Pareto optimal control for mixed Neumann infinite-order parabolic system with state-control constraints

Peer review under responsibility of Taibah University.

Abstract

A distributed Pareto optimal control problem for an infinite order parabolic system is considered. The performance index has a vector form with two components in integral form. Constraints on controls and on states are imposed. To obtain optimality conditions for the Neumann problem, the generalization of the Dubovitskii–Milyutin theorem was applied.

MSC:

• 35K20
• 49J20
• 49K20
• 93C20

Keywords:

• Pareto optimal control problems
• Parabolic operators with infinite order
• Neumann Problem
• Dubovitskii–Milyutin theorem
• Conical approximations
• Optimality conditions

1 Introduction

The optimal control problems of distributed parameter systems with constraints imposed on controls and on states have been widely discussed in many papers and monographs. A fundamental study of such problems is given by [39] and was next developed by [40]. It was also intensively investigated by [1,4] and [30–36]. In these studies, questions concerning necessary conditions for optimality and existence of optimal controls for these problems have been investigated.

In Refs. [6,7,10,28,33,34] the optimal control problems for systems described by parabolic and hyperbolic operators with infinite order and consisting of one equation have been discussed. Also we extended the discussion in [1–19,24–27] to n × n coupled systems of elliptic, parabolic and hyperbolic types involving different types of operators. To obtain optimality conditions the arguments of [39] have been applied.

Making use of the Dubovitskii–Milyutin theorem from [29], following in [30–36], authors have obtained necessary and sufficient conditions of optimality for similar systems governed by second order operator with an infinite number of variables and with Dirichlet and Neumann boundary conditions. The interest in the study of this class of operators is stimulated by problems in quantum field theory.

In [31,32], Kotarski considered Pareto optimization problem for a parabolic system and obtained necessary and sufficient conditions for optimality by applying the classical Dubovitskii–Milyutin Theorem [22,23,29]. The performance index was more general than the quadratic one and had an integral form. The set representing the constraints on the controls was assumed to have a nonempty interior. This assumption can be easily removed if we apply the generalized version of the Dubovitskii–Milyutin Theorem [38], instead of the classical one [29] (as the approximation of the set of controls, the regular tangent cone is used instead of the regular admissible cone).

In [2] a time optimal control problem for parabolic equations involving second order operator with an infinite number of variables is considered. In [3] a distributed and boundary control problems for cooperative parabolic and elliptic systems governed by Schrödinger operator is considered. In [33] a distributed control problem for a hyperbolic system with mixed control state constraints involving operator of infinite order is imposed. In [35] a distributed control problem for Neumann parabolic problem with time delay is considered. Also in [36], a distributed control problem for a hyperbolic system involving operator of infinite order with Dirichlet conditions is considered.

In this paper the application of the generalized Dubovitskii–Milyutin Theorem will be demonstrated on an distributed Pareto optimization problem for a system described by a parabolic operator of infinite order with Neumann conditions. The cost function has an integral form. Constraints on controls and on states are imposed. A necessary and sufficient conditions for Pareto optimality are given.

This paper is organized as follows. In Section 2, we introduce some preliminaries and definitions such as functional spaces with infinite order also we define Pareto optimal problems and some related theorems. In Section 3, we define a parabolic equation with infinite order. In Section 4, we formulate the Pareto optimal control problem and we introduce the main results of this paper.

2 Preliminaries

The purpose of this section is to give some preliminaries which we need in this paper.

2.1 Infinite order functional spaces

The aim of this subsection is to give the definition of some functional spaces of infinite-order, and the chains of the constructed spaces which will be used later (Refs. [20,21]). We define the Sobolev space $W^{\infty }\{a_{\alpha },2\}({ℝ}^n)$ (which we shall denote by W^∞{a_α, 2}) of infinite order of periodic functions ϕ(x) defined on all boundary Γ of ${ℝ}^n,n\ge 1,$ as follows,$W^{\infty }\{a_{\alpha },2\}=\left\{\varphi (x)\in C^{\infty }({ℝ}^n):\sum _{|\alpha |=0}^{\infty }{a}_{\alpha }{||D^{\alpha }\varphi ||}_2^2<\infty \right\},$where a_α ≥ 0 is a numerical sequence and || . ||₂ is the canonical norm in the space $L^2({ℝ}^n)$(all functions are assumed to be real valued), and$D^{\alpha }=\frac{{\partial }^{|\alpha |}}{{(\partial x_1)}^{{\alpha }_1}\dots {(\partial x_n)}^{{\alpha }_n}},$where α = (α₁, …, α_n) is a multi-index for differentiation, $|\alpha |={\sum }_{i=1}^n{\alpha }_i$.

The space W^−∞{a_α, 2} is defined as the formal conjugate space to the space W^∞{a_α, 2}, namely:$W^{-\infty }\{a_{\alpha },2\}=\{\psi (x):\psi (x)=\sum _{|\alpha |=0}^{\infty }{a}_{\alpha }{D}^{\alpha }{\psi }_{\alpha }(x)\},$where ${\psi }_{\alpha }\in L^2({ℝ}^n)$ and ${\sum }_{|\alpha |=0}^{\infty }{a}_{\alpha }{||{\psi }_{\alpha }||}_2^2<\infty .$

The duality pairing of the spaces W^∞{a_α, 2} and W^−∞{a_α, 2} is postulated by the formula$(\varphi ,\psi )=\sum _{|\alpha |=0}^{\infty }{a}_{\alpha }{\int }_{{ℝ}^n}{\psi }_{\alpha }(x)D^{\alpha }\varphi (x)\mathrm{dx},$where$\varphi \in W^{\infty }\{a_{\alpha },2\},\psi \in W^{-\infty }\{a_{\alpha },2\}.$

From above, W^∞{a_α, 2} is everywhere dense in $L^2({ℝ}^n)$ with topological inclusions and W^−∞{a_α, 2} denotes the topological dual space with respect to $L^2({ℝ}^n)$, so we have the following chain:$W^{\infty }\{a_{\alpha },2\}\subseteq L^2({ℝ}^n)\subseteq W^{-\infty }\{a_{\alpha },2\}.$

We now introduce $L^2(0,T;L^2({ℝ}^n))$ which we shall denote by L²(Q), where $Q={ℝ}^n\times ]0,T[,$ denotes the space of measurable functions t → ϕ(t) such that${||\varphi ||}_{L^2(Q)}={({\int }_0^T{||\varphi (t)||}_2^2\mathrm{dt})}^{12}<\infty ,$endowed with the scalar product $(f,g)={\int }_0^T{(f(t),g(t))}_{L^2({ℝ}^n)}\mathrm{dt}$, L²(Q) is a Hilbert space. In the same manner we define the spaces L²(0, T ; W^∞{a_α, 2}), and L²(0, T ; W^−∞ {a_α, 2}), as its formal conjugate.

Finally we have the following chains:$L^2(0,T;W^{\infty }\{a_{\alpha },2\})\subseteq L^2(Q)\subseteq L^2(0,T;W^{-\infty }\{a_{\alpha },2\}),$Next, let us introduce the space$W(0,T):=\left\{y;y\in L^2(0,T;W^{\infty }\{a_{\alpha },2\}),\frac{\partial y}{\partial t}\in L^2(0,T;W^{-\infty }\{a_{\alpha },2\})\right\},$in which a solution of a parabolic equation with infinite-order will be contained.

2.2 Definitions Of cones and separation theorem

At first we recall definitions of conical approximations and cones of the same sense or of the opposite sense [29,32,41,42]. Let A be a set contained in a Banach space X and $F:X\to ℝ$ be a given functional. Note that we will state in this section all theorems without proofs and we refer to [32] for the details of the proofs.

Definition 2.1

A set TC(A, x⁰) : = {h ∈ X : ∃ ϵ₀ > 0, ∀ ϵ ∈ (0, ϵ₀), ∃ r(ϵ) ∈ X ; x⁰ + ϵh + r(ϵ) ∈ A}, where $\frac{r(ϵ)}{ϵ}\to 0$ as ϵ → 0 is called the tangent cone to the set A at the point x⁰ ∈ A.

Definition 2.2

A set $\mathrm{AC}(A,x^0):=\{h\in X:\exists {ϵ}_0>0,\exists U(h),\forall ϵ\in (0,{ϵ}_0),\forall \overline{h}\in U(h);x^0+ϵ\overline{h}\in A\}$, where U(h) is a neighborhood of h, is called the admissible cone to the set A at the point x⁰ ∈ A.

Definition 2.3

A set $\mathrm{FC}(F,x^0):=\{h\in X:\exists {ϵ}_0>0,\exists U(h),\forall ϵ\in (0,{ϵ}_0),\forall \overline{h}\in U(h);F(x^0+ϵ\overline{h})<F(x^0)\}$, is called the cone of decrease of the functional F at the point x⁰ ∈ X.

Definition 2.4

A set $\mathrm{NC}(F,x^0):=\{h\in X:\exists {ϵ}_0>0,\exists U(h),\forall ϵ\in (0,{ϵ}_0),\forall \overline{h}\in U(h);F(x^0+ϵ\overline{h})\le F(x^0)\}$, is called the cone of nonincrease of the functional F at the point x⁰ ∈ X.

All the cones defined above are cones with vertices at the origin. The cones AC(A, x⁰), FC(F, x⁰) and NC(F, x⁰) are open while the cone TC(A, x⁰) is closed. If intA ≠ ∅ , then AC(A, x⁰) does not exist. Moreover, if $A_1,\dots ,A_n\in X,x^0\in {\bigcap }_{i=1}^n{A}_i,$ then${\bigcap }_{i=1}^n\mathrm{TC}(A_i,x^0)\supset \mathrm{TC}({\bigcap }_{i=1}^n{A}_i,x^0)\mathrm{and}{\bigcap }_{i=1}^n\mathrm{AC}(A_i,x^0)=\mathrm{AC}({\bigcap }_{i=1}^n{A}_i,x^0).$

If the cones TC(A, x⁰), AC(A, x⁰), FC(F, x⁰) and NC(F, x⁰) are convex, then they are called regular cones and we denote them by RTC(A, x⁰), RAC(A, x⁰), RFC(F, x⁰) and RNC(F, x⁰), respectively.

Let C_i, i = 1, …, n be a system of cones and B_M be a ball with center 0 and radius M > 0 in the space X.

Definition 2.5

The cones C_i, i = 1, …, n are of the same sense if ∀M > 0 , ∃ M₁, …, M₂ > 0 so that $\forall x\in B_M\cap {\sum }_{i=1}^n{C}_i,x={\sum }_{i=1}^n{x}_i,x_i\in C_i,i=1,\dots ,n,$ we have $x_i\in B_{M_i}\cap C_i,i=1,\dots ,n$ (or equivalently the inequality ||x|| ≤ M implies the inequalities ||x_i|| ≤ M_i, i = 1, …, n).

Definition 2.6

The cones C_i, i = 1, …, n are of the opposite sense if ∃(x₁, …, x_n) ≠ (0, …, 0), x_i ∈ C_i, i = 1, …, n so that $0={\sum }_{i=1}^n{x}_i$.

Remark 2.1

From Definitions 2.5 and 2.6 it follows that the set of cones of the same sense is disjoint with the set of cones of the opposite sense. If a certain subsystem of cones is of the opposite sense, then the whole system is also of the opposite sense.

In finite dimensional spaces only the cones of the two types mentioned above may exist while in arbitrary infinite dimensional normed spaces the situation is more complicated. In [42] the conditions under which a system of cones is of the same sense are given.

Definition 2.7

Let K be a cone in X. The adjoint cone K^* of K is defined as$K^{*}:=\{f\in X^{*};f(x)\ge 0\forall x\in K\},$where X^* denotes the dual space of X.

Definition 2.8

Let Q be a set in X, x⁰ ∈ Q . A functional f ∈ X^* is said to be a support functional to the set Q at x⁰ if f(x) ≥ f(x⁰) ∀ x ∈ Q.

Now we are give a theorem on separation of convex cones.

Theorem 2.1

Assuming that:

(i)	the cones K1, …, kp ⊂ X are open and convex with vertices at 0,

(ii)	the cones Kp+1, …, Kn ⊂ X, n > p, are closed and convex with vertices at 0,

(iii)

the adjoint cones $K_{p+1}^{*},\dots ,K_n^{*}$ to Kp+1, …, Kn respectively are either of the same sense or of the opposite sense, then${\bigcap }_{i=1}^n{K}_i=\varnothing$if and only if there exist linear continuous functionals fi, i = 1, …, n not all equal to zero simultaneously; so that(2.1) $\sum _{i=1}^n{f}_i=0(\mathrm{theso}-\mathrm{calledEuler}--\mathrm{LagrangeEquation})$(2.1) where $f_i\in K_i^{*},i=1,\dots ,n.$

2.3 Statement of Pareto optimal problems

Let X be Banach space, Q_k ⊂ X, intQ_k ≠ ∅ , k = 1, …, p represent inequality constraints, Q_k ⊂ X, intQ_k = ∅ , k = p + 1, …, n represent equality constraints, $I_i:=X\to ℝ,i=1,\dots ,s$ are given functionals I = (I₁, …, I_s)^T i.e. $I:X\to {ℝ}^s$ be vector performance index. We are interested in the following problem:

Problem (P): find x⁰ ∈ Q such that

(2.2) $\mathrm{Pareto}\underset{x\in Q\cap U(x^0)}{min}I(x)=I(x^0),$(2.2) where $Q={\bigcap }_{k=1}^n{Q}_k$ and U(x⁰) is some neighborhood of x⁰.

If we define equality constraints in the operator form:$Q_k:=\{x\in X:F_k(x)=0\}$where F_k : X → Y_k are given operators, Y_k are Banach spaces, k = p + 1, …, n, then we obtain Problem (P1) instead of Problem (P).

Definition 2.9

A point x⁰ ∈ X is called global (local) Pareto optimal for Problem (P) or (P1) if x⁰ ∈ Q and there is no x⁰ ≠ x ∈ Q(Q ∩ U(x⁰)) with I_i(x) ≤ I(x⁰) for i = 1, …, s with strict inequality for at least one i, 1 ≤ i ≤ s.

2.4 Necessary conditions for local Pareto optimum generalized Dubovitskii–Milyutin theorem

In the sequel we denote by K_i, i = 1, …, s, D_j, j = 1, …, s, C_k, k = 1, …, p any nonempty open cone contained in the cones FC(I_i, x⁰), NC(I_j, x⁰), and AC(Q_k, x⁰), respectively. C_k, k = p + 1, …, n stands for a nonempty cone contained in the cone TC(Q_k, x⁰) and $\tilde{C}$ is a nonempty cone contained in $\mathrm{TC}({\bigcap }_{k=p+1}^n{Q}_k,x^0)$. All these cones are those with vertices at zero.

For problem (P) or (P1) we have the following necessary condition for Pareto optimality:

Lemma 2.1

If x⁰ ∈ Q is a local Pareto optimum for problem (P) or (P1), Then

(2.3) $k_i\cap \left({\bigcap }_{j=1,j\ne i}^s{D}_j\right)({\bigcap }_{k=1}^p{C}_k)\cap \tilde{C}=\varnothing ,i=1,\dots ,s.$(2.3)

Condition (2.3)(2.3) $k_i\cap \left({\bigcap }_{j=1,j\ne i}^s{D}_j\right)({\bigcap }_{k=1}^p{C}_k)\cap \tilde{C}=\varnothing ,i=1,\dots ,s.$(2.3) in Lemma 2.1 can be formulated in a more convenient form:

Theorem 2.2

Generalized Dubovitskii–Milyutin Theorem

We assume for problem (P) that:

(i)	the cones Ki, i = 1, …, s, Dj, j = 1, …, s, Ck, k = 1, …, p, are open and convex,

(ii)	the cones Ck, k = p + 1, …, n are convex and closed,

(iii)

the cone $\tilde{C}={\bigcap }_{k=p+1}^n{C}_k$ is contained in the cone tangent to the set ${\bigcap }_{k=p+1}^n{Q}_k,$

(iv)	the cones $C_k^{*},k=p+1,\dots ,n$ are either of the same sense or of the opposite sense,

(v)

x0 ∈ Q is a local Pareto optimum for problem (P), then the following “s” equations (the so-called Euler-Lagrange equations) must hold:(2.4) $f_i+\sum _{j=1,j\ne i}{f}_j^{(i)}+\sum _{k=1}^n{\phi }_k^{(i)}=0,i=1,2,\dots ,s,$(2.4) where $f_i\in K_i^{*}$, $f_j^{(i)}\in D_j^{*}$,$j=1,\dots ,s,j\ne i,{\phi }_k^{(i)}\in C_k^{*},k=1,\dots ,n$, with not all functionals equal to zero simultaneously.

From Theorem 2.2 as a particular case follows

Theorem 2.3

We assume for problem (P)that:

(i)	there exists cones: RAC(Qk, x0), k = 1, …, p, RTC(Qk, x0), k = p + 1, …, n, RFC(Ii, x0), i = 1, …, s, and

(ii)	$\mathrm{RTC}({\bigcap }_{k=p+1}^n{Q}_k,x^0)={\bigcap }_{k=p+1}^n\mathrm{RTC}(Q_k,x^0),$

(iii)

the cones ${[\mathrm{RTC}(Q_k,x^0)]}^{*}$ are either of the same sense or of the opposite sense,

(iv)

$x^0\in {\bigcap }_{k=1}^n{Q}_k$ is a local Pareto optimum to the problem (P), then the following “s” equations (the so-called Euler-Lagrange equations) must hold:(2.5) $f_i+\sum _{j=1,j\ne i}{f}_j^{(i)}+\sum _{k=1}^n{\phi }_k^{(i)}=0,i=1,2,\dots ,s,$(2.5) where $f_i\in {[\mathrm{RFC}(I_i,x^0)]}^{*}$, $f_j^{(i)}\in {[\mathrm{RNC}(I_j,x^0)]}^{*}$,j = 1, …, s, j ≠ i, ${\phi }_k^{(i)}\in {[\mathrm{RAC}(Q_k,x^0)]}^{*},k=1,\dots ,p$,and ${\phi }_k^{(i)}\in {[\mathrm{RAC}(Q_k,x^0)]}^{*},k=p+1,\dots ,n$ and all functionals are not equal to zero, simultaneously.

To find conditions ensuring the equality ${[\mathrm{RFC}(I_i,x^0)]}^{*}={[\mathrm{RNC}(I_i,x^0)]}^{*}$ we need.

Definition 2.10

A functional $F:X\to ℝ$ will be called Ponstein convex if$F(x_2)\le F(x_1)ℝ\mathrm{ightarrowF}(\lambda x_1+\nu x_2)<F(x_1),\forall x_1\ne x_2,\lambda ,\nu >0,\lambda +\nu =1.$Strictly convex functionals are also Ponstein convex but not every convex functional is Ponstein convex.

The example below shows that the notations of convexity and Ponstein convexity generally are independent of each other.

Example 2.1

Let us consider the functionals:$f_1,f_2,f_3:ℝ\to ℝ,$ f₁(x) = 0, f₂(x) = − x² − x and f₃(x) = − x, x ∈ [0, 1]. The function f₁ is convex but not Ponstein convex, f₂ is Ponstein convex, but not convex, while f₃ is both convex and Ponstein convex.

3 Mixed Neumann infinite-order parabolic problem

The aim of this section is to give some definitions of the infinite-order operator and the bilinear forms with its coerciveness. Also we formulate the mixed Neumann problem.

Definition 3.1

We define our infinite-order operator with finite dimension in the form:(3.1) $\mathcal{A}\mathrm{\Phi }(x,t)=\sum _{|\alpha |=0}^{\infty }{(-1)}^{|\alpha |}{a}_{\alpha }{D}^{2\alpha }\mathrm{\Phi }(x,t).$(3.1) The operator $\mathcal{A}$ is a bounded self-adjoint elliptic operator with infinite order mapping W^∞{a_α, 2} onto W^−∞{a_α, 2}.

Mixed Neumann problem: We consider the following mixed Neumann evolution equation:(3.2) $\frac{\partial y}{\partial t}+\mathcal{A}y=f,x\in {ℝ}^n,t\in (0,T),$(3.2) (3.3) $y(x,0)=y_p(x),x\in {ℝ}^n,$(3.3) (3.4) $\frac{{\partial }^{\omega }y(x,t)}{\partial {\nu }_{\mathcal{A}}^{\omega }}=0,x\in \mathrm{\Gamma },t\in (0,T),$(3.4) where$f\in L^2(0,T;W^{-\infty }\{a_{\alpha },2\}({ℝ}^n)),y_p\in L^2({ℝ}^n),$are given functions and $\frac{{\partial }^{\omega }}{\partial {\nu }_{\mathcal{A}}^{\omega }}$ is the co-normal derivatives with respect to A, i.e.$\frac{{\partial }^{\omega }}{\partial {\nu }_{\mathcal{A}}^{\omega }}=\frac{{\partial }^{\omega }}{\partial {\nu }^{\omega }}\cos (\nu ;x_k);$ cos(ν ; x_k) = k - th direction cosine of ν ; ν being the normal to the boundary Γ of Rⁿ for |ω| = 0, 1, 2, …, |ω| ≤ α − 1, $\mathcal{A}$ is given by (3.1)(3.1) $\mathcal{A}\mathrm{\Phi }(x,t)=\sum _{|\alpha |=0}^{\infty }{(-1)}^{|\alpha |}{a}_{\alpha }{D}^{2\alpha }\mathrm{\Phi }(x,t).$(3.1) .

Definition 3.2

The bilinear form

For each t ∈]0, T[, we define the following bilinear form on W^∞{a_α, 2}:$\pi (t;\varphi ,\psi )={(\mathcal{A}\varphi ,\psi )}_{L^2({ℝ}^n)},\varphi ,\psi \in W^{\infty }\{a_{\alpha },2\}.$Then

$\begin{array}{cc}\pi (t;\varphi ,\psi )\hfill & ={\left(\mathcal{A}\varphi ,\psi \right)}_{L^2({ℝ}^n)}\hfill \\ \hfill & ={\left(\mathcal{A}\varphi (x),\psi (x)\right)}_{L^2({ℝ}^n)}\hfill \\ \hfill & ={\left(\sum _{|\alpha |=0}^{\infty }{(-1)}^{|\alpha |}{a}_{\alpha }{D}^{2\alpha }\varphi (x,t),\psi (x)\right)}_{L^2({ℝ}^n)}\hfill \\ \hfill & ={\int }_{{ℝ}^n}\sum _{|\alpha |=0}^{\infty }{(-1)}^{|\alpha |}{D}^{\alpha }\varphi (x)D^{\alpha }\psi (x)\mathrm{dx}.\hfill \end{array}$

i.e.(3.5) $\pi (t;\varphi ,\psi )={\int }_{{ℝ}^n}\sum _{|\alpha |=0}^{\infty }{(-1)}^{|\alpha |}{D}^{\alpha }\varphi (x)D^{\alpha }\psi (x)\mathrm{dx}$(3.5)

Lemma 3.1

The bilinear form (3.5)(3.5) $\pi (t;\varphi ,\psi )={\int }_{{ℝ}^n}\sum _{|\alpha |=0}^{\infty }{(-1)}^{|\alpha |}{D}^{\alpha }\varphi (x)D^{\alpha }\psi (x)\mathrm{dx}$(3.5) is coercive on W^∞{a_α, 2} that is, there exists $\eta \in ℝ,$ such that:(3.6) $\pi (t;\varphi ,\varphi )=\eta {||\varphi ||}_{W^{\infty }\{a_{\alpha },2\}}^2,\eta >0.$(3.6)

Proof

It is well known that the ellipticity of $\mathcal{A}$ is sufficient for the coercitivness of π(t ; ϕ, ψ) on W^∞{a_α, 2}. In fact,$\begin{array}{cc}\hfill \pi (t;\varphi ,\varphi )& =\left(\sum _{|\alpha |=0}^{\infty }{(-1)}^{|\alpha |}{a}_{\alpha }{D}^{2\alpha }\varphi (x,t),\varphi (x,t)\right)\hfill \\ \hfill & \ge \left(\sum _{|\alpha |=0}^{\infty }{(-1)}^{|\alpha |}{a}_{\alpha }{||D^{\alpha }\varphi (x)||}_{L^2({ℝ}^n)}^2\right)\hfill \\ \hfill & =\eta {||\varphi (x)||}_{W^{\infty }\{a_{\alpha },2\}}^2.\hfill \end{array}$□

Also we have:

(i)	∀ϕ, ψ ∈ W∞{aα, 2}, the function t → π(t ; ϕ, ψ) is continuously differentiable in ]0, T[ and π(t ; ϕ, ψ) is symmetric i.e.(3.7) $\pi (t;\varphi ,\psi )=\pi (t;\psi ,\varphi ).$(3.7)

(ii)	The operator $\frac{\partial }{\partial t}+\mathcal{A}$ is parabolic operator with an infinite order which maps L2(0, T ; W∞{aα, 2}) onto L2(0, T ; W−∞{aα, 2}) .

Under the above consideration, using the theorems of [39], we can formulate the following mixed Neumann problem, which define the state of our control problem.

4 Pareto optimal control problem

This section is devoted to state the distributed mixed Neumann Pareto optimal control problem and to give several mathematical examples for derived the optimality conditions as follows:

The state equations:(4.1) $\frac{\partial y}{\partial t}+\mathcal{A}y=u,x\in {ℝ}^n,t\in (0,T),$(4.1) (4.2) $y(x,0)=y_p(x),x\in {ℝ}^n,$(4.2) (4.3) $\frac{{\partial }^{\omega }y(x,t)}{\partial {\nu }_{\mathcal{A}}^{\omega }}=0,x\in \mathrm{\Gamma },t\in (0,T).$(4.3)

Then the state is given by the solution of mixed Neumann problem for infinite-order parabolic system and the control u being exercised through in the distributed domain ${ℝ}^n$.

The performance index (The cost function):

(4.4) $I(y,u)=\left[\begin{array}{cc}\hfill & I_1(u)\hfill \\ \hfill & I_2(y)\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill & {\int }_0^T{\int }_{{ℝ}^n}{u}^2\mathrm{dxdt}\hfill \\ \hfill & {\int }_{{ℝ}^n}{(y(T,x)-z_d(x))}^2\mathrm{dx}\hfill \end{array}\right]\to \text{Pareto min}.$(4.4)

Control constraints.

We assume the following constraints on controls:

(4.5) $u\in {\mathcal{U}}_{\mathrm{ad}}\subset \mathcal{U}:=L^2(0,T;W^{\infty }\{a_{\alpha },2\}),{\mathcal{U}}_{\mathrm{ad}}\text{is closed and convex.}$(4.5)

State constraints.

We assume the following constraints on states:

(4.6) $y\in Y_{\mathrm{ad}}\subset Y:=L^2(0,T;W^{\infty }\{a_{\alpha },2\}),Y_{\mathrm{ad}}\text{is closed convex with a non-empty interior in}Y$(4.6) where $y_p,z_d\in L^2({ℝ}^n)$ are given. $\mathcal{A}$ is the same operator defined in Section 3. The control time T is assumed to be fixed in our problem.

We also assume that there exists $(\tilde{y},\tilde{u})$ such as $\tilde{u}\in {\mathcal{U}}_{\mathrm{ad}},\tilde{y}\in \text{int}{Y}_{\mathrm{ad}}$ and $(\tilde{y},\tilde{u})$ satisfy equations (4.1)–(4.3) (Slater's condition).

The solution of the stated Pareto optimal control problem (4.1)–(4.6) is equivalent to seeking of a pair $(y^0,u^0)\in E:=Y\times \mathcal{U},$ which satisfies Eqs. (4.1)–(4.3) and minimizes in the Pareto sense the vector functional (4.4)(4.4) $I(y,u)=\left[\begin{array}{cc}\hfill & I_1(u)\hfill \\ \hfill & I_2(y)\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill & {\int }_0^T{\int }_{{ℝ}^n}{u}^2\mathrm{dxdt}\hfill \\ \hfill & {\int }_{{ℝ}^n}{(y(T,x)-z_d(x))}^2\mathrm{dx}\hfill \end{array}\right]\to \text{Pareto min}.$(4.4) under constraints (4.5(4.5) $u\in {\mathcal{U}}_{\mathrm{ad}}\subset \mathcal{U}:=L^2(0,T;W^{\infty }\{a_{\alpha },2\}),{\mathcal{U}}_{\mathrm{ad}}\text{is closed and convex.}$(4.5) –(4.6)(4.6) $y\in Y_{\mathrm{ad}}\subset Y:=L^2(0,T;W^{\infty }\{a_{\alpha },2\}),Y_{\mathrm{ad}}\text{is closed convex with a non-empty interior in}Y$(4.6) .

We formulate the necessary and sufficient conditions of optimality for the problem (4.1)–(4.6) in the following optimization theorem.

Theorem 4.1

For every λ₁, λ₂ > 0 such as λ₁ + λ₂ = 1 there is the unique solution (y⁰, u⁰) to the Pareto optimal control problem (4.1)–(4.6). Moreover, there are two adjoint states p and ξ such as p ∈ W(0, T), and ξ ∈ L²(0, T ; W^−∞{a_α, 2})). Besides, p and u⁰ satisfy (in the weak sense) the adjoint equations given below. The necessary and sufficient conditions of optimality are characterized by the the following system of partial differential equations and inequalities:

State equations:

(4.7) $\frac{\partial y^0}{\partial t}+\mathcal{A}{y}^0=u^0,x\in {ℝ}^n,t\in (0,T),$(4.7) (4.8) $y^0(x,0)=y_p(x),x\in {ℝ}^n.$(4.8) (4.9) $\frac{{\partial }^{\omega }{y}^0(x,t)}{\partial {\nu }_{\mathcal{A}}^{\omega }}=0,x\in \mathrm{\Gamma },t\in (0,T),$(4.9)

Adjoint equations:

(4.10) $-\frac{\partial p}{\partial t}+{\mathcal{A}}^{*}p=0,x\in {ℝ}^n,t\in (0,T),$(4.10) (4.11) $p(x,T)={\lambda }_2[y^0(x,T)-z_d],x\in {ℝ}^n,$(4.11) (4.12) $\frac{{\partial }^{\omega }p(x,t)}{\partial {\nu }_{\mathcal{A}}^{\omega }}=0,x\in \mathrm{\Gamma },t\in (0,T).$(4.12) (4.13) $-\frac{\partial u^0}{\partial t}+{\mathcal{A}}^{*}{u}^0=\xi ,x\in {ℝ}^n,t\in (0,T),$(4.13) (4.14) $u^0(x,T)=-\frac{1}{{\lambda }_1}p(x,T),x\in {ℝ}^n,$(4.14) (4.15) $\frac{{\partial }^{\omega }{u}^0(x,t)}{\partial {\nu }_{\mathcal{A}}^{\omega }}=0,x\in \mathrm{\Gamma },t\in (0,T).$(4.15)

Maximum conditions:

(4.16) ${\int }_0^T{\int }_{{ℝ}^n}(p+{\lambda }_1{u}^0)(u-u^0)\mathrm{dxdt}\ge 0\forall u\in U_{\mathrm{ad}},$(4.16) (4.17) ${\int }_0^T{\int }_{{ℝ}^n}\xi (y-y^0)\mathrm{dxdt}\ge 0\forall y\in Y_{\mathrm{ad}}.$(4.17)

Proof

Note that the conditions $\underset{(y,u)}{inf}{I}_i(y,u)<I_i(y^0,u^0),i=1,2$ hold, I₁, I₂ are strictly convex, hence they are Ponstein convex (strict convexity implies the Ponstein convexity). I₁, I₂ are also Frèchet differentiable.

The stated Pareto optimal control problem (4.1)–(4.6) is equivalent to the one with the scalar performance functional I = λ₁I₁ + λ₂I₂, λ₁, λ₂ > 0, λ₁ + λ₂ = 1. To this scalar problem we apply Theorem 1.8.1 in [32]. We approximate the set ${\mathcal{U}}_{\mathrm{ad}}$ by the admissible cone, the set Y_ad and the constraints given by equations (4.1)–(4.3) by the tangent cones and the scalar functional by the cone of decrease.

(a.) Analysis of constraints on controls.

The set $Q_1=Y\times {\mathcal{U}}_{\mathrm{ad}}\subset E$ represents equality constraints. Using Theorem 10.5 [29] we find the functional belonging to the adjoint tangent cone i.e.$f_1(\overline{y},\overline{u})\in {[\mathrm{RTC}(Q_1,(y^0,u^0))]}^{*}.$The functional $f_1(\overline{u},\overline{u})$ can be expressed as follows$f_1(\overline{u},\overline{u})=f_1^1(\overline{y})+f_1^2(\overline{u})$where $f_1^1(\overline{y})=0\forall \overline{y}\in Y$ (Theorem 10.1 [29]) and $f_1^2(\overline{u})$ is the support functional to the set U_ad at the point u⁰ (Theorem 10.5 [29]).

(b.) Analysis of constraints on states.

The set Q₂ = Y_ad × Y ⊂ E represents inequality constraints. Using Theorem 10.5 [29] we find the functional belonging to the adjoint regular admissible cone i.e.$f_2(\overline{y},\overline{u})\in {[\mathrm{RAC}(Q_2,(y^0,u^0))]}^{*}.$

Similarly as above we have that $f_2(\overline{y},\overline{u})=f_2^1(\overline{y})$ is equal to the support functional to the set Y_ad at the point y⁰.

(c.) Analysis of state equations (4.1)–(4.3).

The set$Q_3:=\left\{(y,u)\in E;\begin{array}{cc}\hfill \frac{\partial y}{\partial t}+\mathcal{A}y& =u,x\in {ℝ}^n,t\in (0,T),\hfill \\ \hfill y(x,0)& =y_p(x),x\in {ℝ}^n,\hfill \\ \hfill \frac{{\partial }^{\omega }y(x,t)}{\partial {\nu }_{\mathcal{A}}^{\omega }}& =0,x\in \mathrm{\Gamma },t\in (0,T)\hfill \end{array}\right\}$

represents the equality constraints. On the basis of Lusternik's theorem (Theorem 9.1 [29]) the regular tangent cone has the form

$\begin{array}{cc}\mathrm{RTC}(Q_2,(y^0,u^0))\hfill & =\left\{(\overline{y},\overline{u})\in E;P^{\prime }(y^0,u^0)(\overline{y},\overline{u})=0\right\}\hfill \\ \hfill & =\left\{(\overline{y},\overline{u})\in E;\begin{array}{cc}\hfill \frac{\partial \overline{y}}{\partial t}+\mathcal{A}\overline{y}& =\overline{u},x\in {ℝ}^n,t\in (0,T)\hfill \\ \hfill \overline{y}(x,0)& =0,x\in {ℝ}^n,\hfill \\ \hfill \frac{{\partial }^{\omega }\overline{y}(x,t)}{\partial {\nu }_{\mathcal{A}}^{\omega }}& =0,x\in \mathrm{\Gamma },t\in (0,T)\hfill \end{array}\right\}\hfill \end{array}$where $P^{\prime }(y^0,u^0)(\overline{y},\overline{u})$ is the Frèchet differential of the operator$P(y,u):=\left(\frac{\partial y}{\partial t}+\mathcal{A}y-u,y(x,0)-y_p(x)\right)$mapping from the space$\Im :=L^2(0,T;W^{\infty }\{a_{\alpha },2\})\times L^2(0,T;W^{\infty }\{a_{\alpha },2\})$into the space$\mathcal{Z}:=L^2(0,T;W^{-\infty }\{a_{\alpha },2\})\times L^2({ℝ}^n).$Knowing that there exists a unique solution to the equation (4.2)–(4.3) for every u and y_p it is easy to prove that P′(y⁰, u⁰) is the mapping from the space ℑ onto $\mathcal{Z}$ as required in the Lusternik theorem.

(d.) Analysis of the performance functional.

Applying Theorem 7.5 [29] we find the cone$\mathrm{RFC}(I,(y^0,u^0))=\left\{(\overline{y},\overline{u})\in E;\sum _{i=1}^2{\lambda }_i{I^{\prime }}_i(y^0,u^0)(\overline{y},\overline{u})<0\right\},$where ${I^{\prime }}_i$ denotes the Frèchet differential of I_i.

It is easily seen that$\begin{array}{cc}\hfill {I^{\prime }}_1(\overline{y},\overline{i})& =2{\int }_0^T{\int }_{{ℝ}^n}{u}^0\overline{u}\mathrm{dxdt},\hfill \\ \hfill {I^{\prime }}_2(\overline{y},\overline{u})& =2{\int }_{{ℝ}^n}(y^0(T)-z_d)\overline{y}(T)\mathrm{dx}.\hfill \end{array}$From Theorem 19.2 [29] we find the functional belonging to the adjoint cone. It has the form$f_4(\overline{y},\overline{u})=-\mu {\lambda }_1{\int }_0^T{\int }_{{ℝ}^n}{u}^1\overline{u}\mathrm{dxdt}-\mu {\lambda }_2{\int }_{{ℝ}^n}(y^0(T)-z_d)\overline{y}(T)\mathrm{dx},$where μ ≥ 0. From Remark 1.5.1 [32] it follows that μ ≠ 0 .

To write down the Euler-Lagrange Equation, we need to check the assumption $(v)$ of Theorem 1.8.1 [32].

It is known that the tangent cones are closed [38]. Following the idea of [41], we shall show that:-$\mathrm{RTC}(Q_1\cap Q_3,(y^0,u^0))=\mathrm{RTC}(Q_1,(y^0,u^0))\bigcap \mathrm{RTC}(Q_3,(y^0,u^0)).$We only need to show the inclusion ″ ⊂ ″, because we always have ″ ⊃ ″ [38].

It can be easily checked that in the neighborhood V₁ of the point (y⁰, u⁰) the operator P satisfies the assumptions of the implicit function theorem [41]. Consequently, the set Q₃ can be represented in the neighborhood V₀ in the form(4.18) $\left\{(y,u)\in E;y=\phi (l)\right\},$(4.18) where φ : U → Y is an operator of the class C¹ satisfying the condition P(φ(u), u) = 0 for u such as (φ(u), u) ∈ V₀. From this we know that(4.19) $\mathrm{RTC}(Q_3,(k^0,u^0))=\left\{(\overline{y},\overline{u})\in E;\overline{y}={\phi }_u(u^6)\overline{u}\right\}.$(4.19)

Let $(\overline{y},\overline{u})$ be any element of the set$\mathrm{RTC}(Q_1,(y^0,u^4))\bigcap \mathrm{RTC}(Q_3,(y^0,u^0)).$

From the definition of the tangent cone we can see that there exists the operator $r_u^1:={ℝ}^1\to U$ such as $\frac{r_u^1(ϵ)}{ϵ}\to 0$ with ϵ → 0⁺ and(4.20) $(y^0,u^2)+ϵ(\overline{y},\overline{u})+(r_y^1,r_u^1)\in Q_1$(4.20) for a sufficiently small ϵ and with any $r_y^4(ϵ)$.

From (4.18)(4.18) $\left\{(y,u)\in E;y=\phi (l)\right\},$(4.18) follows that for sufficiently small ϵ, we have$\left(\phi (u^0+ϵ\overline{u}+r_u^1(ϵ)),u^0+ϵ\overline{u}+r_u^1(ϵ)\right)\in Q_3.$Since φ is a differentiable operator, therefore$\phi (u^0+ϵ\overline{u}+r_u^1(ϵ))=\phi (u^0)+ϵ{\phi }_u(u^9)\overline{u}+r_y^3(ϵ)$for some $r_y^3(ϵ)$ such as $\frac{r_y^3(ϵ)}{ϵ}\to 0$ with ϵ → 0⁺.

Taking into account (4.18(4.18) $\left\{(y,u)\in E;y=\phi (l)\right\},$(4.18) ) and (4.19(4.19) $\mathrm{RTC}(Q_3,(k^0,u^0))=\left\{(\overline{y},\overline{u})\in E;\overline{y}={\phi }_u(u^6)\overline{u}\right\}.$(4.19) ), we get(4.21) $(y^0,u^2)+ϵ(\overline{y},\overline{u})+(r_y^3(ϵ),r_u^1(ϵ))\in Q_3.$(4.21)

If in (4.20)(4.20) $(y^0,u^2)+ϵ(\overline{y},\overline{u})+(r_y^1,r_u^1)\in Q_1$(4.20) we have $r_u^1(ϵ)=r_y^3(ϵ)$, then it follows from (4.20(4.20) $(y^0,u^2)+ϵ(\overline{y},\overline{u})+(r_y^1,r_u^1)\in Q_1$(4.20) ) and (4.21(4.21) $(y^0,u^2)+ϵ(\overline{y},\overline{u})+(r_y^3(ϵ),r_u^1(ϵ))\in Q_3.$(4.21) ) that $(\overline{y},\overline{u})$ is an element of the cone tangent to the set Q₁ ∩ Q₃ at (y⁰, u⁰). It completes the proof of the inclusion ″ ⊃ ″. Further applying Theorem 3.3 [32] we can prove that the adjoint cones ${[\mathrm{RTC}(Q_1,(y^0,u^0))]}^{*}$ and ${[\mathrm{RTC}(Q_3,(y^0,u^0))]}^{*}$ are of the same sense.

(e.) Analysis of the Euler-Lagrange Equation.

The Euler-Lagrange Equation for our optimization problem has the form(4.22) $\sum _{i=1}^4{f}_i(\overline{y},\overline{u})=0.$(4.22)

Taking into account the form of functionals in (4.22)(4.22) $\sum _{i=1}^4{f}_i(\overline{y},\overline{u})=0.$(4.22) , we get(4.23) $\begin{array}{c}\hfill f_1^2(\overline{u})+f_2^1(\overline{y})=\mu {\lambda }_1{\int }_0^T{\int }_{{ℝ}^n}{u}^0\overline{u}\mathrm{dxdt}+\mu {\lambda }_2{\int }_{{ℝ}^n}(y^0(T)-z_d)\overline{y}(T)\mathrm{dx},\hfill \\ \hfill \forall (\overline{y},\overline{u})\in \mathrm{RTC}(Q_3,(y^0,u^0)).\hfill \end{array}$(4.23)

Wa transform the component with $\overline{y}(T)$ in (4.23)(4.23) $\begin{array}{c}\hfill f_1^2(\overline{u})+f_2^1(\overline{y})=\mu {\lambda }_1{\int }_0^T{\int }_{{ℝ}^n}{u}^0\overline{u}\mathrm{dxdt}+\mu {\lambda }_2{\int }_{{ℝ}^n}(y^0(T)-z_d)\overline{y}(T)\mathrm{dx},\hfill \\ \hfill \forall (\overline{y},\overline{u})\in \mathrm{RTC}(Q_3,(y^0,u^0)).\hfill \end{array}$(4.23) using the adjoint equations (4.10)–(4.12) and the fact that $(\overline{y},\overline{u})\in \mathrm{RTC}(Q_3,(y^0,u^0)).$

In turn, we get(4.24) $\begin{array}{cc}\hfill 0& ={\int }_0^T{\int }_{{ℝ}^n}\left(-\frac{\partial p}{\partial t}+{\mathcal{A}}^{*}p\right)\overline{y}\mathrm{dxdt}\hfill \\ \hfill & ={\int }_0^T{\int }_{{ℝ}^n}\left(\frac{\partial \overline{y}}{\partial t}+\mathcal{A}\overline{y}\right)\mathrm{pdxdt}\hfill \\ \hfill & +{\int }_{{ℝ}^n}p(0)\overline{y}(0)\mathrm{dx}-{\int }_{{ℝ}^n}p(T)\overline{y}(T)\mathrm{dx}\hfill \\ \hfill & ={\int }_0^T{\int }_{R^n}p\overline{u}\mathrm{dxdt}-{\int }_{{ℝ}^n}p(T)\overline{y}(T)\mathrm{dx}.\hfill \end{array}$(4.24)

From (4.24(4.24) $\begin{array}{cc}\hfill 0& ={\int }_0^T{\int }_{{ℝ}^n}\left(-\frac{\partial p}{\partial t}+{\mathcal{A}}^{*}p\right)\overline{y}\mathrm{dxdt}\hfill \\ \hfill & ={\int }_0^T{\int }_{{ℝ}^n}\left(\frac{\partial \overline{y}}{\partial t}+\mathcal{A}\overline{y}\right)\mathrm{pdxdt}\hfill \\ \hfill & +{\int }_{{ℝ}^n}p(0)\overline{y}(0)\mathrm{dx}-{\int }_{{ℝ}^n}p(T)\overline{y}(T)\mathrm{dx}\hfill \\ \hfill & ={\int }_0^T{\int }_{R^n}p\overline{u}\mathrm{dxdt}-{\int }_{{ℝ}^n}p(T)\overline{y}(T)\mathrm{dx}.\hfill \end{array}$(4.24) ) and 4.11(4.11) $p(x,T)={\lambda }_2[y^0(x,T)-z_d],x\in {ℝ}^n,$(4.11) ), we obtain${\lambda }_2{\int }_{{ℝ}^n}(y^0(T)-z_d)\overline{y}(T)\mathrm{dx}={\int }_0^T{\int }_{{ℝ}^n}p\overline{u}\mathrm{dxdt}.$Transforming the component with $\overline{u}$ in (4.23)(4.23) $\begin{array}{c}\hfill f_1^2(\overline{u})+f_2^1(\overline{y})=\mu {\lambda }_1{\int }_0^T{\int }_{{ℝ}^n}{u}^0\overline{u}\mathrm{dxdt}+\mu {\lambda }_2{\int }_{{ℝ}^n}(y^0(T)-z_d)\overline{y}(T)\mathrm{dx},\hfill \\ \hfill \forall (\overline{y},\overline{u})\in \mathrm{RTC}(Q_3,(y^0,u^0)).\hfill \end{array}$(4.23) with the help of the adjoint equations (4.13)–(4.15) and having in mind that $(\overline{y},\overline{u})\in \mathrm{RTC}(Q_3,(y^0,u^0)),$ we get(4.25) $\begin{array}{cc}\hfill {\int }_0^T{\int }_{{ℝ}^n}{u}^0\overline{u}\mathrm{dxdt}& ={\int }_0^T{\int }_{{ℝ}^n}{u}^0\left(\frac{\partial \overline{y}}{\partial t}+\mathcal{A}\overline{y}\right)\mathrm{dxdt}\hfill \\ \hfill & ={\int }_0^T{\int }_{{ℝ}^n}\left(-\frac{\partial u^0}{\partial t}+{\mathcal{A}}^{*}{u}^0\right)\overline{y}\mathrm{dxdt}-{\int }_{{ℝ}^n}{u}^0(0)\overline{y}(0)\mathrm{dx}\hfill \\ \hfill & +{\int }_{{ℝ}^n}{u}^0(T)\overline{y}(T)\mathrm{dx}={\int }_0^T{\int }_{{ℝ}^n}\xi \overline{y}\mathrm{dxdt}+{\int }_{{ℝ}^n}{u}^0(T)\overline{y}(T)\mathrm{dx}\hfill \\ \hfill & ={\int }_1^T{\int }_{{ℝ}^n}\xi \overline{y}\mathrm{dxdt}-\frac{{\lambda }_2}{{\lambda }_1}{\int }_{{ℝ}^n}(y^0(T)-z_d)\overline{y}\mathrm{dx}.\hfill \end{array}$(4.25)

Replacing the right-hand side of (4.23)(4.23) $\begin{array}{c}\hfill f_1^2(\overline{u})+f_2^1(\overline{y})=\mu {\lambda }_1{\int }_0^T{\int }_{{ℝ}^n}{u}^0\overline{u}\mathrm{dxdt}+\mu {\lambda }_2{\int }_{{ℝ}^n}(y^0(T)-z_d)\overline{y}(T)\mathrm{dx},\hfill \\ \hfill \forall (\overline{y},\overline{u})\in \mathrm{RTC}(Q_3,(y^0,u^0)).\hfill \end{array}$(4.23) by (4.24)(4.24) $\begin{array}{cc}\hfill 0& ={\int }_0^T{\int }_{{ℝ}^n}\left(-\frac{\partial p}{\partial t}+{\mathcal{A}}^{*}p\right)\overline{y}\mathrm{dxdt}\hfill \\ \hfill & ={\int }_0^T{\int }_{{ℝ}^n}\left(\frac{\partial \overline{y}}{\partial t}+\mathcal{A}\overline{y}\right)\mathrm{pdxdt}\hfill \\ \hfill & +{\int }_{{ℝ}^n}p(0)\overline{y}(0)\mathrm{dx}-{\int }_{{ℝ}^n}p(T)\overline{y}(T)\mathrm{dx}\hfill \\ \hfill & ={\int }_0^T{\int }_{R^n}p\overline{u}\mathrm{dxdt}-{\int }_{{ℝ}^n}p(T)\overline{y}(T)\mathrm{dx}.\hfill \end{array}$(4.24) and (4.25(4.25) $\begin{array}{cc}\hfill {\int }_0^T{\int }_{{ℝ}^n}{u}^0\overline{u}\mathrm{dxdt}& ={\int }_0^T{\int }_{{ℝ}^n}{u}^0\left(\frac{\partial \overline{y}}{\partial t}+\mathcal{A}\overline{y}\right)\mathrm{dxdt}\hfill \\ \hfill & ={\int }_0^T{\int }_{{ℝ}^n}\left(-\frac{\partial u^0}{\partial t}+{\mathcal{A}}^{*}{u}^0\right)\overline{y}\mathrm{dxdt}-{\int }_{{ℝ}^n}{u}^0(0)\overline{y}(0)\mathrm{dx}\hfill \\ \hfill & +{\int }_{{ℝ}^n}{u}^0(T)\overline{y}(T)\mathrm{dx}={\int }_0^T{\int }_{{ℝ}^n}\xi \overline{y}\mathrm{dxdt}+{\int }_{{ℝ}^n}{u}^0(T)\overline{y}(T)\mathrm{dx}\hfill \\ \hfill & ={\int }_1^T{\int }_{{ℝ}^n}\xi \overline{y}\mathrm{dxdt}-\frac{{\lambda }_2}{{\lambda }_1}{\int }_{{ℝ}^n}(y^0(T)-z_d)\overline{y}\mathrm{dx}.\hfill \end{array}$(4.25) ), we get(4.26) $f_1^2(\overline{u})+f_2^1(\overline{y})=\frac{1}{2}\mu {\int }_0^T{\int }_{{ℝ}^n}(p+{\lambda }_1{u}^0)\overline{u}\mathrm{dxdt}+\frac{1}{2}\mu {\int }_0^T{\int }_{{ℝ}^n}\xi \overline{y}\mathrm{dxdt}.$(4.26)

Further from (4.26)(4.26) $f_1^2(\overline{u})+f_2^1(\overline{y})=\frac{1}{2}\mu {\int }_0^T{\int }_{{ℝ}^n}(p+{\lambda }_1{u}^0)\overline{u}\mathrm{dxdt}+\frac{1}{2}\mu {\int }_0^T{\int }_{{ℝ}^n}\xi \overline{y}\mathrm{dxdt}.$(4.26) and the definition of the support functional to ${\mathcal{U}}_{\mathrm{ad}}$ and Y_ad, respectively at the point u⁰ or y⁰, we obtain maximum conditions (4.16)–(4.17). This last remark ends the proof of necessity.

The conditions (4.16)(4.16) ${\int }_0^T{\int }_{{ℝ}^n}(p+{\lambda }_1{u}^0)(u-u^0)\mathrm{dxdt}\ge 0\forall u\in U_{\mathrm{ad}},$(4.16) –4.17(4.17) ${\int }_0^T{\int }_{{ℝ}^n}\xi (y-y^0)\mathrm{dxdt}\ge 0\forall y\in Y_{\mathrm{ad}}.$(4.17) ) are also sufficient for the Pareto optimality for the problem (4.1)–(4.6). It follows immediately from the fact that the stated optimization problem is convex, I₁, I₂ are convex, continuous and so the Slater condition is fulfilled. The uniqueness of the optimal pair y⁰, u⁰ follows from the strict convexity of the scalar performance index.▪

Comments

The main result of the paper contains necessary and sufficient conditions of optimality (of Pontryagin's type) for infinite order parabolic system that give characterization of Pareto optimal control. But it is easily seen that obtaining analytical formulas for optimal control is very difficult. This results from the fact that state equations (4.7)–(4.9), adjoint equations (4.10)–(4.15) and maximum conditions (4.16)–(4.17) are mutually connected that cause that the usage of derived conditions is difficult. Therefore we must resign from the exact determining of the optimal control and therefore we are forced to use approximations methods. Those problems need further investigations and form tasks for future research.

Also it is evident that by modifying:

•	the boundary conditions, (Dirichlet, Neumann, mixed, etc.),

•	the nature of the control (distributed, boundary, etc.),

•	the nature of the observation (distributed, boundary, etc.),

•	the initial differential system,

•	the time delays (constant time delays, time-varying delays, multiple time-varying delays, time delays given in the integral form, etc.),

•	the number of variables (finite number of variables, infinite number of variables systems, etc.),

•	the type of equation (elliptic, parabolic, hyperbolic, etc.),

•	the order of equation (second order, Schrödinger, infinite order, etc.),

•

the type of control (optimal control problem, time-optimal control problem, etc.), many infinity of variations on the above problems are possible to study with the help of [39] and Dubovitskii–Milyutin formalisms see [1–19,37]. Those problems need further investigations and form tasks for future research. These ideas mentioned above will be developed in forthcoming papers.

Acknowledgements

The research presented here was carried out within the research programme of the Taibah University-Dean of Scientific Research under project number 6006/1435 H. The authors would like to express their gratitude to the anonymous reviewers for their very valuable remarks.

Notes

Peer review under responsibility of Taibah University.