \in [1, L]$ and a.e. $t \in (0, T);$

(iv) conditions of the types (Equation32(32) $\begin{aligned} | a (u_{1}) - a (u_{2}) | \leq L_{a} | u_{1} (t) - u_{2} (t) |, \end{aligned}$ (32) ), (Equation33(33) $\begin{aligned} | b (u_{1}) - b (u_{2}) | \leq L_{b} | u_{1} (t) - u_{2} (t) |, \end{aligned}$ (33) ) and (Equation34(34) $\begin{aligned} | f (u_{1}) - f (u_{2}) | \leq L_{f} | u_{1} (t) - u_{2} (t) |, \end{aligned}$ (34) ) also occur for the derivatives $a^{'},$ $b^{'}$ and $f^{'}$ with Lipschitz constants $L_{a^{'}},$ $L_{b^{'}}$ and $L_{f^{'}}$ respectively.

The structure of this section is the following: first we prove the existence of the solution of the state system (Equation18(18) $\frac{\partial C}{\partial t} = \frac{\partial}{\partial x} (P e^{- 1} x^{- β} u^{β} (t) \frac{\partial C}{\partial x}) + (P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t)) \frac{\partial C}{\partial x},$ (18) ), (Equation14(14) $\begin{aligned} C (0, x) = C_{0}, x \in (1, L), \end{aligned}$ (14) ), (Equation15(15) $\begin{aligned} C (t, 1) = 1, C (t, L) = 0, t \in (0, T), \end{aligned}$ (15) ). Then, we prove the existence of an optimal control $u^{*}$ that minimizes the functional cost (Equation30(30) $J (u) = \frac{1}{2} {‖ y^{u} (T, \cdot) - y_{r e f} (\cdot) ‖}_{L^{2} (1, L)}^{2},$ (30) ). In order to obtain the necessary condition of optimality, we introduce the system of first-order variations, we prove the existence and the unicity of the solution of this system and we highlight the connection that exists between the state system and the first order variations system. At the end of this section, we introduce the dual system and we obtain the necessary condition of optimality.

2.1. Existence for the state system

Let us denote $V = H_{0}^{1} (1, L),$ $H = L^{2} (1, L)$ and $V^{'} = H^{- 1} (1, L)$ the dual of V. We identify H with its own dual and we have $V ↪ H \equiv H^{'} ↪ V^{'},$ with continuous and dense embeddings. In what follows, we denote $φ_{t}, φ_{x}$ the partial derivative of the function $φ (t, x)$ with respect to t and x and $‖ φ ‖_{L^{\infty} (Q)} = ‖ φ ‖_{\infty}$ for $φ \in L^{\infty} (Q)$ . For convenience, we shall not write the arguments of the function in the integrands.

Definition 2.1

A function $y \in L^{2} (0, T; V)$ with $\frac{d y}{d t} \in L^{2} (0, T; V^{'})$ is a solution of problem (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ) if (35) $\begin{aligned} \int_{0}^{T} {⟨ \frac{d y}{d t} (t), ψ (t) ⟩}_{V^{'}, V} d t + \int_{0}^{T} \int_{1}^{L} (a (u) y_{x} ψ_{x} - b (u) y_{x} ψ) d x d t \\ = \int_{0}^{T} \int_{1}^{L} f (u) ψ d x d t, for all ψ \in L^{2} (0, T; V) \end{aligned}$ (35) and (36) $y (0) = y_{0} .$ (36)

We note that the conditions $φ \in L^{2} (0, T; V)$ and $\frac{d φ}{d t} \in L^{2} (0, T; V^{'})$ imply that $φ \in C ([0, T], H)$ and therefore (Equation36(36) $y (0) = y_{0} .$ (36) ) makes sense.

We consider the family of operators $A (t) : V \to V^{'},$ defined on V, a.e. $t \in (0, T)$ by (37) ${⟨ A (t) y, ψ ⟩}_{V^{'}, V} = \int_{1}^{L} (a (u) y_{x} ψ_{x} (t) - b (u) y_{x} ψ) d x, (\forall) ψ \in V$ (37) and introduce the Cauchy problem (38) $\begin{aligned} \frac{d y}{d t} (t) + A (t) y (t) = f (u) (t), a . e . t \in (0, T) \end{aligned}$ (38) (39) $\begin{aligned} y (0) = y_{0} . \end{aligned}$ (39) Equation (Equation38(38) $\begin{aligned} \frac{d y}{d t} (t) + A (t) y (t) = f (u) (t), a . e . t \in (0, T) \end{aligned}$ (38) ) can be written as (40) ${⟨ \frac{d y}{d t} (t), ψ ⟩}_{V^{'}, V} + {⟨ A (t), ψ ⟩}_{V^{'}, V} = {⟨ f (u), ψ ⟩}_{V^{'}, V} a . e . t \in (0, T), (\forall) ψ \in V .$ (40) The problems (Equation38(38) $\begin{aligned} \frac{d y}{d t} (t) + A (t) y (t) = f (u) (t), a . e . t \in (0, T) \end{aligned}$ (38) )–(Equation39(39) $\begin{aligned} y (0) = y_{0} . \end{aligned}$ (39) ) and (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ) are equivalent [Citation6,Citation7].

Theorem 2.1

Let $y_{0} \in H$ . Then for every $u \in U,$ the problem (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ) has a unique solution $y^{u},$ (41) $y^{u} \in C ([0, T]; H) \cap L^{2} (0, T; V) \cap H^{1} (0, T; V^{'})$ (41) which satisfies the following estimate: (42) $\begin{aligned} sup_{t \in [0, T]} {‖ y^{u} (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ y^{u} (t) ‖}_{V}^{2} d t \leq C, \end{aligned}$ (42) (43) $\begin{aligned} \int_{0}^{T} {‖ \frac{d y^{u}}{d t} (t) ‖}_{V^{'}}^{2} d t \leq C_{1}, \end{aligned}$ (43) where the constants C and $C_{1}$ depend on T, on the structure of the system (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ) and on the norm of the initial data (Equation24(24) $\begin{aligned} y (0, x) = y_{0} (x), x \in (1, L), \end{aligned}$ (24) ). Moreover if $u_{i} \in U, i = 1, 2$ and $y^{u_{i}}, i = 1, 2$ are the corresponding solutions of the state system then the estimate (44) ${‖ y^{u_{1}} - y^{u_{2}} ‖}_{L^{\infty} (0, T; H)}^{2} + {‖ y^{u_{1}} - y^{u_{2}} ‖}_{L^{2} (0, T; V)}^{2} \leq C_{2} {‖ u_{1} - u_{2} ‖}_{\infty}^{2},$ (44) takes place, where $C_{2}$ depends only on L, T and on the structure of the system (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ).

Proof.

From (Equation28(28) $\begin{aligned} f (u) = P e^{- 1} (β - 1) {(L - 1)}^{- 1} x^{- β - 1} u^{β} (t) - {(L - 1)}^{- 1} x^{- 1} u (t), \end{aligned}$ (28) ) and (Equation19(19) $\begin{aligned} U = {u \in W^{1, \infty} (0, T); u_{m} \leq u (t) \leq 1, \\ u (0) = 1, u (T) = u_{m}, | u^{'} (t) | \leq u_{\infty}, a . e . t \in (0, T)}, \end{aligned}$ (19) ), we obtain that $f^{u} \in W^{1, \infty} (0, T; C^{\infty} [1, L]),$ for all $u \in U$ . It is easy to see that the family of operators defined by (Equation37(37) ${⟨ A (t) y, ψ ⟩}_{V^{'}, V} = \int_{1}^{L} (a (u) y_{x} ψ_{x} (t) - b (u) y_{x} ψ) d x, (\forall) ψ \in V$ (37) ) has the following properties:

(i) $A (t)$ is linear and continuous from V to $V^{'}$ , (45) ${‖ A (t) y ‖}_{V^{'}} \leq C_{3} {‖ y ‖}_{V}, forall t \in [0, T],$ (45) where $C_{3}$ is a positive constant which depends only on the norms of the coefficients $a (u)$ and $b (u)$ (see Remark 2.1, i);

(ii) $ϖ, γ > 0$ exist such that we have (46) ${⟨ A (t) y, y ⟩}_{V^{'}, V} \geq ϖ {‖ y ‖}_{V}^{2} - γ {‖ y ‖}_{H}^{2},$ (46) for all $y \in V$ and a.e. $t \in (0, T)$ . The existence and uniqueness of the solution of the problem (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ) is now obtained from a standard result [Citation8, Theorem 2, p.513]. In order to obtain the estimate (Equation42(42) $\begin{aligned} sup_{t \in [0, T]} {‖ y^{u} (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ y^{u} (t) ‖}_{V}^{2} d t \leq C, \end{aligned}$ (42) ) we test (Equation38(38) $\begin{aligned} \frac{d y}{d t} (t) + A (t) y (t) = f (u) (t), a . e . t \in (0, T) \end{aligned}$ (38) ) at $y^{u} \in L^{2} (0, T; V),$ we integrate on $(0, t)$ and we use Gronwall's lemma.

In order to obtain the estimate (Equation43(43) $\begin{aligned} \int_{0}^{T} {‖ \frac{d y^{u}}{d t} (t) ‖}_{V^{'}}^{2} d t \leq C_{1}, \end{aligned}$ (43) ), we start from Equation (Equation38(38) $\begin{aligned} \frac{d y}{d t} (t) + A (t) y (t) = f (u) (t), a . e . t \in (0, T) \end{aligned}$ (38) ) and we obtain (47) $\int_{0}^{T} {‖ \frac{d y^{u}}{d t} (t) ‖}_{V^{'}}^{2} d t \leq 2 \int_{0}^{T} {‖ f (u) ‖}_{V^{'}}^{2} d t + 2 \int_{0}^{T} {‖ A (t) y^{u} (t) ‖}_{V^{'}}^{2} d t .$ (47) The estimate (Equation43(43) $\begin{aligned} \int_{0}^{T} {‖ \frac{d y^{u}}{d t} (t) ‖}_{V^{'}}^{2} d t \leq C_{1}, \end{aligned}$ (43) ) results from (Equation47(47) $\int_{0}^{T} {‖ \frac{d y^{u}}{d t} (t) ‖}_{V^{'}}^{2} d t \leq 2 \int_{0}^{T} {‖ f (u) ‖}_{V^{'}}^{2} d t + 2 \int_{0}^{T} {‖ A (t) y^{u} (t) ‖}_{V^{'}}^{2} d t .$ (47) ) if we take into account the estimates (Equation45(45) ${‖ A (t) y ‖}_{V^{'}} \leq C_{3} {‖ y ‖}_{V}, forall t \in [0, T],$ (45) ) and (Equation42(42) $\begin{aligned} sup_{t \in [0, T]} {‖ y^{u} (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ y^{u} (t) ‖}_{V}^{2} d t \leq C, \end{aligned}$ (42) ) and the Remark 2.1i.

The goal next is to establish the estimate (Equation44(44) ${‖ y^{u_{1}} - y^{u_{2}} ‖}_{L^{\infty} (0, T; H)}^{2} + {‖ y^{u_{1}} - y^{u_{2}} ‖}_{L^{2} (0, T; V)}^{2} \leq C_{2} {‖ u_{1} - u_{2} ‖}_{\infty}^{2},$ (44) ). If $u_{1}, u_{2} \in U$ and $y^{u_{1}}, y^{u_{2}}$ are the corresponding solutions of the state system (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ), we have (48) $\begin{aligned} \frac{d y^{u_{i}}}{d t} (t) + A_{i} (t) y^{u_{i}} (t) = f (u_{i}), a . e . t \in (0, T), i = 1, 2 \end{aligned}$ (48) (49) $\begin{aligned} y^{u_{i}} (0) = y_{0}, i = 1, 2, \end{aligned}$ (49) where $A_{j} (t) = \int_{1}^{L} (a (u_{j}) y_{x} ψ_{x} (t) - b (u_{j}) y_{x} ψ) d x, (\forall) ψ \in V, j = 1, 2.$ Subtracting the equations and the initial conditions corresponding to these two Cauchy problems (Equation48(48) $\begin{aligned} \frac{d y^{u_{i}}}{d t} (t) + A_{i} (t) y^{u_{i}} (t) = f (u_{i}), a . e . t \in (0, T), i = 1, 2 \end{aligned}$ (48) )–(Equation49(49) $\begin{aligned} y^{u_{i}} (0) = y_{0}, i = 1, 2, \end{aligned}$ (49) ) we get (50) $\begin{aligned} \frac{d}{d t} (y^{u_{1}} - y^{u_{2}}) (t) + A_{1} (t) (y^{u_{1}} - y^{u_{2}}) (t) + (A_{1} (t) - A_{2} (t)) y^{u_{2}} (t) \\ = f (u_{1}) - f (u_{2}), a . e . t \in (0, T), \end{aligned}$ (50) (51) $\begin{aligned} (y^{u_{1}} - y^{u_{2}}) (0) = 0. \end{aligned}$ (51) We test (Equation50(50) $\begin{aligned} \frac{d}{d t} (y^{u_{1}} - y^{u_{2}}) (t) + A_{1} (t) (y^{u_{1}} - y^{u_{2}}) (t) + (A_{1} (t) - A_{2} (t)) y^{u_{2}} (t) \\ = f (u_{1}) - f (u_{2}), a . e . t \in (0, T), \end{aligned}$ (50) ) for $y^{u_{1}} - y^{u_{2}}$ and integrate on $(0, t)$ . We perform the same calculations we made when we got the estimate (Equation42(42) $\begin{aligned} sup_{t \in [0, T]} {‖ y^{u} (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ y^{u} (t) ‖}_{V}^{2} d t \leq C, \end{aligned}$ (42) ) and we obtain (Equation44(44) ${‖ y^{u_{1}} - y^{u_{2}} ‖}_{L^{\infty} (0, T; H)}^{2} + {‖ y^{u_{1}} - y^{u_{2}} ‖}_{L^{2} (0, T; V)}^{2} \leq C_{2} {‖ u_{1} - u_{2} ‖}_{\infty}^{2},$ (44) ) where (52) $C_{2} = \frac{\exp (a_{m} + {‖ b_{x}^{u} ‖}_{\infty} + 2)}{min {1; a_{m}}} {(\frac{L_{a}^{2}}{a_{m} + L_{b}^{2}}) C + L_{f}^{2} (L - 1) T},$ (52) and C is the constant which appears in (Equation42(42) $\begin{aligned} sup_{t \in [0, T]} {‖ y^{u} (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ y^{u} (t) ‖}_{V}^{2} d t \leq C, \end{aligned}$ (42) ).

2.2. Existence in (P)

In this section, we show that the minimization problem (P) subject to (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ) where the cost functional is given by (Equation30(30) $J (u) = \frac{1}{2} {‖ y^{u} (T, \cdot) - y_{r e f} (\cdot) ‖}_{L^{2} (1, L)}^{2},$ (30) ) and the set U is given by (Equation19(19) $\begin{aligned} U = {u \in W^{1, \infty} (0, T); u_{m} \leq u (t) \leq 1, \\ u (0) = 1, u (T) = u_{m}, | u^{'} (t) | \leq u_{\infty}, a . e . t \in (0, T)}, \end{aligned}$ (19) ) has a solution.

Theorem 2.2

The problem (P) has at least one solution $u^{*}$ with the corresponding state (53) $y^{*} \in C ([0, T]; H) \cap L^{2} (0, T; V) \cap H^{1} (0, T; V^{'}) .$ (53)

Proof.

The functional J has an infimum because $J (u) \geq 0$ for all $u \in U$ . Let us denote (54) $d = inf_{u \in U} J (u) .$ (54) It is clear that $d \geq 0.$ Let us consider a minimizing sequence $(u_{n})_{n \geq 1},$ $u_{n} \in U$ and $(y^{u_{n}})_{n \geq 1}$ the corresponding sequence state. Then we have (55) $d \leq \frac{1}{2} {‖ y^{u_{n}} (T, \cdot) - y_{r e f} (\cdot) ‖}_{L^{2} (1, L)}^{2} \leq d + \frac{1}{n} .$ (55) The function $y_{n}$ is the unique solution of the Cauchy problem (56) $\begin{aligned} \frac{d y^{u_{n}}}{d t} (t) + A (t) y^{u_{n}} (t) = f (u_{n}), a . e . t \in (0, T) \\ y^{u_{n}} (0) = y_{0} \end{aligned}$ (56) and verifies the conditions: (57) $\begin{aligned} \int_{0}^{T} {⟨ \frac{d y^{u_{n}}}{d t} (t), ψ (t) ⟩}_{V^{'}, V} d t + \int_{0}^{T} \int_{1}^{L} (a (u_{n}) y_{x}^{u_{n}} ψ_{x} - b (u_{n}) y_{x}^{u_{n}} ψ) d x d t \\ = \int_{0}^{T} \int_{1}^{L} f (u_{n}) ψ d x d t, for all ψ \in L^{2} (0, T; V), \end{aligned}$ (57) (58) $\begin{aligned} y^{u_{n}} (0) = y_{0} . \end{aligned}$ (58) From Theorem 2.1 we obtain that for each $n \in N$ the solution $y^{u_{n}}$ of (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ) exists and is unique and from (Equation42(42) $\begin{aligned} sup_{t \in [0, T]} {‖ y^{u} (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ y^{u} (t) ‖}_{V}^{2} d t \leq C, \end{aligned}$ (42) ) and (Equation43(43) $\begin{aligned} \int_{0}^{T} {‖ \frac{d y^{u}}{d t} (t) ‖}_{V^{'}}^{2} d t \leq C_{1}, \end{aligned}$ (43) ) we obtain (59) ${‖ y^{u_{n}} (t) ‖}_{H}^{2} + \int_{0}^{t} {‖ y^{u_{n}} ‖}_{V}^{2} d s \leq C, a . e . t \in (0, T)$ (59) and (60) $\int_{0}^{T} {‖ \frac{d y^{u_{n}}}{d t} (t) ‖}_{V^{'}}^{2} d t \leq C_{1},$ (60) where $C, C_{1}$ are positive constants that do not depend on $u_{n}$ . We obtain from (Equation59(59) ${‖ y^{u_{n}} (t) ‖}_{H}^{2} + \int_{0}^{t} {‖ y^{u_{n}} ‖}_{V}^{2} d s \leq C, a . e . t \in (0, T)$ (59) ) that the sequence $(y^{u_{n}})_{n}$ is bounded in $L^{\infty} (0, T; H) \cap L^{2} (0, T; V)$ and from (Equation60(60) $\int_{0}^{T} {‖ \frac{d y^{u_{n}}}{d t} (t) ‖}_{V^{'}}^{2} d t \leq C_{1},$ (60) ) we obtain that the sequence $(\frac{d y^{u_{n}}}{d t})_{n}$ is bounded in $L^{2} (0, T; V^{'})$ . From these and from $(u_{n})_{n} \subset U$ we obtain that (eventually passing to subsequences): (61) $\begin{aligned} u_{n} \to u^{*} weak - star in L^{\infty} (0, T), as n \to \infty, \end{aligned}$ (61) (62) $\begin{aligned} \frac{d u_{n}}{d t} \to \frac{d u^{*}}{d t} weak - starin L^{\infty} (0, T), as n \to \infty . \end{aligned}$ (62) (63) $\begin{aligned} y^{u_{n}} \to y^{*} weak - star in L^{\infty} (0, T; H), as n \to \infty, \end{aligned}$ (63) (64) $\begin{aligned} y^{u_{n}} \to y^{*} weakly in L^{2} (0, T; V), as n \to \infty, \end{aligned}$ (64) (65) $\begin{aligned} \frac{d y^{u_{n}}}{d t} \to \frac{d y^{*}}{d t} weakly in L^{2} (0, T; V^{'}), as n \to \infty . \end{aligned}$ (65) From the definition of the set U we obtain that $u^{*} \in U$ . From (Equation61(61) $\begin{aligned} u_{n} \to u^{*} weak - star in L^{\infty} (0, T), as n \to \infty, \end{aligned}$ (61) ), (Equation62(62) $\begin{aligned} \frac{d u_{n}}{d t} \to \frac{d u^{*}}{d t} weak - starin L^{\infty} (0, T), as n \to \infty . \end{aligned}$ (62) ) and Arzelà's theorem we obtain that (66) $u_{n} (t) \to u^{*} (t) uniformly on [0, T] .$ (66) From (Equation64(64) $\begin{aligned} y^{u_{n}} \to y^{*} weakly in L^{2} (0, T; V), as n \to \infty, \end{aligned}$ (64) ), (Equation65(65) $\begin{aligned} \frac{d y^{u_{n}}}{d t} \to \frac{d y^{*}}{d t} weakly in L^{2} (0, T; V^{'}), as n \to \infty . \end{aligned}$ (65) ) and from Aubin–Lions' lemma (Théorème 1.20, [Citation8, p.58], [Citation9]) we reach the conclusion that (67) $y^{u_{n}} \to y^{*} strongly in L^{2} (0, T; H) .$ (67) In what follows, we show that $y^{*}$ is the solution of the problem (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) ), (Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ) which corresponds to $u = u^{*}$ i.e. $y^{*}$ verifies (68) $\begin{aligned} \int_{0}^{T} {⟨ \frac{d y^{*}}{d t} (t), ψ (t) ⟩}_{V^{'}, V} d t + \int_{0}^{T} \int_{1}^{L} (a (u^{*}) y_{x}^{*} ψ_{x} - b (u^{*}) y_{x}^{*} ψ) d x d t \\ = \int_{0}^{T} \int_{1}^{L} f (u^{*}) ψ d x, d t, for all ψ \in L^{2} (0, T; V), \end{aligned}$ (68) (69) $\begin{aligned} y^{*} (0) = y_{0} . \end{aligned}$ (69) First, we can easily see that we obtain from (Equation65(65) $\begin{aligned} \frac{d y^{u_{n}}}{d t} \to \frac{d y^{*}}{d t} weakly in L^{2} (0, T; V^{'}), as n \to \infty . \end{aligned}$ (65) ) that (70) $lim_{n \to \infty} \int_{0}^{T} {⟨ \frac{d y^{u_{n}}}{d t} (t), ψ (t) ⟩}_{V^{'}, V} d t = \int_{0}^{T} {⟨ \frac{d y^{*}}{d t} (t), ψ (t) ⟩}_{V^{'}, V} d t .$ (70) We write (71) $\begin{aligned} \int_{0}^{T} \int_{1}^{L} (a (u_{n}) y_{x}^{u_{n}} - a (u^{*}) y_{x}^{*}) ψ_{x} d x d t = \int_{0}^{T} \int_{1}^{L} (a (u_{n}) - a (u^{*})) y_{x}^{u_{n}} ψ_{x} d x d t \\ + \int_{0}^{T} \int_{1}^{L} a (u^{*}) (y_{x}^{u_{n}} - y_{x}^{*}) ψ_{x} d x d t, \end{aligned}$ (71) in order to show that the second term in the left-hand side of (Equation57(57) $\begin{aligned} \int_{0}^{T} {⟨ \frac{d y^{u_{n}}}{d t} (t), ψ (t) ⟩}_{V^{'}, V} d t + \int_{0}^{T} \int_{1}^{L} (a (u_{n}) y_{x}^{u_{n}} ψ_{x} - b (u_{n}) y_{x}^{u_{n}} ψ) d x d t \\ = \int_{0}^{T} \int_{1}^{L} f (u_{n}) ψ d x d t, for all ψ \in L^{2} (0, T; V), \end{aligned}$ (57) ) converges for $n \to \infty$ to the left-hand side of (Equation68(68) $\begin{aligned} \int_{0}^{T} {⟨ \frac{d y^{*}}{d t} (t), ψ (t) ⟩}_{V^{'}, V} d t + \int_{0}^{T} \int_{1}^{L} (a (u^{*}) y_{x}^{*} ψ_{x} - b (u^{*}) y_{x}^{*} ψ) d x d t \\ = \int_{0}^{T} \int_{1}^{L} f (u^{*}) ψ d x, d t, for all ψ \in L^{2} (0, T; V), \end{aligned}$ (68) ). From (Equation64(64) $\begin{aligned} y^{u_{n}} \to y^{*} weakly in L^{2} (0, T; V), as n \to \infty, \end{aligned}$ (64) ) we obtain that the second term in the right-hand side of (Equation71(71) $\begin{aligned} \int_{0}^{T} \int_{1}^{L} (a (u_{n}) y_{x}^{u_{n}} - a (u^{*}) y_{x}^{*}) ψ_{x} d x d t = \int_{0}^{T} \int_{1}^{L} (a (u_{n}) - a (u^{*})) y_{x}^{u_{n}} ψ_{x} d x d t \\ + \int_{0}^{T} \int_{1}^{L} a (u^{*}) (y_{x}^{u_{n}} - y_{x}^{*}) ψ_{x} d x d t, \end{aligned}$ (71) ) converges to zero. From (Equation32(32) $\begin{aligned} | a (u_{1}) - a (u_{2}) | \leq L_{a} | u_{1} (t) - u_{2} (t) |, \end{aligned}$ (32) ), we obtain $\begin{aligned} | \int_{0}^{T} \int_{1}^{L} (a (u_{n}) - a (u^{*})) y_{x}^{u_{n}} ψ_{x} d x d t | \leq \int_{0}^{T} \int_{1}^{L} | a (u_{n}) - a (u^{*}) | | y_{x}^{u_{n}} | | ψ_{x} | d x d t \\ \leq L_{a} ‖ u_{n} - u^{*} ‖_{\infty} ‖ y_{x}^{u_{n}} ‖_{L^{2} (0, T; V)} ‖ ψ ‖_{L^{2} (0, T; V)}, \end{aligned}$ for every $ψ \in L^{2} (0, T; V)$ . We obtain from (Equation42(42) $\begin{aligned} sup_{t \in [0, T]} {‖ y^{u} (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ y^{u} (t) ‖}_{V}^{2} d t \leq C, \end{aligned}$ (42) ) and from the previous inequality that $lim_{n \to \infty} \int_{0}^{T} \int_{1}^{L} (a (u_{n}) - a (u^{*})) y_{x}^{u_{n}} ψ_{x} d x d t = 0,$ and from (Equation71(71) $\begin{aligned} \int_{0}^{T} \int_{1}^{L} (a (u_{n}) y_{x}^{u_{n}} - a (u^{*}) y_{x}^{*}) ψ_{x} d x d t = \int_{0}^{T} \int_{1}^{L} (a (u_{n}) - a (u^{*})) y_{x}^{u_{n}} ψ_{x} d x d t \\ + \int_{0}^{T} \int_{1}^{L} a (u^{*}) (y_{x}^{u_{n}} - y_{x}^{*}) ψ_{x} d x d t, \end{aligned}$ (71) ) we reach the conclusion that (72) $lim_{n \to \infty} \int_{0}^{T} \int_{1}^{L} a (u_{n}) y_{x}^{u_{n}} ψ_{x} d x d t = \int_{0}^{T} \int_{1}^{L} a (u^{*}) y_{x}^{*} ψ_{x} d x d t .$ (72) Similarly we get (73) $\begin{aligned} lim_{n \to \infty} \int_{0}^{T} \int_{1}^{L} b (u_{n}) y_{x}^{u_{n}} ψ_{x} d x d t = \int_{0}^{T} \int_{1}^{L} b (u^{*}) y_{x}^{*} ψ_{x} d x d t, \end{aligned}$ (73) (74) $\begin{aligned} lim_{n \to \infty} \int_{0}^{T} \int_{1}^{L} f (u_{n}) ψ d x d t = \int_{0}^{T} \int_{1}^{L} f (u^{*}) ψ d x d t . \end{aligned}$ (74) Passing to the limit in (Equation57(57) $\begin{aligned} \int_{0}^{T} {⟨ \frac{d y^{u_{n}}}{d t} (t), ψ (t) ⟩}_{V^{'}, V} d t + \int_{0}^{T} \int_{1}^{L} (a (u_{n}) y_{x}^{u_{n}} ψ_{x} - b (u_{n}) y_{x}^{u_{n}} ψ) d x d t \\ = \int_{0}^{T} \int_{1}^{L} f (u_{n}) ψ d x d t, for all ψ \in L^{2} (0, T; V), \end{aligned}$ (57) ) for $n \to \infty$ we obtain (Equation68(68) $\begin{aligned} \int_{0}^{T} {⟨ \frac{d y^{*}}{d t} (t), ψ (t) ⟩}_{V^{'}, V} d t + \int_{0}^{T} \int_{1}^{L} (a (u^{*}) y_{x}^{*} ψ_{x} - b (u^{*}) y_{x}^{*} ψ) d x d t \\ = \int_{0}^{T} \int_{1}^{L} f (u^{*}) ψ d x, d t, for all ψ \in L^{2} (0, T; V), \end{aligned}$ (68) ) and thus $y^{*}$ is the solution of the problem (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )-(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ) corresponding to $u = u^{*}$ . The fact that $y^{*}$ verifies the initial condition (Equation24(24) $\begin{aligned} y (0, x) = y_{0} (x), x \in (1, L), \end{aligned}$ (24) ) follows from the Arzelà–Ascoli theorem [Citation10, Theorem 3.1, p.55]. From (Equation44(44) ${‖ y^{u_{1}} - y^{u_{2}} ‖}_{L^{\infty} (0, T; H)}^{2} + {‖ y^{u_{1}} - y^{u_{2}} ‖}_{L^{2} (0, T; V)}^{2} \leq C_{2} {‖ u_{1} - u_{2} ‖}_{\infty}^{2},$ (44) ) and (Equation66(66) $u_{n} (t) \to u^{*} (t) uniformly on [0, T] .$ (66) ) we obtain that $y^{n} \to y^{*}$ strongly in $C ([0, T]; H) .$ We reach the conclusion that $y^{n} (t) \to y^{*} (t) strongly in H, uniformly on [0, T],$ from where we obtain that (75) $y^{*} (T) = lim_{n \to \infty} y^{n} (T) .$ (75) We pass to the limit in (Equation55(55) $d \leq \frac{1}{2} {‖ y^{u_{n}} (T, \cdot) - y_{r e f} (\cdot) ‖}_{L^{2} (1, L)}^{2} \leq d + \frac{1}{n} .$ (55) ) as $n \to \infty$ . From (Equation75(75) $y^{*} (T) = lim_{n \to \infty} y^{n} (T) .$ (75) ) and from the weakly lower semicontinuity of the norm $‖ \cdot ‖_{H}$ we get (76) $\begin{aligned} J (u^{*}) & \leq \underset{n \to \infty}{lim inf} \frac{1}{2} {‖ y^{u_{n}} (T, \cdot) - y_{r e f} (\cdot) ‖}_{L^{2} (1, L)}^{2} \\ = lim_{n \to \infty} \frac{1}{2} {‖ y^{u_{n}} (T, \cdot) - y_{r e f} (\cdot) ‖}_{L^{2} (1, L)}^{2} = d \end{aligned}$ (76) so $u^{*}$ is a solution of problem (P).

2.3. The system of first-order variations

The results given by Theorem 2.1 allow us to introduce the control-to-state mapping S [Citation11]. We define (77) $\begin{aligned} X := L^{\infty} (0, T), Y := C ([0, T]; H) \cap L^{2} (0, T; V), \end{aligned}$ (77) (78) $\begin{aligned} S : U \subset X \to Y, S (u) := y^{u}, \end{aligned}$ (78) where $y^{u}$ is the unique solution of the problem (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ). Let $u^{*}$ be the solution of the problem (P), $y^{*} = S (u^{*})$ the solution of the problem (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )-(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ) corresponding to $u = u^{*},$ $λ \in (0, 1),$ $u \in U$ and let us denote (79) $u^{λ} (t) = u^{*} (t) + λ w (t), a . e . t \in (0, T),$ (79) where $w = u - u^{*}$ . It is clear that $u^{λ} \in U,$ $w \in W^{1, \infty} (0, T)$ and $| w (t) | \leq 2,$ a.e. $t \in (0, T)$ .

Let us consider the system: (80) $\begin{aligned} z_{t} - {(a (u^{*}) z_{x})}_{x} - b (u^{*}) z_{x} = {(a^{'} (u^{*}) y_{x}^{*})}_{x} w + b^{'} (u^{*}) y_{x}^{*} w + f^{'} (u^{*}) w, (t, x) \in Q, \\ z (0, x) = 0, x \in (1, L), \\ z (t, 1) = z (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (80) We denote (Equation80(80) $\begin{aligned} z_{t} - {(a (u^{*}) z_{x})}_{x} - b (u^{*}) z_{x} = {(a^{'} (u^{*}) y_{x}^{*})}_{x} w + b^{'} (u^{*}) y_{x}^{*} w + f^{'} (u^{*}) w, (t, x) \in Q, \\ z (0, x) = 0, x \in (1, L), \\ z (t, 1) = z (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (80) ) the first-order variation system.

Theorem 2.3

For every $w \in W^{1, \infty} (0, T)$ the system (Equation80(80) $\begin{aligned} z_{t} - {(a (u^{*}) z_{x})}_{x} - b (u^{*}) z_{x} = {(a^{'} (u^{*}) y_{x}^{*})}_{x} w + b^{'} (u^{*}) y_{x}^{*} w + f^{'} (u^{*}) w, (t, x) \in Q, \\ z (0, x) = 0, x \in (1, L), \\ z (t, 1) = z (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (80) ) has a unique solution (81) $z \in C ([0, T]; H) \cap L^{2} (0, T; V) \cap H^{1} (0, T; V^{'}),$ (81) which satisfies the estimate (82) $sup_{t \in [0, T]} {‖ z (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ z (t) ‖}_{V}^{2} d t \leq C_{4} {‖ w ‖}_{X}^{2},$ (82) where the positive constant $C_{4}$ depends only on the structure of the system (Equation80(80) $\begin{aligned} z_{t} - {(a (u^{*}) z_{x})}_{x} - b (u^{*}) z_{x} = {(a^{'} (u^{*}) y_{x}^{*})}_{x} w + b^{'} (u^{*}) y_{x}^{*} w + f^{'} (u^{*}) w, (t, x) \in Q, \\ z (0, x) = 0, x \in (1, L), \\ z (t, 1) = z (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (80) ).

Proof.

Taking into account that $y^{*} \in L^{2} (0, T; V)$ we have $\int_{0}^{T} \int_{1}^{L} {(a^{'} (u^{*}) w y_{x}^{*})}^{2} \leq 4 u_{M} {‖ a^{'} (u^{*}) ‖}_{L^{\infty} (Q)} \int_{0}^{T} {‖ y_{x}^{*} ‖}_{H} d t,$ from where we obtain that $a^{'} (u^{*}) w y_{x}^{*} \in L^{2} (0, T; H)$ . In a similar way, we prove that $b^{'} (u^{*}) w y_{x}^{*} \in L^{2} (0, T; H)$ and $f^{'} (u^{*}) w \in L^{2} (0, T; H)$ . The family of operators $A (t) : V \to V^{'},$ defined on V, a.e. $t \in (0, T),$ (83) ${⟨ A (t) y, ψ ⟩}_{V^{'}, V} = \int_{1}^{L} (a (u^{*}) y_{x} ψ_{x} - b (u^{*}) y_{x} ψ) d x, (\forall) ψ \in V$ (83) has the properties (i) and (ii) from Theorem 2.1. The existence and uniqueness of the solution of the system (Equation80(80) $\begin{aligned} z_{t} - {(a (u^{*}) z_{x})}_{x} - b (u^{*}) z_{x} = {(a^{'} (u^{*}) y_{x}^{*})}_{x} w + b^{'} (u^{*}) y_{x}^{*} w + f^{'} (u^{*}) w, (t, x) \in Q, \\ z (0, x) = 0, x \in (1, L), \\ z (t, 1) = z (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (80) ) now result from Lions' theorem [Citation8, Theorem 2, p.513]. In order to obtain the estimate (Equation82(82) $sup_{t \in [0, T]} {‖ z (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ z (t) ‖}_{V}^{2} d t \leq C_{4} {‖ w ‖}_{X}^{2},$ (82) ) we follow the steps that we made when proving the estimate (Equation42(42) $\begin{aligned} sup_{t \in [0, T]} {‖ y^{u} (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ y^{u} (t) ‖}_{V}^{2} d t \leq C, \end{aligned}$ (42) ).

Remark 2.2

Let $X = L^{\infty} (0, T),$ $Y = C ([0, T]; H) \cap L^{2} (0, T; V)$ . We define the linear mapping (84) $G : X \to Y, X ∋ w \to G (w) := z$ (84) where z is the unique solution of the system of first-order variations (Equation80(80) $\begin{aligned} z_{t} - {(a (u^{*}) z_{x})}_{x} - b (u^{*}) z_{x} = {(a^{'} (u^{*}) y_{x}^{*})}_{x} w + b^{'} (u^{*}) y_{x}^{*} w + f^{'} (u^{*}) w, (t, x) \in Q, \\ z (0, x) = 0, x \in (1, L), \\ z (t, 1) = z (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (80) ). Then the inequality (Equation82(82) $sup_{t \in [0, T]} {‖ z (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ z (t) ‖}_{V}^{2} d t \leq C_{4} {‖ w ‖}_{X}^{2},$ (82) ) leads us to the inequality (85) ${‖ G (w) ‖}_{Y} \leq C_{4} {‖ w ‖}_{X} .$ (85) In particular (Equation85(85) ${‖ G (w) ‖}_{Y} \leq C_{4} {‖ w ‖}_{X} .$ (85) ) means that the linear map D is continuous from X to Y.

Let $y^{λ}$ be the solution of (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ) corresponding to $u = u^{λ}$ and let (86) $z^{λ} = \frac{y^{λ} - y^{*}}{λ} .$ (86) From (Equation44(44) ${‖ y^{u_{1}} - y^{u_{2}} ‖}_{L^{\infty} (0, T; H)}^{2} + {‖ y^{u_{1}} - y^{u_{2}} ‖}_{L^{2} (0, T; V)}^{2} \leq C_{2} {‖ u_{1} - u_{2} ‖}_{\infty}^{2},$ (44) ), we obtain (87) $\int_{0}^{t} {‖ z_{x}^{λ} ‖}_{H}^{2} d τ \leq \frac{1}{λ} C_{2} {‖ u^{λ} - u^{*} ‖}_{X}^{2} = C {‖ w ‖}_{X}^{2},$ (87) and it is easy to see that $z^{λ} \in C ([0, T]; H) \cap L^{2} (0, T; V) \cap H^{1} (0, T; V^{'})$ is the unique solution of the following problem: $\begin{aligned} z_{t}^{λ} & = {(a (u^{λ}) z_{x}^{λ})}_{x} + b (u^{λ}) z_{x}^{λ} + {(\frac{a (u^{λ}) - a (u^{*})}{λ} y_{x}^{*})}_{x} \\ + \frac{b (u^{λ}) - b (u^{*})}{λ} y_{x}^{*} + \frac{f (u^{λ}) - f (u^{*})}{λ}, \end{aligned}$ (88) $z^{λ} (0, x) = 0, x \in (1, L),$ (88) $z^{λ} (t, 1) = z^{λ} (t, L) = 0, a . e . t \in (0, T) .$

Proposition 2.1

We have (89) $lim_{λ \to 0} {‖ z^{λ} - z^{*} ‖}_{L^{\infty} (0, T; H) \cap L^{2} (0, T; V)} = 0,$ (89) where $z^{*}$ is the unique solution of the system (Equation80(80) $\begin{aligned} z_{t} - {(a (u^{*}) z_{x})}_{x} - b (u^{*}) z_{x} = {(a^{'} (u^{*}) y_{x}^{*})}_{x} w + b^{'} (u^{*}) y_{x}^{*} w + f^{'} (u^{*}) w, (t, x) \in Q, \\ z (0, x) = 0, x \in (1, L), \\ z (t, 1) = z (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (80) ).

Proof.

We obtain from (Equation80(80) $\begin{aligned} z_{t} - {(a (u^{*}) z_{x})}_{x} - b (u^{*}) z_{x} = {(a^{'} (u^{*}) y_{x}^{*})}_{x} w + b^{'} (u^{*}) y_{x}^{*} w + f^{'} (u^{*}) w, (t, x) \in Q, \\ z (0, x) = 0, x \in (1, L), \\ z (t, 1) = z (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (80) ) and (Equation88(88) $z^{λ} (0, x) = 0, x \in (1, L),$ (88) ) that the function $z^{λ} - z^{*}$ is the solution of the system $\begin{aligned} {(z^{λ} - z^{*})}_{t} & = {((a (u^{λ}) - a (u^{*})) z_{x}^{λ})}_{x} + {(a (u^{*}) {(z^{λ} - z^{*})}_{x})}_{x} \\ + (b (u^{λ}) - b (u^{*})) z_{x}^{λ} + b (u^{*}) {(z^{λ} - z^{*})}_{x} \\ + {((\frac{a (u^{λ}) - a (u^{*})}{λ} - a^{'} (u^{*}) w) y_{x}^{*})}_{x} \\ + (\frac{b (u^{λ}) - b (u^{*})}{λ} - b^{'} (u^{*}) w) y_{x}^{*} + \frac{f (u^{λ}) - f (u^{*})}{λ} - f^{'} (u^{*}) w, \end{aligned}$ (90) $z^{λ} (0, x) = 0, x \in (1, L),$ (90) $z^{λ} (t, 1) = z^{λ} (t, L) = 0, a . e . t \in (0, T) .$ The system (Equation90(90) $z^{λ} (0, x) = 0, x \in (1, L),$ (90) ) is equivalent to the following Cauchy problem: (91) $\begin{aligned} \frac{d}{d t} (z^{λ} - z^{*}) (t) + A^{*} (t) (z^{λ} - z^{*}) (t) = g (t), a . e . t \in (0, T) \end{aligned}$ (91) (92) $\begin{aligned} (z^{λ} - z^{*}) (0) = 0, \end{aligned}$ (92) where ${⟨ A^{*} (t) y (t), ψ ⟩}_{V^{'}, V} = \int_{1}^{L} (a (u^{*}) y_{x} ψ_{x} (t) - b (u^{*}) y_{x} ψ) d x, (\forall) ψ \in V$ and $g \in L^{2} (0, T; V^{'})$ is given by $\begin{aligned} ⟨ g (t), ψ ⟩_{V^{'}, V} = \int_{1}^{L} (b (u^{λ}) - b (u^{*}) - a (u^{λ}) + a (u^{*})) z_{x}^{λ} ψ d x \\ + \int_{1}^{L} (\frac{b (u^{λ}) - b (u^{*})}{λ} - b^{'} (u^{*}) w - \frac{a (u^{λ}) - a (u^{*})}{λ} + a^{'} (u^{*}) w) y_{x}^{*} ψ d x \\ + \int_{1}^{L} (\frac{f (u^{λ}) - f (u^{*})}{λ} - f^{'} (u^{*}) w) ψ d x, (\forall) ψ \in V . \end{aligned}$ We test (Equation91(91) $\begin{aligned} \frac{d}{d t} (z^{λ} - z^{*}) (t) + A^{*} (t) (z^{λ} - z^{*}) (t) = g (t), a . e . t \in (0, T) \end{aligned}$ (91) ) for $z^{λ} - z^{*} \in L^{2} (0, T; V)$ and we integrate on $(0, t) .$ We obtain: $\begin{aligned} \frac{1}{2} ‖ z^{λ} - z^{*} ‖_{H}^{2} + \int_{0}^{t} \int_{1}^{L} a (u^{*}) ((z^{λ} - z^{*})_{x})^{2} d x d τ \\ = - \int_{0}^{t} \int_{1}^{L} (a (u^{λ}) - a (u^{*})) z_{x}^{λ} (z^{λ} - z^{*})_{x} d x d τ \\ + \int_{0}^{t} \int_{1}^{L} (b (u^{λ}) - b (u^{*})) z_{x}^{λ} (z^{λ} - z^{*}) d x d τ \\ - \frac{1}{2} \int_{0}^{t} \int_{1}^{L} b_{x} (u^{*}) (z^{λ} - z^{*})^{2} d x d τ \\ - \int_{0}^{t} \int_{1}^{L} (\frac{a (u^{λ}) - a (u^{*})}{λ} - a^{'} (u^{*}) w) y_{x}^{*} (z^{λ} - z^{*})_{x} d x d τ \\ + \int_{0}^{t} \int_{1}^{L} (\frac{b (u^{λ}) - b (u^{*})}{λ} - b^{'} (u^{*}) w) y_{x}^{*} (z^{λ} - z^{*}) d x d τ \\ + \int_{0}^{t} \int_{1}^{L} (\frac{f (u^{λ}) - f (u^{*})}{λ} - f^{'} (u^{*}) w) (z^{λ} - z^{*}) d x d τ . \end{aligned}$ From (Equation32(32) $\begin{aligned} | a (u_{1}) - a (u_{2}) | \leq L_{a} | u_{1} (t) - u_{2} (t) |, \end{aligned}$ (32) ), (Equation33(33) $\begin{aligned} | b (u_{1}) - b (u_{2}) | \leq L_{b} | u_{1} (t) - u_{2} (t) |, \end{aligned}$ (33) ), (Equation34(34) $\begin{aligned} | f (u_{1}) - f (u_{2}) | \leq L_{f} | u_{1} (t) - u_{2} (t) |, \end{aligned}$ (34) ), (Equation87(87) $\int_{0}^{t} {‖ z_{x}^{λ} ‖}_{H}^{2} d τ \leq \frac{1}{λ} C_{2} {‖ u^{λ} - u^{*} ‖}_{X}^{2} = C {‖ w ‖}_{X}^{2},$ (87) ) and after a few calculations involving Cauchy's inequality we obtain (93) $\begin{aligned} {‖ z^{λ} - z^{*} ‖}_{H}^{2} + a_{m} \int_{0}^{t} {‖ z^{λ} - z^{*} ‖}_{V}^{2} d τ \leq (a_{m} + {‖ b_{x} ‖}_{\infty} + \frac{3}{2}) \int_{0}^{t} {‖ z^{λ} - z^{*} ‖}_{H}^{2} d τ \\ + λ^{2} {‖ w ‖}_{X}^{4} (\frac{2 L_{a}^{2}}{a_{m}} + 2 L_{b}^{2} + + \frac{L_{f^{'}}^{2} T}{2} + (\frac{2 L_{a^{'}}^{2}}{a_{m}} + \frac{L_{b^{'}}^{2}}{2}) \int_{0}^{t} {‖ y_{x}^{*} ‖}_{H}^{2} d τ) . \end{aligned}$ (93)

Gronwall's lemma, (Equation42(42) $\begin{aligned} sup_{t \in [0, T]} {‖ y^{u} (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ y^{u} (t) ‖}_{V}^{2} d t \leq C, \end{aligned}$ (42) ), the inequality $| w (t) | \leq 2 u_{M},$ a.e. $t \in (0, T)$ and Remark 2.1 i lead us to the estimate: (94) ${‖ z^{λ} - z^{*} ‖}_{H}^{2} + \int_{0}^{t} {‖ z^{λ} - z^{*} ‖}_{V}^{2} d τ \leq λ^{2} C_{5},$ (94) where the constants $C_{5}$ depend only on the structure of the system (Equation23(23) $\begin{aligned} \frac{\partial y}{\partial t} = \frac{\partial}{\partial x} (a (u) \frac{\partial y}{\partial x}) + b (u) \frac{\partial y}{\partial x} + f (u), (t, x) \in Q, \end{aligned}$ (23) )–(Equation25(25) $\begin{aligned} y (t, 1) = y (t, L) = 0, t \in (0, T) . \end{aligned}$ (25) ). The result (Equation89(89) $lim_{λ \to 0} {‖ z^{λ} - z^{*} ‖}_{L^{\infty} (0, T; H) \cap L^{2} (0, T; V)} = 0,$ (89) ) is obtained by passing to the limit in (Equation94(94) ${‖ z^{λ} - z^{*} ‖}_{H}^{2} + \int_{0}^{t} {‖ z^{λ} - z^{*} ‖}_{V}^{2} d τ \leq λ^{2} C_{5},$ (94) ) for $λ \to 0$ .

Remark 2.3

Proposition 2.1 implies that the map S given by (Equation78(78) $\begin{aligned} S : U \subset X \to Y, S (u) := y^{u}, \end{aligned}$ (78) ) is Gâteaux differentiable and its Gâteaux derivative is the map defined by (Equation84(84) $G : X \to Y, X ∋ w \to G (w) := z$ (84) ) (95) $S^{'} (u^{*}) (w) = G (w), for all w \in X .$ (95)

We use this results in order to obtain the necessary condition of optimality.

2.4. The dual system and the necessary condition of optimality

First, in order to obtain the necessary condition of optimality we introduce the functional (96) $J_{0} : Y \to R, J_{0} (ψ) := \frac{1}{2} \int_{1}^{L} {(ψ (T, x) - y_{r e f} (x))}^{2} d x,$ (96) The functional cost J (Equation30(30) $J (u) = \frac{1}{2} {‖ y^{u} (T, \cdot) - y_{r e f} (\cdot) ‖}_{L^{2} (1, L)}^{2},$ (30) ) is written as (97) $J : U \to R, J = J_{0} \circ S .$ (97) Recall that the map S is Gâteaux differentiable and its Gâteaux derivative $S^{'} (u^{*}) : X \to Y$ is the map G defined by (Equation84(84) $G : X \to Y, X ∋ w \to G (w) := z$ (84) ). The chain rule for the Gâteaux differentiability shows us that the Gâteaux derivative of J calculated in $u^{*},$ $J^{'} (u^{*}) : X \to R$ is (98) ${⟨ J^{'} (u^{*}), w ⟩}_{X^{'}, X} = \int_{1}^{L} (y^{*} (T, x) - y_{r e f} (x)) z (T, x) d x,$ (98) where z is the solution of system of first order variations (Equation80(80) $\begin{aligned} z_{t} - {(a (u^{*}) z_{x})}_{x} - b (u^{*}) z_{x} = {(a^{'} (u^{*}) y_{x}^{*})}_{x} w + b^{'} (u^{*}) y_{x}^{*} w + f^{'} (u^{*}) w, (t, x) \in Q, \\ z (0, x) = 0, x \in (1, L), \\ z (t, 1) = z (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (80) ). Taking into account that U is convex and closed and J is Gâteaux differentiable, the necessary condition of optimality is [Citation12, Theorem 1-2.1, p.1180]: (99) ${⟨ J^{'} (u^{*}), u - u^{*} ⟩}_{X^{'}, X} \geq 0 for every u \in U .$ (99) From (Equation98(98) ${⟨ J^{'} (u^{*}), w ⟩}_{X^{'}, X} = \int_{1}^{L} (y^{*} (T, x) - y_{r e f} (x)) z (T, x) d x,$ (98) ) and (Equation99(99) ${⟨ J^{'} (u^{*}), u - u^{*} ⟩}_{X^{'}, X} \geq 0 for every u \in U .$ (99) ), we obtain that the necessary condition of optimality (100) $\int_{1}^{L} (y^{*} (T, x) - y_{r e f} (x)) z (T, x) d t \geq 0$ (100) holds for every $u \in U,$ where $z = G (w)$ is the solution of system (Equation80(80) $\begin{aligned} z_{t} - {(a (u^{*}) z_{x})}_{x} - b (u^{*}) z_{x} = {(a^{'} (u^{*}) y_{x}^{*})}_{x} w + b^{'} (u^{*}) y_{x}^{*} w + f^{'} (u^{*}) w, (t, x) \in Q, \\ z (0, x) = 0, x \in (1, L), \\ z (t, 1) = z (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (80) ) and $w = u - u^{*}$ .

We introduce the dual system: (101) $\begin{aligned} p_{t} = - {(a (u^{*}) p_{x})}_{x} + {(b (u^{*}) p)}_{x} in Q, \end{aligned}$ (101) (102) $\begin{aligned} p (T, x) = y^{*} (T, x) - y_{r e f} (x) in (1, L), \end{aligned}$ (102) (103) $\begin{aligned} p (t, 1) = 0, p (t, L) = 0 a . e . t \in (0, T) . \end{aligned}$ (103)

Theorem 2.4

The system (Equation101(101) $\begin{aligned} p_{t} = - {(a (u^{*}) p_{x})}_{x} + {(b (u^{*}) p)}_{x} in Q, \end{aligned}$ (101) )–(Equation103(103) $\begin{aligned} p (t, 1) = 0, p (t, L) = 0 a . e . t \in (0, T) . \end{aligned}$ (103) ) has a unique solution (104) $p \in C ([0, T], H) \cap L^{2} (0, T; V) \cap H^{1} (0, T; V^{'}) .$ (104)

Proof.

We make the transformation $τ = T - t$ and we use Lions' theorem [Citation8, Theorem 2, p.513].

Theorem 2.5

Let $u^{*}$ be a solution of the problem (P), $y^{*} = S (u^{*})$ and p the solution of the dual system (Equation101(101) $\begin{aligned} p_{t} = - {(a (u^{*}) p_{x})}_{x} + {(b (u^{*}) p)}_{x} in Q, \end{aligned}$ (101) )–(Equation103(103) $\begin{aligned} p (t, 1) = 0, p (t, L) = 0 a . e . t \in (0, T) . \end{aligned}$ (103) ). Then $u^{*}$ satisfies the necessary condition (105) $\int_{0}^{T} (u^{*} (t) - u (t)) Φ (t) d t \geq 0,$ (105) for all $u \in U,$ where (106) $Φ (t) := \int_{1}^{L} (a^{'} (u^{*}) p_{x} y_{x}^{*} - b^{'} (u^{*}) p y_{x}^{*} - f^{'} (u^{*}) p) d x, a . e . t \in (0, T),$ (106) and p is the solution of the dual system (Equation101(101) $\begin{aligned} p_{t} = - {(a (u^{*}) p_{x})}_{x} + {(b (u^{*}) p)}_{x} in Q, \end{aligned}$ (101) )–(Equation103(103) $\begin{aligned} p (t, 1) = 0, p (t, L) = 0 a . e . t \in (0, T) . \end{aligned}$ (103) ).

Proof.

Let z be the unique solution of the system (Equation80(80) $\begin{aligned} z_{t} - {(a (u^{*}) z_{x})}_{x} - b (u^{*}) z_{x} = {(a^{'} (u^{*}) y_{x}^{*})}_{x} w + b^{'} (u^{*}) y_{x}^{*} w + f^{'} (u^{*}) w, (t, x) \in Q, \\ z (0, x) = 0, x \in (1, L), \\ z (t, 1) = z (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (80) ). We have (107) $\begin{aligned} \int_{0}^{T} {⟨ z_{t} (t), ψ (t) ⟩}_{V^{'}, V} d t + \int_{0}^{T} \int_{1}^{L} (a (u^{*}) z_{x} ψ_{x} - b (u^{*}) z_{x} ψ) d x d t \\ = - \int_{0}^{T} \int_{1}^{L} a^{'} (u^{*}) w y_{x}^{*} ψ_{x} d x d t + \int_{0}^{T} \int_{1}^{L} b^{'} (u^{*}) w y_{x}^{*} ψ d x d t \\ + \int_{0}^{T} \int_{1}^{L} f^{'} (u^{*}) w ψ d x d t, for all ψ \in V, \end{aligned}$ (107) (108) $\begin{aligned} z (0, x) = 0 in (1, L), \end{aligned}$ (108) and (109) $\begin{aligned} \int_{0}^{T} {⟨ p_{t} (t), ψ (t) ⟩}_{V^{'}, V} d t - \int_{0}^{T} \int_{1}^{L} (a (u^{*}) p_{x} ψ_{x} - b (u^{*}) p_{x} ψ) d x d t \\ = 0, for all ψ \in V, \end{aligned}$ (109) (110) $\begin{aligned} p (0, x) = y^{*} (t, x) - y_{r e f} (x) in (1, L) . \end{aligned}$ (110) If we write (Equation107(107) $\begin{aligned} \int_{0}^{T} {⟨ z_{t} (t), ψ (t) ⟩}_{V^{'}, V} d t + \int_{0}^{T} \int_{1}^{L} (a (u^{*}) z_{x} ψ_{x} - b (u^{*}) z_{x} ψ) d x d t \\ = - \int_{0}^{T} \int_{1}^{L} a^{'} (u^{*}) w y_{x}^{*} ψ_{x} d x d t + \int_{0}^{T} \int_{1}^{L} b^{'} (u^{*}) w y_{x}^{*} ψ d x d t \\ + \int_{0}^{T} \int_{1}^{L} f^{'} (u^{*}) w ψ d x d t, for all ψ \in V, \end{aligned}$ (107) ) for $ψ = p,$ we integrate by parts and use (Equation109(109) $\begin{aligned} \int_{0}^{T} {⟨ p_{t} (t), ψ (t) ⟩}_{V^{'}, V} d t - \int_{0}^{T} \int_{1}^{L} (a (u^{*}) p_{x} ψ_{x} - b (u^{*}) p_{x} ψ) d x d t \\ = 0, for all ψ \in V, \end{aligned}$ (109) ) for $ψ = z$ and sum those equalities we obtain (111) $\begin{aligned} \int_{1}^{L} (y^{*} (T, x) - y_{r e f} (x)) z (T, x) d x = - \int_{0}^{T} \int_{1}^{L} a^{'} (u^{*}) w y_{x}^{*} p_{x} d x d t \\ + \int_{0}^{T} \int_{1}^{L} b^{'} (u^{*}) w y_{x}^{*} p d x d t + \int_{0}^{T} \int_{1}^{L} f^{'} (u^{*}) w p d x d t . \end{aligned}$ (111) Condition (Equation105(105) $\int_{0}^{T} (u^{*} (t) - u (t)) Φ (t) d t \geq 0,$ (105) ) is now obtained from (Equation111(111) $\begin{aligned} \int_{1}^{L} (y^{*} (T, x) - y_{r e f} (x)) z (T, x) d x = - \int_{0}^{T} \int_{1}^{L} a^{'} (u^{*}) w y_{x}^{*} p_{x} d x d t \\ + \int_{0}^{T} \int_{1}^{L} b^{'} (u^{*}) w y_{x}^{*} p d x d t + \int_{0}^{T} \int_{1}^{L} f^{'} (u^{*}) w p d x d t . \end{aligned}$ (111) ) and (Equation100(100) $\int_{1}^{L} (y^{*} (T, x) - y_{r e f} (x)) z (T, x) d t \geq 0$ (100) ).

From (Equation106(106) $Φ (t) := \int_{1}^{L} (a^{'} (u^{*}) p_{x} y_{x}^{*} - b^{'} (u^{*}) p y_{x}^{*} - f^{'} (u^{*}) p) d x, a . e . t \in (0, T),$ (106) ) we obtain that $\begin{aligned} \int_{0}^{T} | Φ (t) | d t \leq \int_{0}^{T} \int_{1}^{L} | a^{'} (u^{*}) | | p_{x} | | y_{x}^{*} | d x d t + \int_{0}^{T} \int_{1}^{L} | b^{'} (u^{*}) | | p | | y_{x}^{*} | d x d t \\ + \int_{0}^{T} \int_{1}^{L} | f^{'} (u^{*}) | | p | d x d t \\ \leq (‖ a^{'} (u^{*}) ‖_{\infty} + ‖ b^{'} (u^{*}) ‖_{\infty}) ‖ y^{*} ‖_{L^{2} (0, T; V)} ‖ p ‖_{L^{2} (0, T; V)} \\ + ‖ f^{'} (u^{*}) ‖_{\infty} \sqrt{(L - 1) T} ‖ p ‖_{L^{2} (0, T; V)}, \end{aligned}$ so $Φ \in L^{1} (0, T),$ and inequality (Equation105(105) $\int_{0}^{T} (u^{*} (t) - u (t)) Φ (t) d t \geq 0,$ (105) ) implies that (112) $Φ (t) \in {\tilde{N}}_{U} (u^{*}),$ (112) where (113) ${\tilde{N}}_{U} (u^{*}) = {w \in L^{1} (0, T); \int_{0}^{T} (u^{*} - u) w d t \geq 0 for all u \in U}$ (113) is the normal cone in $L^{1} (0, T)$ to U at $u^{*} \in U$ [Citation13, p.5].

In order to obtain the optimality condition (Equation105(105) $\int_{0}^{T} (u^{*} (t) - u (t)) Φ (t) d t \geq 0,$ (105) ), (Equation106(106) $Φ (t) := \int_{1}^{L} (a^{'} (u^{*}) p_{x} y_{x}^{*} - b^{'} (u^{*}) p y_{x}^{*} - f^{'} (u^{*}) p) d x, a . e . t \in (0, T),$ (106) ) in explicit form we follow the steps presented in [Citation14]. For $θ : R \to [- 1, 1]$ let us denote by signz the graph $sign z = {\begin{array}{cc} 1, & on {z > 0}, \\ [- 1, 1], & on {z = 0}, \\ - 1, & on {z < 0} . \end{array}$ The following result occurs:

Proposition 2.2

[Citation15, Proposition 2.9]

Let $u^{*}$ the solution of problem (P), $y^{*} = S (u^{*})$ the solution of the state system corresponding to $u^{*}$ . Then the optimality condition (Equation105(105) $\int_{0}^{T} (u^{*} (t) - u (t)) Φ (t) d t \geq 0,$ (105) ) reads (114) $Φ \in N_{U} (u^{*}) \cap L^{1} (0, T),$ (114) where Φ is given by (Equation106(106) $Φ (t) := \int_{1}^{L} (a^{'} (u^{*}) p_{x} y_{x}^{*} - b^{'} (u^{*}) p y_{x}^{*} - f^{'} (u^{*}) p) d x, a . e . t \in (0, T),$ (106) ). Moreover $Φ \in N_{U} (u^{*}) \cap L^{1} (0, T)$ if and only if it has the representation (115) $\begin{aligned} Φ (t) = - θ^{'} (t) + μ (t) in D^{'} (0, T), \end{aligned}$ (115) (116) $\begin{aligned} u^{*'} (t) \in u_{\infty} sign (θ (t)) a . e . t \in (0, T), \end{aligned}$ (116) where μ, $θ \in L^{1} (0, T),$ (117) ${\begin{cases} μ (t) \leq 0 & a . e . in {t \in [0, T] : u^{*} (t) = u_{m}}, \\ μ (t) = 0 & a . e . in {t \in [0, T] : u^{*} (t) \in (u_{m}, 1)}, \\ μ (t) \geq 0 & a . e . in {t \in [0, T] : u^{*} (t) = 1}, \end{cases}$ (117) and (118) $θ (t) = {\begin{cases} 0 & a . e . in {t \in [0, T]; | {(u^{*})}^{'} (t) | < u_{\infty}} \\ v (t) u^{*'} (t) & a . e . in {t \in [0, T]; | {(u^{*})}^{'} (t) | = u_{\infty}}, \end{cases}$ (118) with $ν \in L^{1} (0, T),$ $ν \geq 0$ a.e. $t \in (0, T)$ .

In what follows, we will see that in the case of Dirichlet–Neumann boundary conditions an explicit form of the optimal control $u^{*}$ can be obtained.

3. Dirichlet–Neumann boundary conditions

In this section, we consider the following problem: (119) $inf_{u \in U} \int_{1}^{L} C (T, x) d x,$ (119) where $C (t, x)$ is the solution of the system (Equation18(18) $\frac{\partial C}{\partial t} = \frac{\partial}{\partial x} (P e^{- 1} x^{- β} u^{β} (t) \frac{\partial C}{\partial x}) + (P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t)) \frac{\partial C}{\partial x},$ (18) ), (Equation14(14) $\begin{aligned} C (0, x) = C_{0}, x \in (1, L), \end{aligned}$ (14) ), (Equation15(15) $\begin{aligned} C (t, 1) = 1, C (t, L) = 0, t \in (0, T), \end{aligned}$ (15) ). With the above notations, this system can be written under the form: (120) $\begin{aligned} C_{t} = {(a (u) C_{x})}_{x} + b (u) C_{x}, (t, x) \in Q, \end{aligned}$ (120) (121) $\begin{aligned} C (0, x) = 0 in (1, L), \end{aligned}$ (121) (122) $\begin{aligned} C (t, 1) = 1, C_{x} (t, L) = 0 in (0, T), \end{aligned}$ (122) where a and b are given by (Equation26(26) $\begin{aligned} a (u) = P e^{- 1} x^{- β} u^{β} (t), \end{aligned}$ (26) ) and (Equation27(27) $\begin{aligned} b (u) = P e^{- 1} x^{- β - 1} u^{β} (t) - x^{- 1} u (t), \end{aligned}$ (27) ), respectively.

Let us consider the change of function (123) $C = y + 1.$ (123) Problem (Equation119(119) $inf_{u \in U} \int_{1}^{L} C (T, x) d x,$ (119) ) becomes (P2) $inf_{u \in U} J (u),$ (P2) where (124) $J (u) = \int_{1}^{L} (y (T, x) + 1) d x$ (124) and y is the solution of the following problem: (125) $\begin{aligned} y_{t} = {(a (u) y_{x})}_{x} + b (u) y_{x}, \end{aligned}$ (125) (126) $\begin{aligned} y (0, x) = - 1 in (1, L), \end{aligned}$ (126) (127) $\begin{aligned} y (t, 1) = 0; y_{x} (t, L) = 0 in (0, T) . \end{aligned}$ (127) The definition of the control set U is the same as in Section 2.1. We call this minimization problem problem (P2).

We consider the following space: $V_{0} = {v \in H^{1} (1, L); v (1) = 0},$ and we denote its dual by $V_{0}^{'} .$ If $H = L^{2} (1, L),$ then we have the continuous and dense embeddings $V_{0} ↪ H \equiv H^{'} ↪ V_{0}^{'}$ .

Definition 3.1

A function $y \in L^{2} (0, T; V_{0})$ with $y^{'} \in L^{2} (0, T; V_{0}^{'})$ is a solution of problem (Equation125(125) $\begin{aligned} y_{t} = {(a (u) y_{x})}_{x} + b (u) y_{x}, \end{aligned}$ (125) )–(Equation127(127) $\begin{aligned} y (t, 1) = 0; y_{x} (t, L) = 0 in (0, T) . \end{aligned}$ (127) ) if (128) $\int_{0}^{T} {⟨ y_{t} (t), ψ (t) ⟩}_{V_{0}^{'}, V_{0}} d t + \int_{0}^{T} \int_{1}^{L} (a (u) y_{x} ψ_{x} - b (u) y_{x} ψ) d x d t = 0$ (128) for all $ψ \in L^{2} (0, T; V_{0})$ and (129) $y (0) = - 1.$ (129)

The solution of (Equation125(125) $\begin{aligned} y_{t} = {(a (u) y_{x})}_{x} + b (u) y_{x}, \end{aligned}$ (125) )–(Equation127(127) $\begin{aligned} y (t, 1) = 0; y_{x} (t, L) = 0 in (0, T) . \end{aligned}$ (127) ) is the solution of the following Cauchy problem: (130) $\begin{aligned} \frac{d y}{d t} (t) + A (t) y (t) = 0, a . e . t \in (0, T), \end{aligned}$ (130) (131) $\begin{aligned} y (0) = - 1, \end{aligned}$ (131) where the family of operators $A (t) : V_{0} \to V_{0}^{'}$ is given by (Equation37(37) ${⟨ A (t) y, ψ ⟩}_{V^{'}, V} = \int_{1}^{L} (a (u) y_{x} ψ_{x} (t) - b (u) y_{x} ψ) d x, (\forall) ψ \in V$ (37) ).

Theorem 3.1

Let $u \in U$ . The problem (Equation125(125) $\begin{aligned} y_{t} = {(a (u) y_{x})}_{x} + b (u) y_{x}, \end{aligned}$ (125) )–(Equation127(127) $\begin{aligned} y (t, 1) = 0; y_{x} (t, L) = 0 in (0, T) . \end{aligned}$ (127) ) has a unique solution $y^{u} \in C ([0, T]; H) \cap L^{2} (0, T; V_{0}) \cap H^{1} (0, T; V_{0}^{'})$ satisfying the estimates (132) $\begin{aligned} sup_{t \in [0, T]} {‖ y^{u} (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ y^{u} (t) ‖}_{V_{0}}^{2} d t \leq C_{6}, \end{aligned}$ (132) (133) $\begin{aligned} \int_{0}^{T} {‖ \frac{d y^{u}}{d t} (t) ‖}_{V_{0}^{'}}^{2} d t \leq C_{7}, \end{aligned}$ (133) and $- 1 \leq y (t, x) \leq 0$ a.e. in $(0, L)$ for all $t \in [0, T]$ .

Proof.

The existence and uniqueness of the solution of the problem (Equation125(125) $\begin{aligned} y_{t} = {(a (u) y_{x})}_{x} + b (u) y_{x}, \end{aligned}$ (125) )–(Equation127(127) $\begin{aligned} y (t, 1) = 0; y_{x} (t, L) = 0 in (0, T) . \end{aligned}$ (127) ), which has the properties (Equation132(132) $\begin{aligned} sup_{t \in [0, T]} {‖ y^{u} (t) ‖}_{H}^{2} + \int_{0}^{T} {‖ y^{u} (t) ‖}_{V_{0}}^{2} d t \leq C_{6}, \end{aligned}$ (132) ), (Equation133(133) $\begin{aligned} \int_{0}^{T} {‖ \frac{d y^{u}}{d t} (t) ‖}_{V_{0}^{'}}^{2} d t \leq C_{7}, \end{aligned}$ (133) ) is the same as in the proof of Theorem 2.1 from Section 2.1. In order to prove the negativity of the solution we consider its positive part $y^{+} = max {y, 0} .$ We have to show that $y^{+} = 0$ a.e. on $(1, L)$ for each $t \in [0, T]$ . By using Stampacchia's lemma [Citation6, Corollary 2.15, p.289] we have that $(y (t))^{+} \in V_{0}$ a.e. $t \in (0, T)$ and $y_{x}^{+} = {\begin{cases} y_{x} & a . e . on y > 0 \\ 0, & a . e . on y \leq 0 \end{cases} .$ We test (Equation130(130) $\begin{aligned} \frac{d y}{d t} (t) + A (t) y (t) = 0, a . e . t \in (0, T), \end{aligned}$ (130) ) for $y^{+} \in L^{2} (0, T; V_{0})$ and integrate on $(0, t)$ . After some algebra we obtain ${‖ {(y (t))}^{+} ‖}_{H}^{2} + a_{m} \int_{0}^{t} {‖ y_{x}^{+} ‖}_{H}^{2} d τ \leq \frac{{‖ b ‖}_{\infty}^{2}}{a_{m}} \int_{0}^{t} {‖ {(y (t))}^{+} ‖}_{H}^{2} d τ .$ This implies that ${‖ {(y (t))}^{+} ‖}_{H}^{2} \leq \frac{{‖ b ‖}_{\infty}^{2}}{a_{m}} \int_{0}^{t} {‖ {(y (τ))}^{+} ‖}_{H}^{2} d τ$ and the use of Gronwall's lemma leads us to conclude that $(y (t))^{+} = 0$ . This implies that $y (t, x) \leq 0$ a.e. $(t, x) \in Q$ . Similarly we show that $y (t, x) \geq - 1$ $(t, x) \in Q$ . The fact that we have $- 1 \leq y (t, x) \leq 0$ implies $0 \leq C (t, x) \leq 1$ (see (Equation123(123) $C = y + 1.$ (123) )).

Theorem 3.2

Problem (P2) has at least one solution $u^{*}$ with the corresponding state $y^{*} \in C ([0, T]; H) \cap L^{2} (0, T; V_{0}) \cap H^{1} (0, T; V_{0}^{'}) .$

Proof.

The proof is the same as in Theorem 2.2.

We are now able to present the explicit form of the optimal control. In that follows we essentially use the result presented in [Citation15, Proposition 3.2].

Proposition 3.1

If (134) $P e \geq \frac{(β + 1) (β + 2)}{2} u_{M}^{β - 1},$ (134) then a solution $u^{*}$ of the problem (P2) has the form (135) $u^{*} (t) = {\begin{cases} u (0) - u_{\infty} t & for t \in [0, t_{1}) \\ u_{m} & for t \in [t_{1}, t_{2}) \\ u_{\infty} (t - T) + u (T) & for t \in [t_{2}, T], \end{cases}$ (135) where (136) $t_{1} = \frac{u (0) - u_{m}}{u_{\infty}}, t_{2} = T - \frac{u (T) - u_{m}}{u_{\infty}} .$ (136) For $u_{M},$ $u_{m},$ $u_{\infty},$ $u (0),$ $u (T)$ fixed the function $u^{*}$ is unique.

Proof.

The necessary condition of optimality is obtained in the same way as in the case of Dirichlet type boundary conditions. This condition is: $\int_{0}^{T} (u^{*} (t) - u (t)) Φ (t) d t \geq 0,$ for all $u \in U,$ where (137) $Φ (t) := \int_{1}^{L} (a^{'} (u^{*}) p_{x} - b^{'} (u^{*}) p) y_{x}^{*} d x, a . e . t \in (0, T),$ (137) and p is the solution of the dual system: (138) $\begin{aligned} p_{t} = - {(a (u^{*}) p_{x})}_{x} + {(b (u^{*}) p)}_{x}, (t, x) \in Q, \end{aligned}$ (138) (139) $\begin{aligned} p (T, x) = 1, x \in (1, L), \end{aligned}$ (139) (140) $\begin{aligned} p (t, 1) = p_{x} (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (140) According to Proposition 2.2, we have $\begin{aligned} Φ (t) = - θ^{'} (t) + μ (t) a . e . t \in (0, T), \\ u^{*'} (t) \in u_{\infty} sign (θ (t)) a . e . t \in (0, T) . \end{aligned}$ The sign of function Φ can be established as follows. We obtain from Theorem 3.1 that $y^{*} (t, x) \leq 0$ a.e. $(t, x) \in Q$ and from (Equation127(127) $\begin{aligned} y (t, 1) = 0; y_{x} (t, L) = 0 in (0, T) . \end{aligned}$ (127) ) we obtain $y_{x}^{*} (t, 1) \leq 0$ and $y_{x}^{*} (t, L) = 0.$

In order to establish the sign of $y_{x}^{*}$ we consider the sequence $(y_{0, n})_{n \in N^{*}} \subset V_{0}$ so that $y_{0, n} \to y_{0} = - 1$ in H, $y_{0, n} (x) \in [- 1, 0],$ $y_{0, n} (1) = 0$ and $y_{0, n}^{'} (x) \leq 0$ a.e. $x \in (1, L) .$ Let us consider the problem (141) $\begin{aligned} y_{t} = {(a y_{x})}_{x} + b y_{x}, \end{aligned}$ (141) (142) $\begin{aligned} y (0, x) = y_{0, n} (x) in (1, L), \end{aligned}$ (142) (143) $\begin{aligned} y (t, 1) = 0; y_{x} (t, L) = 0 in (0, T) . \end{aligned}$ (143) This problem has a unique solution (144) $y^{n} \in L^{\infty} (0, T; V_{0}) \cap L^{2} (0, T; H^{2} (1, L)) \cap H^{1} (0, T; H)$ (144) (see [Citation16, Theorem 5, p.382]). In addition we have $y^{n} (t, x) \in [- 1, 0]$ a.e. in Q [Citation8, Theorem 2, p.534, Theorem 3, p.535]. As previously done in the demonstration of Theorem 2.2 we show that (eventually passing to a subsequence) the sequence $(y^{n})_{n \in N^{*}}$ weakly converges in $L^{2} (0, T; V_{0})$ to the solution of the problem (Equation125(125) $\begin{aligned} y_{t} = {(a (u) y_{x})}_{x} + b (u) y_{x}, \end{aligned}$ (125) )–(Equation127(127) $\begin{aligned} y (t, 1) = 0; y_{x} (t, L) = 0 in (0, T) . \end{aligned}$ (127) ). Let $y^{n}$ and $y^{m}$ the solutions of (Equation141(141) $\begin{aligned} y_{t} = {(a y_{x})}_{x} + b y_{x}, \end{aligned}$ (141) )–(Equation143(143) $\begin{aligned} y (t, 1) = 0; y_{x} (t, L) = 0 in (0, T) . \end{aligned}$ (143) ) corresponding to $y_{0, n}$ and $y_{0, m} .$ Arguing as in the demonstration of (Equation44(44) ${‖ y^{u_{1}} - y^{u_{2}} ‖}_{L^{\infty} (0, T; H)}^{2} + {‖ y^{u_{1}} - y^{u_{2}} ‖}_{L^{2} (0, T; V)}^{2} \leq C_{2} {‖ u_{1} - u_{2} ‖}_{\infty}^{2},$ (44) ), Theorem 2.1, we obtain (145) ${‖ y^{n} - y^{m} ‖}_{L^{\infty} (0, T; H)} + {‖ y^{n} - y^{m} ‖}_{L^{2} (0, T; V_{0})} \leq C_{8} {‖ y_{0, n} - y_{0, m} ‖}_{H} .$ (145) From $y_{0, n} \to y_{0}$ strongly in H and (Equation145(145) ${‖ y^{n} - y^{m} ‖}_{L^{\infty} (0, T; H)} + {‖ y^{n} - y^{m} ‖}_{L^{2} (0, T; V_{0})} \leq C_{8} {‖ y_{0, n} - y_{0, m} ‖}_{H} .$ (145) ) we obtain that $(y^{n})_{n \in N^{*}}$ is Cauchy in $L^{2} (0, T; V_{0})$ so $y^{n} \to y$ strongly in $L^{2} (0, T; V_{0})$ . Hence, we obtain that $y_{x}^{n} \to y_{x}$ strongly in $L^{2} (0, T; H)$ and so $(y_{x}^{n})^{+} \to (y_{x})^{+} .$ It is easy to see that $y_{x}^{n}$ is the weak solution of the problem (146) $\begin{aligned} z_{t} = {(a (u^{*}) z_{x})}_{x} + (a_{x} (u^{*}) + b (u^{*})) z_{x} + (a_{x x} (u^{*}) + b_{x} (u^{*})) z, \end{aligned}$ (146) (147) $\begin{aligned} z (0, x) = y_{0, n}^{'} (x), \end{aligned}$ (147) (148) $\begin{aligned} z (t, 1) \leq 0, z (t, L) = 0. \end{aligned}$ (148) If we argue as in Theorem 3.1 (or from [Citation8, Theorem 2, p.534]) we obtain $y_{x}^{n} \leq 0,$ a.e. in Q. Hence, we get $0 = (y_{x}^{n})^{+} \to (y_{x})^{+},$ so $y_{x} \leq 0$ a.e. in Q.

If we make the transformation $τ = T - t$ and we use Lions' theorem [Citation8, Theorem 2, p.513] we obtain that (Equation138(138) $\begin{aligned} p_{t} = - {(a (u^{*}) p_{x})}_{x} + {(b (u^{*}) p)}_{x}, (t, x) \in Q, \end{aligned}$ (138) )–(Equation140(140) $\begin{aligned} p (t, 1) = p_{x} (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (140) ) has a unique solution $p \in C (0, T; H) \cap L^{2} (0, T; V_{0}) \cap H^{1} (0, T; V_{0}^{'})$ and $p \geq 0$ a.e. in Q [Citation8, Theorem 2, p.534]. From (Equation140(140) $\begin{aligned} p (t, 1) = p_{x} (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (140) ) we obtain that $p_{x} (t, 1) \leq 0$ and $p_{x} (t, L) = 0$ .

In order to obtain the sign of $p_{x}$ we consider the sequence $(p_{0, n})_{n \in N^{*}} \subset V_{0}$ so that $p_{0, n} \to y_{0} = 1$ in H, $p_{0, n} (x) \in [0, 1],$ $p_{0, n} (1) = 0$ and $p_{0, n}^{'} (x) \geq 0$ a.e. $x \in (1, L)$ . Let us consider the problem (149) $\begin{aligned} p_{t} = - {(a (u^{*}) p_{x})}_{x} + {(b (u^{*}) p)}_{x}, (t, x) \in Q, \end{aligned}$ (149) (150) $\begin{aligned} p (T, x) = p_{0, n} (x), x \in (1, L), \end{aligned}$ (150) (151) $\begin{aligned} p (t, 1) = p_{x} (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (151) This problem has a unique solution (152) $p^{n} \in L^{\infty} (0, T; V_{0}) \cap L^{2} (0, T; H^{2} (1, L)) \cap H^{1} (0, T; H),$ (152) [Citation16, Theorem 5, p.382]. In addition we have $p^{n} (t, x) \in [0, 1]$ a.e. in Q [Citation8, Theorem 2, p.534, Theorem 3, p.535]. Let $v^{n} = p_{x}^{n} .$ Then $v^{n}$ is the solution of the following problem: (153) $\begin{aligned} v_{t}^{n} & = - {(a (u^{*}) v_{x}^{n})}_{x} + (- a_{x} (u^{*}) + b (u^{*})) v_{x}^{n} + (- a_{x x} (u^{*}) + 2 b_{x} (u^{*})) v^{n} \\ + b_{x x} (u^{*}) p^{n}, (t, x) \in Q, \end{aligned}$ (153) (154) $\begin{aligned} v^{n} (T, x) & = p_{0, n} (x), x \in (1, L), \end{aligned}$ (154) (155) $\begin{aligned} v^{n} (t, 1) & \geq 0, v^{n} (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (155) If we take $τ = T - t,$ problem (Equation153(153) $\begin{aligned} v_{t}^{n} & = - {(a (u^{*}) v_{x}^{n})}_{x} + (- a_{x} (u^{*}) + b (u^{*})) v_{x}^{n} + (- a_{x x} (u^{*}) + 2 b_{x} (u^{*})) v^{n} \\ + b_{x x} (u^{*}) p^{n}, (t, x) \in Q, \end{aligned}$ (153) )–(Equation155(155) $\begin{aligned} v^{n} (t, 1) & \geq 0, v^{n} (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (155) ) become (156) $\begin{aligned} v_{t}^{n} & = {(a (u^{*}) v_{x}^{n})}_{x} + (a_{x} (u^{*}) - b (u^{*})) v_{x}^{n} + (a_{x x} (u^{*}) - 2 b_{x} (u^{*})) v^{n} \\ - b_{x x} (u^{*}) p^{n}, (τ, x) \in Q, \end{aligned}$ (156) (157) $\begin{aligned} v^{n} (0, x) & = p_{0, n}^{'} (x), x \in (1, L), \end{aligned}$ (157) (158) $\begin{aligned} v^{n} (τ, 1) & \geq 0, v^{n} (τ, L) = 0, a . e . τ \in (0, T) . \end{aligned}$ (158) We multiply (Equation156(156) $\begin{aligned} v_{t}^{n} & = {(a (u^{*}) v_{x}^{n})}_{x} + (a_{x} (u^{*}) - b (u^{*})) v_{x}^{n} + (a_{x x} (u^{*}) - 2 b_{x} (u^{*})) v^{n} \\ - b_{x x} (u^{*}) p^{n}, (τ, x) \in Q, \end{aligned}$ (156) ) by $v^{-}$ and integrate on $(0, t) \times (1, L)$ . After some algebra we obtain $\begin{aligned} \frac{1}{2} ‖ (v^{n} (t))^{-} ‖_{H}^{2} + a_{m} \int_{0}^{t} ‖ (v_{x}^{n} (τ))^{-} ‖_{H}^{2} d τ \\ \leq ‖ a_{x} (u^{*}) - b (u^{*}) ‖_{\infty} \int_{0}^{t} ‖ (v_{x}^{n} (τ))^{-} ‖_{H} ‖ (v^{n} (τ))^{-} ‖_{H} d τ \\ + ‖ a_{x x} (u^{*}) - 2 b (_{x}) u^{*} ‖_{\infty} \int_{0}^{t} ‖ (v^{n} (τ))^{-} ‖_{H}^{2} d τ + \int_{0}^{t} \int_{1}^{L} b_{x x} (u^{*}) p^{n} (v^{n})^{-} d x d τ . \end{aligned}$

Condition (Equation134(134) $P e \geq \frac{(β + 1) (β + 2)}{2} u_{M}^{β - 1},$ (134) ) assures us that $b_{x x} (u^{*}) \leq 0$ . Due to the fact that $p^{n} \geq 0$ we obtain that the last term in the right member of the last inequality is negative. Cauchy's inequality leads us to (159) $\begin{aligned} {‖ {(v^{n} (t))}^{-} ‖}_{H}^{2} + a_{m} \int_{0}^{t} {‖ {(v_{x}^{n} (τ))}^{-} ‖}_{H}^{2} d τ \\ \leq (\frac{{‖ a_{x} - b ‖}_{\infty}}{a_{m}} + 2 {‖ a_{x x} - 2 b_{x} ‖}_{\infty}) \int_{0}^{t} {‖ {(v^{n} (τ))}^{-} ‖}_{H}^{2} d τ . \end{aligned}$ (159) The use of Gronwall's lemma leads us to conclude from (Equation159(159) $\begin{aligned} {‖ {(v^{n} (t))}^{-} ‖}_{H}^{2} + a_{m} \int_{0}^{t} {‖ {(v_{x}^{n} (τ))}^{-} ‖}_{H}^{2} d τ \\ \leq (\frac{{‖ a_{x} - b ‖}_{\infty}}{a_{m}} + 2 {‖ a_{x x} - 2 b_{x} ‖}_{\infty}) \int_{0}^{t} {‖ {(v^{n} (τ))}^{-} ‖}_{H}^{2} d τ . \end{aligned}$ (159) ) that $(v^{n} (t))^{-} = 0$ . This implies that $p_{x}^{n} = v^{n} \geq 0$ a.e. on Q. We note that from (Equation134(134) $P e \geq \frac{(β + 1) (β + 2)}{2} u_{M}^{β - 1},$ (134) ) we get that $b^{'} (u^{*}) \leq 0$ a.e. on Q. If we argue as before we obtain that $p^{n} \to p$ strongly in $L^{2} (0, T; V_{0})$ , where p is the solution of (Equation138(138) $\begin{aligned} p_{t} = - {(a (u^{*}) p_{x})}_{x} + {(b (u^{*}) p)}_{x}, (t, x) \in Q, \end{aligned}$ (138) )–(Equation140(140) $\begin{aligned} p (t, 1) = p_{x} (t, L) = 0, a . e . t \in (0, T) . \end{aligned}$ (140) ). Hence, we reach the conclusion that $p_{x}^{n} \to p_{x}$ strongly in $L^{2} (0, T; H)$ and so $p_{x}^{n} = (p_{x}^{n})^{+} \to (p_{x})^{+}$ strongly in $L^{2} (0, T; H)$ . We obtain that $p_{x} = (p_{x})^{+}$ so $p_{x} \geq 0$ a.e. in Q. From (Equation137(137) $Φ (t) := \int_{1}^{L} (a^{'} (u^{*}) p_{x} - b^{'} (u^{*}) p) y_{x}^{*} d x, a . e . t \in (0, T),$ (137) ) we obtain that $Φ (t) > 0$ for all $t \in (0, T)$ . The expression (Equation135(135) $u^{*} (t) = {\begin{cases} u (0) - u_{\infty} t & for t \in [0, t_{1}) \\ u_{m} & for t \in [t_{1}, t_{2}) \\ u_{\infty} (t - T) + u (T) & for t \in [t_{2}, T], \end{cases}$ (135) ) of the optimal control $u^{*}$ now results from [Citation15, Proposition 3.2].

4. Numerical results

In this section, we present some numerical results. Our objective is to highlight two main aspects:

presenting the way the mud filtrate dispersion varies in the invaded (damaged) zone and of the mud cake growth;
the use of formula (Equation135(135) $u^{*} (t) = {\begin{cases} u (0) - u_{\infty} t & for t \in [0, t_{1}) \\ u_{m} & for t \in [t_{1}, t_{2}) \\ u_{\infty} (t - T) + u (T) & for t \in [t_{2}, T], \end{cases}$ (135) ) in order to obtain the mud filtrate dispersion.

In order to achieve these goals we consider the wellbore geometry and the physical properties of the reservoir and drilling mud shown in [Citation1,Citation17], for wells no. 1 to 3 and [Citation18], for well no. 4. (see Tables and ). We take the values $α = 51.7$ and $β = 1.25$ for the coefficients α and β from formula (Equation9(9) $D = α {(\frac{u (t)}{2 π x h})}^{β},$ (9) ). Thus we obtain $(β + 1) (β + 2) u_{M}^{β - 1} / 2 = 3.656$ (we use dimensionless values so $u_{M} = 1$ ). In order to obtain the mud filtrate dispersion and the mudcake growth we use the linear single phase mud filtrate invasion model (Equation1(1) $\begin{aligned} \frac{d ξ_{c}}{d t} & = c \frac{u (t)}{x_{w} - ξ_{c} (t)}, \end{aligned}$ (1) ), (Equation2(2) $\begin{aligned} ξ_{c} (0) & = 0, \end{aligned}$ (2) ), (Equation5(5) $\frac{\partial C}{\partial t} = \frac{1}{x} \frac{\partial}{\partial x} (x D \frac{\partial C}{\partial x}) - \frac{u (t)}{2 π x h Φ (1 - S_{w i r} - S_{o r})} \frac{\partial C}{\partial x},$ (5) ), (Equation6(6) $\begin{aligned} C (0, x) = C_{0} (x), x \in (x_{w}, x_{e}), \end{aligned}$ (6) ) with Dirichlet–Neumann boundary conditions (Equation8(8) $C (t, x_{w}) = C_{w}, \frac{\partial C}{\partial x} (t, x_{e}) = 0, t \in (0, T) .$ (8) ). We made the calculations only for the first well and for T = 528 hours. The results are shown in Figures and . In order to use the formula (Equation135(135) $u^{*} (t) = {\begin{cases} u (0) - u_{\infty} t & for t \in [0, t_{1}) \\ u_{m} & for t \in [t_{1}, t_{2}) \\ u_{\infty} (t - T) + u (T) & for t \in [t_{2}, T], \end{cases}$ (135) ) we need the values of parameters $u_{M},$ $u_{m},$ and $u_{\infty}$ . From the four wells we obtained these values from formula (Equation3(3) $\begin{aligned} u (t) = u_{M} \frac{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}}}{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}} - \ln (1 - \frac{ξ_{c} (t)}{r_{w}})}, \end{aligned}$ (3) ) and they are presented in Table . From $β = 1.25$ we obtain $(β + 1) (β + 2) u_{M}^{β - 1} / 2 = 3.656$ (we use dimensionless values so $u_{M} = 1)$ . Condition (Equation134(134) $P e \geq \frac{(β + 1) (β + 2)}{2} u_{M}^{β - 1},$ (134) ) is satisfied only by the first three wells. The values of filtration rate u are presented in Figures , and . We calculate the mud filtrate dispersion by solving the mixed problem (Equation5(5) $\frac{\partial C}{\partial t} = \frac{1}{x} \frac{\partial}{\partial x} (x D \frac{\partial C}{\partial x}) - \frac{u (t)}{2 π x h Φ (1 - S_{w i r} - S_{o r})} \frac{\partial C}{\partial x},$ (5) ), (Equation6(6) $\begin{aligned} C (0, x) = C_{0} (x), x \in (x_{w}, x_{e}), \end{aligned}$ (6) ), (Equation8(8) $C (t, x_{w}) = C_{w}, \frac{\partial C}{\partial x} (t, x_{e}) = 0, t \in (0, T) .$ (8) ). These values are presented in Figures , and . Let $C_{1}$ and $C_{2}$ the mud filtrate dispersion determined by using for the filtration rate u, the formulas (Equation135(135) $u^{*} (t) = {\begin{cases} u (0) - u_{\infty} t & for t \in [0, t_{1}) \\ u_{m} & for t \in [t_{1}, t_{2}) \\ u_{\infty} (t - T) + u (T) & for t \in [t_{2}, T], \end{cases}$ (135) ) and (Equation3(3) $\begin{aligned} u (t) = u_{M} \frac{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}}}{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}} - \ln (1 - \frac{ξ_{c} (t)}{r_{w}})}, \end{aligned}$ (3) ), respectively. In order to compare these values of mud filtrate dispersion we use the formula (160) $Δ C = max_{t \in [0, T]} | C_{1} (T, x) - C_{2} (T, x) | .$ (160) The results obtained are presented in Table . We performed all the calculations using Octave.

Figure 2. Mud filtrate dispersion: Dirichlet–Neumann boundary condition, well no. 1.

Figure 3. Mudcake growth: Dirichlet–Neumann boundary conditions, well no. 1.

Figure 4. The optimal control (filtration rate) for Dirichlet–Neumann boundary conditions, well no 1: red dashed line – optimal control obtained with formulas (Equation3(3) $\begin{aligned} u (t) = u_{M} \frac{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}}}{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}} - \ln (1 - \frac{ξ_{c} (t)}{r_{w}})}, \end{aligned}$ (3) ); blue line – optimal control obtained with formula (Equation135(135) $u^{*} (t) = {\begin{cases} u (0) - u_{\infty} t & for t \in [0, t_{1}) \\ u_{m} & for t \in [t_{1}, t_{2}) \\ u_{\infty} (t - T) + u (T) & for t \in [t_{2}, T], \end{cases}$ (135) ).

Figure 5. Mud filtrate dispersion values, well no 1: red dashed line – optimal control obtained with formulas (Equation3(3) $\begin{aligned} u (t) = u_{M} \frac{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}}}{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}} - \ln (1 - \frac{ξ_{c} (t)}{r_{w}})}, \end{aligned}$ (3) ); blue line – optimal control obtained with formula (Equation135(135) $u^{*} (t) = {\begin{cases} u (0) - u_{\infty} t & for t \in [0, t_{1}) \\ u_{m} & for t \in [t_{1}, t_{2}) \\ u_{\infty} (t - T) + u (T) & for t \in [t_{2}, T], \end{cases}$ (135) ).

Figure 5. Mud filtrate dispersion values, well no 1: red dashed line – optimal control obtained with formulas (Equation3(3) u(t)=uMkckln⁡rerwkckln⁡rerw−ln⁡(1−ξc(t)rw),(3) ); blue line – optimal control obtained with formula (Equation135(135) u∗(t)={u(0)−u∞tfort∈[0,t1)umfort∈[t1,t2)u∞(t−T)+u(T)fort∈[t2,T],(135) ).

Figure 6. The optimal control (filtration rate) for Dirichlet–Neumann boundary conditions, well no 2: red dashed line – optimal control obtained with formulas (Equation3(3) $\begin{aligned} u (t) = u_{M} \frac{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}}}{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}} - \ln (1 - \frac{ξ_{c} (t)}{r_{w}})}, \end{aligned}$ (3) ); blue line – optimal control obtained with formula (Equation135(135) $u^{*} (t) = {\begin{cases} u (0) - u_{\infty} t & for t \in [0, t_{1}) \\ u_{m} & for t \in [t_{1}, t_{2}) \\ u_{\infty} (t - T) + u (T) & for t \in [t_{2}, T], \end{cases}$ (135) ).

Figure 7. Mud filtrate dispersion values, well no 2: red dashed line – optimal control obtained with formulas (Equation3(3) $\begin{aligned} u (t) = u_{M} \frac{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}}}{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}} - \ln (1 - \frac{ξ_{c} (t)}{r_{w}})}, \end{aligned}$ (3) ); blue line – optimal control obtained with formula (Equation135(135) $u^{*} (t) = {\begin{cases} u (0) - u_{\infty} t & for t \in [0, t_{1}) \\ u_{m} & for t \in [t_{1}, t_{2}) \\ u_{\infty} (t - T) + u (T) & for t \in [t_{2}, T], \end{cases}$ (135) ).

Figure 7. Mud filtrate dispersion values, well no 2: red dashed line – optimal control obtained with formulas (Equation3(3) u(t)=uMkckln⁡rerwkckln⁡rerw−ln⁡(1−ξc(t)rw),(3) ); blue line – optimal control obtained with formula (Equation135(135) u∗(t)={u(0)−u∞tfort∈[0,t1)umfort∈[t1,t2)u∞(t−T)+u(T)fort∈[t2,T],(135) ).

Figure 8. The optimal control (filtration rate) for Dirichlet–Neumann boundary conditions, well no 3: red dashed line – optimal control obtained with formulas (Equation3(3) $\begin{aligned} u (t) = u_{M} \frac{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}}}{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}} - \ln (1 - \frac{ξ_{c} (t)}{r_{w}})}, \end{aligned}$ (3) ); blue line – optimal control obtained with formula (Equation135(135) $u^{*} (t) = {\begin{cases} u (0) - u_{\infty} t & for t \in [0, t_{1}) \\ u_{m} & for t \in [t_{1}, t_{2}) \\ u_{\infty} (t - T) + u (T) & for t \in [t_{2}, T], \end{cases}$ (135) ).

Figure 9. Mud filtrate dispersion values, well no 3: red dashed line – optimal control obtained with formulas (Equation3(3) $\begin{aligned} u (t) = u_{M} \frac{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}}}{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}} - \ln (1 - \frac{ξ_{c} (t)}{r_{w}})}, \end{aligned}$ (3) ); blue line – optimal control obtained with formula (Equation135(135) $u^{*} (t) = {\begin{cases} u (0) - u_{\infty} t & for t \in [0, t_{1}) \\ u_{m} & for t \in [t_{1}, t_{2}) \\ u_{\infty} (t - T) + u (T) & for t \in [t_{2}, T], \end{cases}$ (135) ).

Figure 9. Mud filtrate dispersion values, well no 3: red dashed line – optimal control obtained with formulas (Equation3(3) u(t)=uMkckln⁡rerwkckln⁡rerw−ln⁡(1−ξc(t)rw),(3) ); blue line – optimal control obtained with formula (Equation135(135) u∗(t)={u(0)−u∞tfort∈[0,t1)umfort∈[t1,t2)u∞(t−T)+u(T)fort∈[t2,T],(135) ).

Table 1. Well data, Windarto et al. [Citation1], wells 1 to 3, and Wu et al. [Citation18], well 4.

Display Table

Table 2. Additional parameters, Windarto et al. [Citation1], wells 1 to 3, and Wu et al. [Citation18], well 4.

Display Table

Table 3. The values of parameters $u_{M},$ $u_{m},$ $u_{\infty},$ T and Pe.

Display Table

Table 4. Comparison of mud filtrate dispersion values (formula (Equation160(160) $Δ C = max_{t \in [0, T]} | C_{1} (T, x) - C_{2} (T, x) | .$ (160) )).

Display Table

We find that condition (Equation134(134) $P e \geq \frac{(β + 1) (β + 2)}{2} u_{M}^{β - 1},$ (134) ) is fulfilled only for the first three wells. The optimal control $u^{*}$ obtained with formulas (Equation160(160) $Δ C = max_{t \in [0, T]} | C_{1} (T, x) - C_{2} (T, x) | .$ (160) ) (red line) and (Equation135(135) $u^{*} (t) = {\begin{cases} u (0) - u_{\infty} t & for t \in [0, t_{1}) \\ u_{m} & for t \in [t_{1}, t_{2}) \\ u_{\infty} (t - T) + u (T) & for t \in [t_{2}, T], \end{cases}$ (135) ), respectively (blue line), are presented in Figures –.

5. Conclusion

In this paper, we have studied two inverse problems related to the invasion phenomenon. Our goal was to obtain as much information about the invasion rate as possible, if we do not have complete information about the values of all the physical quantities that appear in the mathematical model that describes the invasion phenomenon. We reduced these identification problems to two optimal control problems (minimization problems) in which the control is the invasion rate. We prove the existence of the solution of the state systems and the existence solution of minimization problems. We introduce the first-order variation systems, the dual systems and we prove that these systems have unique solutions. For each of the two problems we obtain the necessary condition of optimality by using the dual system. For the second problem we obtain a simple explicit form of control (filtration rate). We apply this explicit form of control in order to obtain the variation of the invasion rate using the data of three wells. Calculations can be made only if we know the values of Pe, $u_{M},$ $u_{m},$ and $u_{\infty}$ . Regarding the value $u_{\infty}$ we can get information on the upper bound of this quantity (by using the formulas (Equation3(3) $\begin{aligned} u (t) = u_{M} \frac{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}}}{\frac{k_{c}}{k} \ln \frac{r_{e}}{r_{w}} - \ln (1 - \frac{ξ_{c} (t)}{r_{w}})}, \end{aligned}$ (3) ) and (Equation1(1) $\begin{aligned} \frac{d ξ_{c}}{d t} & = c \frac{u (t)}{x_{w} - ξ_{c} (t)}, \end{aligned}$ (1) )).

Acknowledgements

The author thanks the anonymous reviewers for the observations of an earlier form of this paper. The author acknowledges that the work of this paper was done during the doctoral internship at SCOSAAR, Romanian Academy.

Disclosure statement

No potential conflict of interest was reported by the author(s).

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An identification problem related to mud filtrate invasion phenomenon during drilling operations

Abstract

1. Introduction