On the approximate controllability of semilinear control systems

Abstract

In this paper, approximate controllability of an abstract semilinear control system is proved under simple sufficient conditions in Hilbert spaces. The results are obtained when nonlinearity satisfy Lipschitz condition. At the end one example is given to illustrate the results.

Keywords:

• semilinear control systems
• approximate controllability
• solution operator

Mathematics subject classifications:

• 34K30–34K35–93C25

Public Interest Statement

Controllability is an important concept pertaining to any control system. It determines whether the state of the system can be steered to a given target state in a prescribed time interval or not. Therefore, it plays a very important role in the analysis and design of control systems. Also, the noise or stochastic perturbation is omnipresent and unavoidable in nature as well as in man-made systems. So, we have to move from deterministic systems to stochastic systems. Many practical problems contain a delay term in their respective control equations.

Therefore, in this paper, we discuss the approximate controllability of semilinear stochastic systems with multiple delays in control using fixed point theorem technique.

1. Introduction

Let V and be Hilbert spaces and Z = L₂[0, T:V] and $Y=L_2[0,T:\stackrel{\mathrm{\Lambda }}{V}]$ be the corresponding function spaces defined on [0, T], 0 ≤ T < ∞. Consider the semilinear deterministic control system:(1.1) $$\frac{\text{d}{x}_u(t)}{\text{d}t}=A{x}_u(t)+Bu(t)+f(t,x_u(t))$$(1.1) $$x_u(0)=0$$

where A: D(A) ⊆ V → V is a closed linear operator with dense domain D(A) generating a C₀-semigroup S(t) (Mahmudov, Vijayakumar, & Murugesu, 2016, Pazy, 1983), f:[0, t] × V → V is a nonlinear operator which satisfies Caratheodory condition (Joshi & Sukavanam, 1990) and $B:\stackrel{\mathrm{\Lambda }}{V}\to V,$ a bounded linear operator. x_u(t) is state value at time t ∈ [0, T] corresponding to the control u taken from the control space Y. Assume that for any given control u ∈ Y there exists a unique mild solution x_u (t) which is given by the nonlinear integral equation:(1.2) $$x_u(t)=\underset{0}{\overset{t}{\int }}S(t-s)Bu(s)\text{d}s+\underset{0}{\overset{t}{\int }}S(t-s)f(s,x_u(s))\text{d}s$$(1.2)

When f ≡ 0 the system (1.1) is called the corresponding linear system denoted by (*). Let F be the Nemytskii operator from Z into itself defined as [Fx_u](t) = f(t, x_u(t)).

Definition 1.1

The set K_T(F) = {x_u(T) ∈ V:x_u( ⋅ ) ∈ Z is a mild solution of (1.1) for u ∈ Y} is called the Reachable set of the system (1.1)

Definition 1.2

The system (1.1) is said to be approximately controllable if K_T(F) is dense in V.

K_T(0) denotes the reachable set of the linear system (*). It is clear that (*) is approximately controllable if K_T(0) is dense in V.

Shukla, Sukavanam, and Pandey (2015a, 2015b, 2016) studies the controllability of semilinear system with delay in state and control using fixed point theorems in 2015–2016. Sukavanam and Tafesse (2011) obtained some sufficient conditions for approximate controllability of semilinear delay system using fixed point approaches in 2011. The controllability theory for abstract linear control systems is almost complete. The necessary and sufficient conditions for various types of controllability for abstract linear equations have been considered in George (1995). One of the principal results on approximate controllability is that the linear control system (*) is approximate controllable on [0.T] if and only if “[S(t)B]*h = 0” for 0 ≤ t ≤ T implies h = 0 in V, where * denotes the adjoint of an operator (see Balakrishnan, 1976; Fattorini, 1967). Fattorini (1967) has given exact controllability results for linear parabolic equations. Triggiani (1975), Russell (1978) and Lions (1988) have proved some exact and approximate controllability results for linear control systems. Recently authors Russell (1978), Sakthivel, Ganesh, and Anthoni (2013), Seidman and Zhou (1982), Shukla, Arora, and Sukavanam (2015), Shukla, Sukavanam, Pandey, and Arora (2016) established some sufficient conditions for controllability of semilinear systems of integer and fractional order systems using Fixed Point theorems. But, in the case of an abstract semilinear control systems, the controllability has been shown under various complex and restricted conditions in literature. Approximate controllability of abstract semilinear systems of the form (1.1) has been studied by Zhou (1983, 1984). Naito (1987) has given simpler conditions for approximate controllability of the system of the form (1.1). The conditions of Naito (1987) are as follows:

(a)	For every p ∈ Z there exists a $q\in \overline{R(B)}$ such that Lp = Lq where L is the operator defined as in (2.2) and $\overline{R(B)}$ denotes the closure of R(B).
(b)	A generates a compact semigroup.
(c)	The operator f(t, x) satisfies Lipschitz continuity in x, i.e. ||f(t, x) − f(t, y)||V ≤ l||x − y||V for some constant l > 0.
(d)	f is uniformly bounded on V, i.e. || f (t, x)||V ≤ k, a constant. In Mahmudov, Vijayakumar, Ravichandran, and Murugesu (2015), Mahmudov et al. (2016) the uniform boundedness condition (d) was replaced by the growth condition for F, namely,
(e)	||Fx||Z ≤ a||x||Z + b, where a ≥ 0 and b is any nonnegative constant such that MTa(1 + c) < 1; the constants T and c being system constants and M is such that ||S(t)|| ≤ M for t ∈ [0, T]. In Naito (1987), the approximate and exact controllability of the restricted system (1.1) with B = I, the identity operator is proved for any nonlinear function f satisfying condition (c) of which the heat equation considered by Liu and Williams (1997), Klamka (2000) is a particular case. In George (1995), approximate controllability of semilinear system (1.1) with A = A(t) has been proved under certain conditions which are as follows:
(f)	f(t, x) satisfies the monotone condition given by$${(f(t,x)-f(t,y),x-y)}_X\le -\mathit{\beta }{\Vert x-y\Vert }_X^2\forall x,y\in X,t\in [0,T].$$ and the assumptions (a), (b), (e) hold with
(g)	${(-A(t)x,x)}_V\ge 0\forall t\in [0,T].$
(h)	The semigroup S(t, s) generated by A(t) satisfies $\Vert S(t,s\Vert \le M$ for all t, s ∈ [0, T], for some constant M > 0.
(i)	$(1-aM\Vert P\Vert (T)e^{MTa})$ > 0 where P is defined in Section 2. In this paper, we will prove the approximate controllability of the semilinear control system (1.1) for an extended class of nonlinear functions f(t, x) satisfying the monotonicity condition. Thus, it is proved that we no more require the complex inequalities such as (i) to prove the approximate controllability of such systems.

2. Preliminaries

Notations

For a given operator A, D(A), R(A) and N₀(A) denote the domain, range and null space of A, respectively. $\overline{E}$ and E^⊥, respectively, are the closure and orthogonal complement of a set E. Let (.,.)_V and (.,.)_Z, respectively, denote inner products in V and Z and ${(f,g)}_Z=\underset{o}{\overset{T}{\int }}{(f(t),g(t))}_{V}dt$. The norms defined through these inner products are denoted by ||·||_V and ||·||_Z.

Let K be an operator from Z into itself defined as:(2.1) $$[Kz](t)=\underset{0}{\overset{t}{\int }}S(t-s)z(s)\text{d}s$$(2.1)

and L and N be operators from Z into V defined as:(2.2) $$Lz=\underset{0}{\overset{T}{\int }}S(T-s)z(s)\text{d}s$$(2.2) (2.3) $$Nz=\underset{0}{\overset{T}{\int }}S(T-s)[Fz](s)\text{d}s$$(2.3)

It is evident that hypothesis (a) is equivalent to the condition $Z=N_0(L)+\overline{R(B)}$. Moreover, Z can be decomposed as $Z=N_0(L)+N_0^{\perp }(L)$. Also, under hypothesis (a) we can define a map $P:N_0^{\perp }(L)\to \overline{R(B)}$ as follows:

Let $u\in N_0^{\perp }(L)$, P(u) = u₀ where u₀ is the unique minimum norm element in the set $\{u+N_0(L)\}\cap \overline{R(B)}$ satisfying ||Pu|| = ||u_o|| = min $\{\Vert v\Vert :v\in \{u+N_0(L)\}\cap \overline{R(B)}\}$. The operator P is well-defined, linear and continuous (see Mahmudov et al., 2016, Lemma 1). From continuity of P it follows that ||Pu||_Z ≤ c||u||_Z, for some constant c ≥ 0.

3. Controllability results

In this section, we will discuss the approximate controllability of the deterministic semilinear control system (1.1) under the following two cases:

(i)	When B = I, the identity operator, and
(ii)	When B ≠ I in (1.1).

Let M₀ be the subspace of Z such that M₀ = {m ∈ Z:m = K(n), n ∈ N₀(L)}. From (2.1) and (2.2) it is clear that if m ∈ M₀ then m(T) = 0.

Before proving the main theorems, we prove some lemmas.

Lemma 1

Any element z ∈ Z can be uniquely decomposed as z = n + q:n ∈ N₀(L), q ∈ $\overline{R(B)}$ and ||n|| ≤ (1 + c)||z||.

Proof

Let u ∈ $N_0^{\perp }(L)$ then Pu = (n₀ + u) ∈ $\overline{R(B)}$ for some n₀ ∈ N₀(L). Now if z ∈ Z has unique decomposition, namely, z = n₁ + u:n₁ ∈ N₀(L) u ∈ $N_0^{\perp }(L)$, then z can be uniquely decomposed as

z = n + q:n ∈ N₀(L); q ∈ $\overline{R(B)}$ where q = Pu and n = n₁ − n₀.

Now, z = n₁ + u ⇒ ||z||² = ||n₁||² + ||u||²(3.1) $$\text{or}\Vert u\Vert \le \Vert z\Vert .$$(3.1)

Now n = z – q ⇒ ||n|| = ||z − Pu||(3.2) $$\text{Hence}\Vert n\Vert \le \Vert z\Vert +c\Vert u\Vert \le (1+c)\Vert z\Vert (\text{from}(3.1))$$(3.2)

Let A:V → V satisfies the following condition

(j)	${(-Ax,x)}_V\ge \mathit{\mu }{\Vert x\Vert }_V^2:\mathit{\mu }>0.$

Lemma 2

Under the condition (j) the operator K:Z → Z satisfies the condition(3.3) $${(Kx,x)}_Z\ge \mathit{\mu }{\Vert Kx\Vert }_Z^2\text{for; all}x\in X.$$(3.3)

Proof

Define $f(t)=\underset{0}{\overset{t}{\int }}S(t-s)x(s)\text{d}s$

Since S(t) is a strongly continuous semigroup, we have f(t) ∈ D(A) and$$f^{\prime }(t)=x(t)+A(t)\underset{0}{\overset{t}{\int }}S(t-s)x(s)\text{d}s$$(3.4) $${(Kx,x)}_Z=\underset{0}{\overset{T}{\int }}(f(t),f^{\prime }(t)-A(t)\underset{0}{\overset{t}{\int }}{S(t-s)x(s))}_{X}ds=\underset{0}{\overset{T}{\int }}{(f(t),f^{\prime }(t))}_X\text{d}t+\underset{0}{\overset{T}{\int }}{(f(t),-A(t)f(t))}_X\text{d}t$$(3.4) (3.5) $$\text{Also},\underset{0}{\overset{T}{\int }}{(f(t),f^{\prime }(t))}_X\text{d}t=1/2{\Vert f(T)\Vert }_X^2\ge 0$$(3.5) (3.6) $$\text{and}\underset{0}{\overset{T}{\int }}{(f(t),-A(t)f(t))}_X\text{d}t\ge \mathit{\mu }{\Vert f(t)\Vert }_X^2$$(3.6)

From (3.4), (3.5) and (3.6) we get,

${(Kx,x)}_Z\ge \mathit{\mu }{\Vert Kx\Vert }_Z^2$ for all x ∈ Z.

Now the main results of this section follows.

Let $\stackrel{\mathrm{\Lambda }}{V}$ = V and B = I, the identity operator on V, in (1.1) then the resulting system can be written as$$\frac{\text{d}{y}_v(t)}{\text{d}t}=A{y}_v(t)+v(t)+f(t,y_v(t));$$(3.7) $$y_v(0)=0$$(3.7)

and the corresponding linear system can be written as$$\frac{\text{d}{x}_u(t)}{\text{d}t}=A{x}_u(t)+u(t);$$(3.8) $$x_u(0)=0$$(3.8)

Theorem 1

The semilinear control system (3.7) is approximately controllable under the assumptions

(i)	The linear system (3.8) is approximate controllable.
(ii)	The semigroup S(t, s) is compact.
(iii)	$\Vert S(t)\Vert \le M,$ a constant.
(iv)	f(t, x) satisfies the monotone condition (f).

Proof

Let x_u(t) be a mild solution of (3.8) corresponding to a control u and consider the following system$$\frac{\text{d}{y}_v(t)}{\text{d}t}=A{y}_v(t)+f(y_v(t))+u(t)-f(x_u(t));$$(3.9) $$y_v(0)=0.$$(3.9)

The mild solution of (3.8) and (3.9), respectively, can be written as:(3.10) $$x_u(t)=\underset{0}{\overset{t}{\int }}S(t-s)u(s)\text{d}s$$(3.10) (3.11) $$y_v(t)=\underset{0}{\overset{t}{\int }}S(t-s)f(y_v(s))\text{d}s+\underset{0}{\overset{t}{\int }}S(t-s)u(s)\text{d}s-\underset{0}{\overset{t}{\int }}S(t-s)f(x_u(s))\text{d}s$$(3.11)

Subtracting (3.11) from (3.10), we get,(3.12) $$x_u(t)-y_v(t)=\underset{0}{\overset{t}{\int }}S(t-s)\{f(x_u(s))-f(y_v(s))\}\text{d}s$$(3.12)

or x_u − y_v = KNx_u − KNy_v

Taking inner product on both sides with Nx_u − Ny_v, we get,$${(x_u-y_v,N{x}_u-N{y}_v)}_Z={(KN{x}_u-KN{y}_v,N{x}_u-N{y}_v)}_Z={(K(N{x}_u-N{y}_v),N{x}_u-N{y}_v)}_Z$$

Now the LHS satisfies condition (iv) and therefore it is less than or equal to −β||x_u − y_v||² and from Lemma 2 RHS is nonnegative. This is possible only when x_u = y_v in Z which implies that F(x_u) = F(y_v) where F is the Nemytskii operator defined by f. This in turn implies from (3.12) that x_u(t) = y_v(t) for all t ∈ [0, T]. Therefore, the set of all solutions of the linear system is equal to the set of all solutions of the semilinear systems i.e. R_T(F) ⊃ R_T(0) which is dense in V and this implies the approximate controllability of the system (3.7).

Remark

When B = I, the linear system (3.8) is always approximately controllable because for any given initial value x_u(0) and the final value x_u(T) of the state variable x_u(t) we can always find a control u ∈ Y given by u(t) = [x_u(T) + tAx_u(T)]/T. Hence, in this case the system (3.7) is approximately controllable only under the conditions (ii), (iii) and (iv).

In George (1995) the approximate controllability is proved under the conditions given in Section 1. Now we no more require the inequality condition (i) to prove the approximate controllability of the semilinear control systems.

Theorem 2

The semilinear system (1.1) is approximately controllable under the assumptions (a), (b), (f) and (k) $R(F)\subseteq \overline{R(B)}$

Proof

Let y_v(t) be a mild solution of the linear system (*) corresponding to a control v which can be written as $y_v(t)={\int }_0^{t}S(t-s)Bv(s)\text{d}s$. Then, it follows that the system (1.1) will have a unique mild solution for a given control u as f(t, x) satisfies the monotone condition (f).

Let y_v(t) be a mild solution of (1.1) with control v − w. Then $y_v=K(v)$. Let m = Kn where $n\in \overline{R(B)}$ ⋂ N₀(L). Hence $z_w(t)=y_v(t)+m=K(v+n)$ is also a solution of (1.1) with control v + n.

If Fx_u ∈ R(B) then Fx_u = Bw₁(t) for some w₁ ∈ Y, since $R(F)\subseteq \overline{R(B)}$ (given). Also, for a given ε > 0 there exists a w₁ in Y such that ||F(x_u) − Bw₁(t)||_Z ≤ ɛ.

Now let z_w(t) be a mild solution of (1.1) corresponding to a control v + n. Then(3.13) $$y_v(t)-z_w(t)=\underset{0}{\overset{t}{\int }}S(t-s)Bw(s)\text{d}s-\underset{0}{\overset{t}{\int }}S(t-s)[F{y}_v](s)\text{d}s=\underset{0}{\overset{t}{\int }}S(t-s)[Bw-F{x}_u](s)\text{d}s+\underset{0}{\overset{t}{\int }}S(t-s)[F{x}_u-F{y}_v](s)\text{d}s$$(3.13)

From (3.13) we get,$$x_u(T)-y_v(T)=L[Bw-F{x}_u]+L[F{x}_u-F{y}_v]$$$$\Vert x_u(T)-y_v(T)\Vert \le M\Vert Bw-F{x}_u\Vert +M\Vert F{x}_u-F{y}_v\Vert \le \mathit{\epsilon }$$

Thus, K_T(F) is dense in K_T(0) which is dense in V. Hence the result.

Theorem 3

The semilinear system (1.1) is approximately controllable under the assumptions (a), (b), (f) and

(l)	f(t, x) = a(t) + g(t, x) where $a(·)$ ∈ Z, g(t, x) satisfies the monotone condition (f) and R(g) ⊆ R(B) in V.

Proof

Consider the dynamical system $\frac{\text{d}{m}_n(t)}{\text{d}t}=A{m}_n(t)+n(t):m_n(0)=0.$ It has the unique mild solution m = Kn ∈ M₀. Also since a ∈ Z there exists a unique q ∈ $\overline{R(B)}$ and n ∈ N₀(L) as in (Lemma 1) such that(3.14) $$a=q+n$$(3.14)

Now q ∈ $\overline{R(B)}$ implies that for any given ε > 0 there exists a w ∈ Y such that ||q − Bw(t)|| < ɛ. Define the function g₁(t, x) = g(t, x + m(t)) + q(t). It can be easily seen that g₁(t, x) satisfies the monotone condition (f). Let G denotes the Nemytskii operator corresponding to g₁ defined as [Gx](t) = g₁(t, x(t)). As R(g) H ⊆ R(B) in V and $q\in \overline{R(B)}$ in Z it can be easily seen that $R(G)\subseteq \overline{R(B)}$ in Z and hence for every x ∈ Z and δ > 0 there exists a w₁ ∈ Y such that ||Gx − Bw₁|| ≤ δ.

Now consider the pair of systems as follows:(3.15) $$\frac{\text{d}{y}_v(t)}{\text{d}t}=A{y}_v(t)+Bv(t)+g_1(t,y_v(t)):y_v(0)=0;$$(3.15) (3.16) $$\frac{\text{d}{m}_n(t)}{\text{d}t}=A{m}_n(t)+n(t):m_n(0)=0.$$(3.16)

Using Theorem 2 it can be shown that the mild solution of y_v(t) of the first equation and the mild solution x_u(t) of (*) can be chosen such that ||y_v(T) − x_u(T)|| ≤ ε₁ for any ε_1 > 0, by choosing δ depending on ε₁ and the control v as v = u − w₁. By adding (3.15) and (3.16) and using (3.14) and (ℓ) it can be shown that y_v(t) + m_n(t) is the mild solution of (1.1). Since m_n(T) = 0 it follows that approximate controllability of (*) implied that of (1.1).

4. Examples

Example 1

Let V = L₂(0, π) and A = −d²/dx² with D(A) consisting of all y ∈ V with d²y/dx² ∈ V and y(0) = 0 = y(π). Put ${\mathit{\phi }}_n(x)={(2/\mathit{\pi })}^{1/2}sin(nx),0\le x\le \mathit{\pi },n=1,2,\dots ,$ then $\{{\mathit{\phi }}_n:n=1,2,\dots \}$ is an orthonormal basis for V and φ_n is an eigenfunction corresponding to the eigenvalue ${\mathit{\lambda }}_n=-n^2$ of the operator −A, n = 1,2, …. Then the C₀-semigroup S(t) generated by −A has $e^{{\mathit{\lambda }}_{n}t}$ as the eigenvalues and φ_n as their corresponding eigenfunctions (Mahmudov et al., 2016). Now define an infinite dimensional space $\stackrel{\mathrm{\Lambda }}{V}$ by$$\stackrel{\mathrm{\Lambda }}{V}=\left\{\sum _{n=2}^{\infty }{u}_n{\mathit{\phi }}_n,\text{with}\sum _{n=2}^{\infty }{u}_n^2<\infty \right\}$$

The norm in $\stackrel{\mathrm{\Lambda }}{V}$ is defined by $\Vert u\Vert ={\left(\sum _{n=2}^{\infty }{u}_n^2\right)}^{1/2}$

Define a continuous linear mapping B from $\stackrel{\mathrm{\Lambda }}{V}$ to V as follows$$Bu=2u_2{\mathit{\phi }}_1+\sum _{n=2}^{\infty }{u}_n{\mathit{\phi }}_n\text{for}u=\sum _{n=2}^{\infty }{u}_n{\mathit{\phi }}_n\in \stackrel{\mathrm{\Lambda }}{V}$$

Consider the control system governed by the semilinear heat equation(4.1) $$\frac{\mathit{\partial }y(t,x)}{\mathit{\partial }t}=\frac{{\mathit{\partial }}^{2}y}{\mathit{\partial }{x}^2}+Bu(t,x)+f(t,y(t,x));0<t<T,0<x<\mathit{\pi };$$(4.1) $$y(t,0)=y(t,\mathit{\pi })=0;0\le t\le T.$$(4.2) $$y(0,x)=0;0\le x\le \mathit{\pi }.$$(4.2)

Now we define the bounded linear function $\tilde{B}$ from $L_2(0,T:\stackrel{\mathrm{\Lambda }}{V})$ to L₂(0, T:V) by $(\tilde{B}u)(t)=Bu(t)$, for $u\in L_2(0,T:\stackrel{\mathrm{\Lambda }}{V}).$ The nonlinear operator f on [0, T] × V is defined as follows$$f(t,y)=z(t)+g(t,y)\text{where}g(t,y)={2b\Vert y\Vert }_V{\mathit{\varphi }}_1+b{\Vert y\Vert }_V{\mathit{\varphi }}_2+a$$

z(⋅) ∈ L₂(0, T:V), a and b are any positive constants. It can be seen that f(t, y) satisfies condition (l) of Theorem 3. In Klamka (2001), Naito has shown the approximate controllability of the heat control system by assuming uniform boundedness of f(t, y) in the space V. In Mahmudov et al. (2015), approximate controllability was shown under some restrictions on the constants b and T. Now, the approximate controllability of the system follows without any conditions on b and T.

Example 2

Let Ω be a bounded open set in Rⁿ with smooth boundary ∂Ω = Γ. Consider the following distributed parameter system(4.3) $$\frac{\mathit{\partial }}{\mathit{\partial }t}x(t,z)-\sum _{k=1}^n\frac{\mathit{\partial }}{\mathit{\partial }{z}_k}\left(p(t,z)\frac{\mathit{\partial }}{\mathit{\partial }{z}_k}x(t,z)\right)=u(t,z)+g(t,z,x(t,z));$$(4.3) $$x(t,z)=0\text{on}[0,T]\times \mathrm{\Gamma },$$$$x(0,z)=x_0(z)\text{for}z\in \mathrm{\Omega }.$$

where p:[0, T] × Ω → R is such that p(t, z) ≥ c > 0, for some constant c, and is Lipschitz w.r.t. the t variable, C¹ in the z variable and $t\to {\Vert p(t,·)\Vert }_{\infty }\in L_{+}^{\infty }$. Assume that g:[0, T] × Ω × R → R is a nonlinear function such that it is measurable in (t, z) and continuous in x and $|g(t,z,x)|\le {\mathit{\alpha }}_0(t,z)+b(z)|x|$ i.e.

where $\mathit{\alpha }(·,·)\in L_2[0,T;\mathrm{\Omega }]$ and b ∈ L^∞[Ω].

Let V = L₂[Ω] and Y = U = L₂[0, T;V]. Let $D=H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega }).$ For each t ∈ [0, T], define A(t):D ⊂ V → V by (A(t)x, v) = −a(t, x, v) for all x, v ∈ D where a(t, x, v) is given by$$a(t,x,v)=\sum _{k=1}^n\underset{\mathrm{\Omega }}{\int }p(t,z)\frac{\mathit{\partial }x}{\mathit{\partial }{z}_k}\frac{\mathit{\partial }v}{\mathit{\partial }{z}_k}\text{d}z.$$

The assumption on $p(·,·)$ imply that $\Vert A(t)x-A(s)x\Vert \le k|t-s|\Vert x\Vert$ for some constant k > 0. By Poincare’s inequality, there exists μ > 0 such that$$(-A(t)x,x)=\underset{\mathrm{\Omega }}{\int }\sum _{k=1}^{n}p(t,z){\left|\frac{\mathit{\partial }x}{\mathit{\partial }{z}_k}\right|}^2\text{d}z\ge \mathit{\mu }{\Vert x\Vert }^2\text{for ; all}x\in D.$$

For x ∈ D, ||A(t)x|| = sup$\{(A(t)x,v):\Vert v{\Vert }_V$ < 1} $\le {\Vert p(t)\Vert }_{C^1(\mathrm{\Omega })}{\Vert x\Vert }_{H^2(\mathrm{\Omega })}$ (by Cauchy and Poincare’s inequalities). Define f:[0, T] × V → V by$$f(t,x)(z)=g(t,z,x(z)).$$

Thus f satisfies condition (e) with $a={\Vert b(z)\Vert }_{L_2[\mathrm{\Omega }]}$. If −g(t, z, x) is monotonically increasing with respect to x it follows that f satisfies

Now denoting $x(t)=x(t,·)\in L^2[\mathrm{\Omega }]$ and $u(t)=u(t,·)\in L^2[\mathrm{\Omega }]$ the system (4.3) takes the form$$\frac{\text{d}{x}_u(t)}{\text{d}t}=A{x}_u(t)+u(t)+f(t,x_u(t))$$$$x(0)=x_0$$

In George (1995), it is shown that the system (4.33) is approximately controllable if a is sufficiently small and the system satisfies the conditions.

In this paper, the approximate controllability of the system (4.3) follows only under the monotone condition on f.

Additional information

Funding

Funding. The authors received no direct funding for this research.

Notes on contributors

Divya Ahluwalia

Divya Ahluwalia received her PhD degree in Mathematics from Indian Institute of Technology, Roorkee (India) in 2004. Currently, she is an associate professor in the University of Petroleum and Energy Studies, Dehradun, India. Her research area includes controllability of deterministic systems.

N. Sukavanam

N. Sukavanam received his BSc (Maths) degree from University of Madras, India in 1977 and MSc (Maths) from the same university in 1979 and PhD (Maths) from IISC Bangalore, India in 1985. At present, he is working as a professor in the Department of Mathematics IIT Roorkee (India). His research interests include nonlinear analysis, control theory and robotics and control.

Anurag Shukla

Anurag Shukla received his MSc (Maths) degree from IIT Roorkee, India in 2011. He received his PHD degree from IIT Roorkee, India in 2016. At present, he is working as an assistant professor in the University of Petroleum and Energy Studies, Dehradun. His research interests are nonlinear analysis, control theory and optimal controls.