Approximate controllability and regularity for nonlinear differential equations

Abstract

In this article, we deal with the existence, uniqueness, and a variation of solutions of the nonlinear control system with nonlinear monotone hemicontinuous and coercive operator. Moreover, the approximate controllability for the given nonlinear control system is studied.

1 Introduction

Let H and V be two real separable Hilbert spaces such that V is a dense subspace of H. We are interested in the following nonlinear differential control system on H:

$$\left\{\begin{array}{c}{x}^{\prime }\left(t\right)+Ax\left(t\right)=g\left(t,x_t,{\int }_0^{t}k\left(t,s,x_s\right)ds\right)+\left(Bu\right)\left(t\right),0<t,\\ x\left(0\right)={\varphi }^0,x\left(s\right)={\varphi }^1\left(s\right)-h\le s\le 0,\end{array}\right.$$

(SE)

where the nonlinear term, which is a Lipschitz continuous operator, is a semilinear version of the quasi-linear form. The principal operator A is assumed to be a single valued, monotone operator, which is hemicontinuous and coercive from V to V*. Here, V* stands for the dual space of V . Let U be a Banach space of control variables. The controller B is a linear-bounded operator from a Banach space L ²(0, T; U) to L ²(0, T; H) for any T > 0. Let the nonlinear mapping k be Lipschitz continuous from ℝ × [- h, 0] × V into H. If the right-hand side of the equation (SE) belongs to L ²(0, T; V* ), then it is well known as the quasi-autonomous differential equation(see Theorem 2.6 of Chapter III in [1]).

The problem of existence for solutions of semilinear evolution equations in Banach spaces has been established by several authors [1–3]. We refer to [2, 4, 5] to see the existence of solutions for a class of nonlinear evolution equations with monotone perturbations

First, we begin with the existence, and a variational constant formula for solutions of the equation (SE) on L ²(0, T; V ) ∩ W ^1,2(0, T; V* ), which is also applicable to optimal control problem. We prove the existence and uniqueness for solution of the equation by converting the problem into a fixed point problem. Thereafter, based on the regularity results for solutions of (SE), we intend to establish the approximate controllability for (SE). The controllability results for linear control systems have been proved by many authors, and several authors have extended these concepts to infinite dimensional semilinear system (see [6–8]). In recent years, as for the controllability for semilinear differential equations, Carrasco and Lebia [9] discussed sufficient conditions for approximate controllability of parabolic equations with delay, and Naito [10] and other authors [6–8, 11] proved the approximate controllability under the range conditions of the controller B.

The previous results on the approximate controllability of a semilinear control system have been proved as a particular case of sufficient conditions for the approximate solvability of semilinear equations, assuming either that the semigroup generated by A is a compact operator or that the corresponding linear system (SE) when g ≡ 0 is approximately controllable. However, Triggian [12] proved that the abstract linear system is never exactly controllable in an infinite dimensional space when the semigroup generated by A is compact. Thus, we will establish the approximate controllability under more general conditions on the nonlinear term and the controller.

Our aim in this article is to establish the approximate controllability for (SE) under a stronger assumption that {y : y(t) = (Bu)(t), u ∈ L ²(0, T; U)} is dense subspace of L ²(0, T, H), which is reasonable and widely used in case of the nonlinear system. We show that the input to solution (control to state) map is compact by using the fact that L ²(0, T; V ) ∩ W ^1,2(0, T; V* ) furnished with the usual topology is compactly embedded in L ²(0, T, H), provided that the injection V ⊂ H is compact.

In the last section, we give a simple example to which the range conditions of the controller can be applied.

2 Nonlinear functional equations

Let H and V be two real Hilbert spaces. Assume and V is dense subspace in H and the injection of V into H is continuous. If H is identified with its dual space, then we may write V ⊂ H ⊂ V* densely, and the corresponding injections are continuous. The norm on V (resp. H) will be denoted by || · || (resp. |· |). The duality pairing between the element v ₁ of V* and the element v ₂ of V is denoted by (v ₁ , v ₂), which is the ordinary inner product in H if v ₁ , v ₂ ∈ H. For the sake of simplicity, we may consider

$$\parallel u{\parallel }_{*}\le \mid u\mid \le \parallel u\parallel ,u\in V$$

where || · ||_* is the norm of the element of V*. If an operator A is bounded linear from V to V* and generates an analytic semigroup, then it is easily seen that

$$H=\left\{x\in V^{*}:{\int }_0^T\parallel A{e}^{tA}x{\parallel }_{*}^{2}dt<\infty \right\},$$

for the time T > 0. Therefore, in terms of the intermediate theory we can see that

$${{\left(V,V^{*}\right)}_{\frac{1}{2}}}_{,2}=H$$

where ${{\left(V,V^{*}\right)}_{\frac{1}{2}}}_{,2}$ denotes the real interpolation space between V and V*.

We note that a nonlinear operator A is said to be hemicontinuous on V if

$$w-\underset{t\to 0}{lim}A\left(x+ty\right)=Ax$$

for every x, y ∈ V where "w - lim" indicates the weak convergence on V*. Let A : V → V* be given a single-valued, monotone operator and hemicontinuous from V to V* such that

(A1) A(0) = 0, (Au - Av, u - v) ≥ ω ₁ ||u - v||² - ω ₂ |u - v| ²,

(A2) ||Au||_* ≤ ω ₃(||u|| + 1)

for every u, v ∈ V where ω ₂ ∈ ℝ and ω ₁ , ω ₃ are some positive constants.

Here, we note that if 0 ≠ A(0), then we need the following assumption:

$$\left(Au,u\right)\ge {\omega }_1\parallel u{\parallel }^2-{\omega }_2\mid u{\mid }^2$$

for every u ∈ V . It is also known that A is maximal monotone, and R(A) = V* where R(A) denotes the range of A.

Let the controller B is a bounded linear operator from a Banach space L ²(0, T; U) to L ²(0, T; H) where U is a Banach space.

For each t ∈ [0, T], we define x _t : [ -h, 0] → H as

$$x_t\left(s\right)=x\left(t+s\right),-h\le s\le 0.$$

We will set

$$\Pi =L^2\left(-h,0;V\right)\mathsf{\text{and}}{ℝ}^{+}=\left[0,\infty \right).$$

Let ℒ and ℬ be the Lebesgue σ-field on [0, ∞) and the Borel σ-field on [- h, 0] respectively. Let k : ℝ⁺ × ℝ⁺ × ∏ → H be a nonlinear mapping satisfying the following:

(K1) For any x. ∈ ∏ the mapping k(·, ·, x.) is strongly ℒ × ℬ -measurable;

(K2) There exist positive constants K ₀, and K ₁ such that

$$\begin{array}{c}\mid k\left(t,s,x.\right)-k\left(t,s,y.\right)\mid \le K_1\parallel x.-y.\parallel {}_{\Pi },\\ \mid k\left(t,s,0\right)\mid \le K_0\end{array}$$

for all (t, s) ∈ ℝ⁺ × [ -h, 0] and x., y. ∈ ∏.

Let g : ℝ⁺ × ∏ × H → H be a nonlinear mapping satisfying the following:

(G1) For any x ∈ ∏, y ∈ H the mapping g(·, x., y) is strongly ℒ -measurable;

(G2) There exist positive constants L ₀ , L ₁, and L ₂ such that

$$\begin{array}{c}\mid g\left(t,x.,y\right)-g\left(t,\widehat{x}.,ŷ\right)\mid \le L_1\parallel x.-\widehat{x}.{\parallel }_{\Pi }+L_2\mid y-ŷ\mid ,\\ \mid g\left(t,0,0\right)\mid \le L_0\end{array}$$

for all t ∈ ℝ⁺, x, $\widehat{x}.\in \Pi$, and y, $ŷ\in H$.

Remark 2.1. The above operator g is the semilinear case of the nonlinear part of quasi-linear equations considered by Yong and Pan[13].

For x ∈ L ²(-h, T; V ), T > 0 we set

$$G\left(t,x\right)=g\left(t,x_t,{\int }_0^{t}k\left(t,s,x_s\right)ds\right).$$

Here, as in[13], we consider the Borel measurable corrections of x(·).

Lemma 2.1. Let x ∈ L ²(- h, T; V ). Then, the mapping t ↦ x _t belongs to C([0, T ]; ∏) and

$$\parallel x.{\parallel }_{L^2\left(0.T;\Pi \right)}\le \sqrt{T}\parallel x{\parallel }_{L^2\left(-h,T;V\right)}.$$

(2.1)

Proof. It is easy to verify the first paragraph and (2.1) is a consequence of the estimate:

$$\begin{array}{cc}\hfill \parallel x.{\parallel }_{L^2\left(0.T;\Pi \right)}^2& \le {\int }_0^T\parallel x_t{\parallel }_{\Pi }^{2}dt\le {\int }_0^T{\int }_{-h}^0\parallel x\left(t+s\right){\parallel }^{2}dsdt\hfill \\ \le {\int }_0^{T}dt{\int }_{-h}^T\parallel x\left(s\right){\parallel }^{2}ds\le T\parallel x{\parallel }_{L^2\left(-h,T;V\right)}^2.\hfill \end{array}$$

Lemma 2.2. Let x ∈ L ²(- h, T; V ), T > 0. Then, G(·, x) ∈ L ²(0, T; H) and

$$\begin{array}{c}\parallel G\left(\cdot ,x\right){\parallel }_{L^2\left(0,T;H\right)}\le L_0\sqrt{T}+L_2{K}_0{T}^{3∕2}∕\sqrt{3}\\ +\left(L_1\sqrt{T}+L_2{K}_1{T}^{3∕2}∕\sqrt{2}\right)\parallel x{\parallel }_{L^2\left(-h,T;V\right)}.\end{array}$$

(2.2)

Moreover, if x ₁ , x ₂ ∈ L ²(- h, T; V ), then

$$\parallel G\left(\cdot ,x_1\right)-G\left(\cdot ,x_2\right){\parallel }_{L^2\left(0,T;H\right)}\le \left(L_1\sqrt{T}+L_2{K}_1{T}^{3∕2}∕\sqrt{2}\right)\parallel x_1-x_2{\parallel }_{L^2\left(-h,T;V\right)}.$$

(2.3)

Proof. It follows from (K2) and (2.1) that

$$\begin{array}{c}\parallel {\int }_0^{\cdot }k\left(\cdot ,s,x_s\right)ds{\parallel }_{L^2\left(0,T;H\right)}\le \parallel {\int }_0^{\cdot }k\left(\cdot ,s,0\right)ds{\parallel }_{L^2\left(0,T;H\right)}\\ +\parallel {\int }_0^{\cdot }\left(k\left(\cdot ,s,x_s\right)-k\left(\cdot ,s,0\right)\right)ds{\parallel }_{L^2\left(0,T;H\right)}\\ \le K_0{T}^{3∕2}∕\sqrt{3}+{\left\{{\int }_0^T\mid {\int }_0^t{K}_1\parallel x_s{\parallel }_{\Pi }ds{\mid }^{2}dt\right\}}^{1∕2}\\ \le K_0{T}^{3∕2}∕\sqrt{3}+{\left\{{\int }_0^T{K}_1^{2}t{\int }_0^t\parallel x_s{\parallel }_{\Pi }^{2}dsdt\right\}}^{1∕2}\\ \le K_0{T}^{3∕2}∕\sqrt{3}+K_{1}T∕\sqrt{2}\parallel x_{\cdot }{\parallel }_{L^2\left(0,T;\Pi \right)}\\ \le K_0{T}^{3∕2}∕\sqrt{3}+K_1{T}^{3∕2}∕\sqrt{2}\parallel x{\parallel }_{L^2\left(-h,T;V\right)}\end{array}$$

and hence, from (G2), (2.1), and the above inequality, it is easily seen that

$$\begin{array}{c}\parallel G\left(\cdot ,x\right){\parallel }_{L^2\left(0,T;H\right)}\le \parallel G\left(\cdot ,0\right)\parallel +\parallel G\left(\cdot ,x\right)-G\left(\cdot ,0\right)\parallel \\ \le L_0\sqrt{T}+L_1\parallel x_{\cdot }{\parallel }_{L^2\left(0,T;\Pi \right)}+L_2\parallel {\int }_0^{\cdot }k\left(\cdot ,s,x_s\right)ds{\parallel }_{L^2\left(0,T;H\right)}\\ \le L_0\sqrt{T}+L_1\sqrt{T}\parallel x{\parallel }_{L^2\left(-h,T;V\right)}\\ +L_2\left(K_0{T}^{3∕2}∕\sqrt{3}+K_1{T}^{3∕2}∕\sqrt{2}\parallel x{\parallel }_{L^2\left(-h,T;V\right)}\right).\end{array}$$

Similarly, we can prove (2.3).

Let us consider the quasi-autonomous differential equation

$$\left\{\begin{array}{c}{x}^{\prime }\left(t\right)+Ax\left(t\right)=f\left(t\right),0<t\le T,\\ x\left(0\right)={\varphi }^0\end{array}\right.$$

(E)

where A satisfies the hypotheses mentioned above. The following result is from Theorem 2.6 of Chapter III in [1].

Proposition 2.1. Let Φ ⁰ ∈ H and f ∈ L ²(0, T; V* ). Then, there exists a unique solution x of (E) belonging to

$$C\left(\left[0,T\right];H\right)\cap L^2\left(0,T;V\right)\cap W^{1,2}\left(0,T;V^{*}\right)$$

and satisfying

$$\mid x\left(t\right){\mid }^2+{\int }_0^t\parallel x\left(s\right){\parallel }^{2}ds\le C_1\left(\mid {\varphi }^0{\mid }^2+{\int }_0^t\parallel f\left(s\right){\parallel }_{*}^{2}ds\right),$$

(2.4)

$${\int }_0^t\parallel \frac{dx\left(s\right)}{ds}{\parallel }_{*}^{2}dt\le C_1\left(\mid {\varphi }^0{\mid }^2+{\int }_0^t\parallel f\left(s\right){\parallel }_{*}^{2}ds\right)$$

(2.5)

where C ₁ is a constant.

Acting on both sides of (E) by x(t), we have

$$\frac{1}{2}\frac{d}{dt}\mid x\left(t\right){\mid }^2+{\omega }_1\parallel x\left(t\right){\parallel }^2\le {\omega }_2\mid x\left(t\right){\mid }^2+\left(f\left(t\right),x\left(t\right)\right).$$

As is seen Theorem 2.6 in [1], integrating from 0 to t, we can determine the constant C ₁ in Proposition 2.1.

We establish the following result on the solvability of the equation (SE).

Theorem 2.1. Let A and the nonlinear mapping g be given satisfying the assumptions mentioned above. Then, for any (Φ ⁰ , Φ ¹) ∈ H × L ²(- h, 0; V ) and f ∈ L ²(0, T; V*), T > 0, the following nonlinear equation

$$\left\{\begin{array}{c}{x}^{\prime }\left(t\right)+Ax\left(t\right)=G\left(t,x\right)+f\left(t\right),0<t\le T,\\ x\left(0\right)={\varphi }^0,x\left(s\right)={\varphi }^1\left(s\right),-h\le s\le 0\end{array}\right.$$

(2.6)

has a unique solution x belonging to

$$L^2\left(-h,T;V\right)\cap W^{1,2}\left(0,T;V^{*}\right)\subset C\left(\left[0,T\right];H\right)$$

and satisfying that there exists a constant C ₂ such that

$$\parallel x{\parallel }_{L^2\left(-h,T;V\right)\cap W^{1,2}\left(0,T;V*\right)}\le C_2\left(1+\mid {\varphi }^0\mid +\parallel {\varphi }^1{\parallel }_{L^2\left(-h,0;V\right)}+\parallel f{\parallel }_{L^2\left(0,T;V*\right)}\right).$$

(2.7)

Proof. Let y ∈ L ²(0, T; V ). Then, we extend it to the interval (-h, 0) by setting y(s) = Φ ¹(s) for s ∈ (-h, 0), and hence, G(·, y(·)) ∈ L ²(0, T; H) from Lemma 2.2. Thus, by virtue of Proposition 2.1, we know that the problem

$$\left\{\begin{array}{c}{x}^{\prime }\left(t\right)+Ax\left(t\right)=G\left(t,y\right)+f\left(t\right),0<t,\\ x\left(0\right)={\varphi }^0,x\left(s\right)={\varphi }^1\left(s\right)-h\le s\le 0\end{array}\right.$$

(2.8)

has a unique solution x _y ∈ L ²(0, T; V ) ∩ W ^1,2(0, T; V* ) corresponding to y. Let us fix T ₀ > 0 so that

$${\omega }_1^{-1}{e}^{{\omega }_2{T}_0}\left(L_1{\sqrt{T}}_0+L_2{K}_1{T}_0^{3∕2}∕\sqrt{2}\right)<1.$$

(2.9)

Let x _i , i = 1, 2, be the solution of (2.8) corresponding to y _i . Multiplying by x ₁(t) - x ₂(t), we have that

$$\begin{array}{c}\left({ẋ}_1\left(t\right)-{ẋ}_2\left(t\right),x_1\left(t\right)-x_2\left(t\right)\right)+\left(A{x}_1\left(t\right)-A{x}_2\left(t\right),x_1\left(t\right)-x_2\left(t\right)\right)\\ =\left(G\left(t,y_1\right)-G\left(t,y_2\right),x_1\left(t\right)-x_2\left(t\right)\right),\end{array}$$

and hence it follows that

$$\begin{array}{l}\frac{1}{2}\frac{d}{dt}\mid x_1(t)-x_2(t){\mid }^2+{\omega }_1\parallel x_1(t)-x_2(t){\parallel }^2\\ \le {\omega }_2\mid x_1(t)-x_2(t){\mid }^2+\parallel G(t,y_1))-G(t,y_2){\parallel }_{*}\parallel x_1(t)-x_2(t)\parallel .\end{array}$$

From Lemma 2.2 and integrating over [0,t], it follows

$$\begin{array}{c}\frac{1}{2}\mid x_1\left(t\right)-x_2\left(t\right){\mid }^2+{\omega }_1{\int }_0^t\parallel x_1\left(s\right)-x_2\left(s\right){\parallel }^{2}ds\\ \le \frac{1}{2c}{\int }_0^t\parallel G\left(s,y_1\right)-G\left(s,y_2\right){\parallel }_{*}^{2}ds\\ +\frac{c}{2}{\int }_0^t\parallel x_1\left(s\right)-x_2\left(s\right){\parallel }^{2}ds+{\omega }_2{\int }_0^t\mid x_1\left(s\right)ds-x_2\left(s\right)ds{\mid }^{2}ds,\end{array}$$

where c is a positive constant satisfying 2ω ₁ - c > 0. Here, we used that

$$ab\le \frac{a^p}{p}+\frac{b^q}{q},p^{-1}+q^{-1}=1\left(1<p<\infty \right)$$

for any pair of nonnegative numbers a and b. Thus, from (2.3) it follows that

$$\begin{array}{c}\mid x_1\left(t\right)-x_2\left(t\right){\mid }^2+\left(2{\omega }_1-c\right){\int }_0^t\parallel x_1\left(s\right)ds-x_2\left(s\right)ds{\parallel }^{2}ds\\ \le c^{-1}{\left(L_1{\sqrt{T}}_0+L_2{K}_1{T}_0^{3∕2}∕\sqrt{2}\right)}^2{\int }_0^t\parallel y_1\left(s\right)-y_2\left(s\right){\parallel }^{2}ds\\ +2{\omega }_2{\int }_0^t\mid x_1\left(s\right)-x_2\left(s\right){\mid }^{2}ds.\end{array}$$

By using Gronwall's inequality, we get

$$\begin{array}{c}\mid x_1\left(T_0\right)-x_2\left(T_0\right){\mid }^2+\left(2{\omega }_1-c\right){\int }_0^{T_0}\parallel x_1\left(s\right)-x_2\left(s\right){\parallel }^{2}ds\\ \le c^{-1}{\left(L_1{\sqrt{T}}_0+L_2{K}_1{T}_0^{3∕2}∕\sqrt{2}\right)}^2{e}^{2{\omega }_2{T}_0}{\int }_0^{T_0}\parallel y_1\left(s\right)-y_2\left(s\right){\parallel }^{2}ds.\end{array}$$

Taking c = ω ₁, it holds that

$$\begin{array}{l}\parallel x_1-x_2{\parallel }_{L^2(0,T_0;V)}\le {\omega }_1^{-1}{e}^{{\omega }_2{T}_0}(L_1{\sqrt{T}}_0\\ +L_2{K}_1{T}_0^{3/2}/\sqrt{2})\parallel y_1-y_2{\parallel }_{L^2(0,T_0;V)}.\end{array}$$

Hence, we have proved that y ↦ x is a strictly contraction from L ²(0, T ₀; V ) to itself if the condition (2.9) is satisfied. It shows that the equation (2.6) has a unique solution in [0, T ₀].

From now on, we derive the norm estimates of solution of the equation (2.6). Let y be the solution of

$$\left\{\begin{array}{c}{y}^{\prime }\left(t\right)+Ay\left(t\right)=f\left(t\right),0<t\le T_0,\\ y\left(0\right)={\varphi }^0.\end{array}\right.$$

(2.10)

Then,

$$\frac{d}{dt}\left(x\left(t\right)-y\left(t\right)\right)+\left(Ax\left(t\right)-Ay\left(t\right)\right)=G\left(t,x\right),$$

by multiplying by x(t) - y(t) and using the assumption (A1), we obtain

$$\begin{array}{c}\frac{1}{2}\frac{d}{dt}\mid x\left(t\right)-y\left(t\right){\mid }^2+{\omega }_1\parallel x\left(t\right)-y\left(t\right){\parallel }^2\\ \le {\omega }_2\mid x\left(t\right)-y\left(t\right){\mid }^2+\parallel G\left(t,x\right){\parallel }_{*}\parallel x\left(t\right)-y\left(t\right)\parallel .\end{array}$$

By integrating over [0, t] and using Gronwall's inequality, we have

$$\begin{array}{l}\parallel x-y{\parallel }_{L^2(0,T_0;V)}\le {\omega }_1^{-1}{e}^{{\omega }_2{T}_0}\parallel G(\eta ,x){\parallel }_{L^2(0,T_0;V^{*})}\\ \le {\omega }_1^{-1}{e}^{{\omega }_2{T}_0}\{L_0{\sqrt{T}}_0+L_2{K}_0{T}_0^{3/2}/\sqrt{3}\\ +(L_1{\sqrt{T}}_0+L_2{K}_1{T}_0^{3/2}/\sqrt{2})(\parallel x{\parallel }_{L^2(0,T_0;V)}+\parallel {\varphi }^1{\parallel }_{L^2(-h,0;V})\},\end{array}$$

and hence, putting

$$N={\omega }_1^{-1}{e}^{{\omega }_2{T}_0}\mathsf{\text{and}}L=L_1{\sqrt{T}}_0+L_2{K}_1{T}_0^{3∕2}∕\sqrt{2},$$

it holds

$$\begin{array}{l}\parallel x{\parallel }_{L^2(0,T_0;V)}\le \frac{N}{1-NL}(L_0{\sqrt{T}}_0+L_2{K}_0{T}_0^{3/2}/\sqrt{3})\\ +\frac{1}{1-NL}\parallel y{\parallel }_{L^2(0,T_0;V)}+\frac{NL}{1-NL}\parallel {\varphi }^1{\parallel }_{L^2(-h,0;V})\\ \le \frac{N}{1-NL}(L_0{\sqrt{T}}_0+L_2{K}_0{T}_0^{3/2}/\sqrt{3})\\ +\frac{C_1}{1-NL}(\mid {\varphi }^0\mid +\parallel f{\parallel }_{L^2(0,T_0;V^{*})})\\ +\frac{NL}{1-NL}\parallel {\varphi }^1{\parallel }_{L^2(-h,0;V)}\\ \le C_2(1+\mid {\varphi }^0\mid +\parallel {\varphi }^1{\parallel }_{L^2(-h,0;V)}+\parallel f{\parallel }_{L^2(0,T_0;V^{*})})\end{array}$$

(2.11)

for some positive constant C ₂. Since the condition (2.9) is independent of initial values, the solution of (2.6) can be extended to the internal [0, nT ₀] for natural number n, i.e., for the initial value (x(nT ₀), x _n T ₀ ) in the interval [nT ₀, (n + 1)T ₀], as analogous estimate (2.11) holds for the solution in [0, (n + 1)T ₀].

Theorem 2.2. If (Φ ⁰, Φ ¹) ∈ H × L ²(-h, 0, V )) and f ∈ L ²(0, T; V* ), then x ∈ L ²(-h, T; V ))∩ W ^1,2(0, T; V* ), and the mapping

$$H\times L^2(-h,0;V)\times L^2(0,T;V^{*})\ni ({\varphi }^0,{\varphi }^1,f)\mapsto x\in L^2(-h,T;V))\cap W^{1,2}(0,T;V^{*})$$

is continuous.

Proof. It is easy to show that if (Φ ⁰ , Φ ¹) ∈ H × L ²(-h, 0; V )) and f ∈ L ²(0, T; V* ) for every T > 0, then x belongs to L ²(- h, T; V )∩W ^1,2(0, T; V*). Let

$$\left({\varphi }_i^0,{\varphi }_i^1,f_i\right)\in H\times L^2\left(-h,0;V\right)\times L^2\left(0,T_1;V^{*}\right)$$

and x _i be the solution of (2.6) with $\left({\varphi }_i^0,{\varphi }_i^1,f_i\right)$ in place of $\left({\varphi }^0,{\varphi }^1,f\right)$ for i = 1,2.

Then, in view of Proposition 2.1 and Lemma 2.2, we have

$$\begin{array}{c}\frac{1}{2}\frac{d}{dt}\mid x_1\left(t\right)-x_2\left(t\right){\mid }^2+{\omega }_1\parallel x_1\left(t\right)-x_2\left(t\right){\parallel }^2\\ \le {\omega }_2\mid x_1\left(t\right)-x_2\left(t\right){\mid }^2+\parallel G\left(t,x_1\right)-G\left(t,x_2\right){\parallel }_{*}\parallel x_1\left(t\right)-x_2\left(t\right)\parallel \\ +\parallel f_1\left(t\right)-f_2\left(t\right){\parallel }_{*}\parallel x_1\left(t\right)-x_2\left(t\right)\parallel \end{array}$$

(2.12)

If ω ₁ - c/ 2 > 0, we can choose a constant c ₁ > 0 so that

$${\omega }_1-c∕2-c_1∕2>0$$

and

$$\begin{array}{l}\parallel f_1(t)-f_2(t)){\parallel }_{*}\parallel x_1(t)-x_2(t)\parallel \le \frac{1}{2c_1}\parallel f_1(t)-f_2(t){\parallel }_{*}^2\\ +\frac{c_1}{2}\parallel x_1(t)-x_2(t){\parallel }^2.\end{array}$$

Let T ₁ < T be such that

$$2{\omega }_1-c-c_1-c^{-1}{e}^{2{\omega }_2{T}_1}{\left(L_1{\sqrt{T}}_1+L_2{K}_1{T}_1^{3∕2}∕\sqrt{2}\right)}^2>0.$$

Integrating on (2.12) over [0, T ₁] and as is seen in the first part of proof, it follows

$$\begin{array}{l}(2{\omega }_1-c-c_1)\parallel x_1-x_2{\parallel }_{L^2(0,T_1;V)}^2\le e^{2{\omega }_2{T}_1}\{\mid {\varphi }_1^0-{\varphi }_2^0{\mid }^2\\ +\frac{1}{c}\parallel G(t,x_1)-G(t,x_2){\parallel }_{L^2(0,T_1;V^{*})}^2+1c_1\parallel f_1-f_2{\parallel }_{L^2(0,T_1;V^{*})}^2\}\\ \le e^{2{\omega }_2{T}_1}\{\mid {\varphi }_1^0-{\varphi }_2^0{\mid }^2\\ +\frac{1}{c}(L_1{\sqrt{T}}_1+L_2{K}_1{T}_1^{3/2}/\sqrt{2}{)}^2\parallel x_1-x_2{\parallel }_{L^2(-h,T_1;V)}^2\\ +\frac{1}{c_1}\parallel f_1-f_2{\parallel }_{L^2(0,T_1;V^{*})}^2\}.\end{array}$$

Putting that

$$N_1=2{\omega }_1-c-c_1-c^{-1}{e}^{2{\omega }_2{T}_1}{\left(L_1{\sqrt{T}}_1+L_2{K}_1{T}_1^{3∕2}∕\sqrt{2}\right)}^2$$

we have

$$\begin{array}{c}\parallel x_1-x_2{\parallel }_{L^2\left(0,T_1;V\right)}\le \frac{e^{{\omega }_2{T}_1}}{N_1^{1∕2}}\left(\mid {\varphi }_1^0-{\varphi }_2^0\mid +\frac{1}{c_1}\parallel f_1-f_2{\parallel }_{L^2\left(0,T_1;V*\right)}\right)\\ +\frac{c^{-1∕2}{e}^{{\omega }_2{T}_1}\left(L_1\sqrt{T_1}+L_2{K}_1{T}_1^{3∕2}∕\sqrt{2}\right)}{N_1^{1∕2}}\parallel {\varphi }_1^1-{\varphi }_2^1{\parallel }_{L^2\left(-h,0;V\right)}.\end{array}$$

(2.13)

Suppose that

$$({\varphi }_n^0,{\varphi }_n^1,f_n)\to ({\varphi }^0,{\varphi }^1,f)\text{in}H\times L^2(-h,0;V))\times L^2(0,T;V^{*}),$$

and let x _n and x be the solution (2.6) with $\left({\varphi }_n^0,{\varphi }_n^1,f_n\right)$ and $\left({\varphi }^0,{\varphi }^1,f\right)$ respectively.

By virtue of (2.13) with T being replaced by T ₁, we see that

$$x_n\to x\text{ℯin}{L}^2(-h,T_1;V))\cap W^{1,2}(0,T_1;V^{*})\subset C([0,T_1];H).$$

This implies that $\left(x_n\left(T_1\right),{\left(x_n\right)}_{T_1}\right)\to \left(x\left(T_1\right),x_{T_1}\right)$ in H × L ² (-h, 0; V). Hence, the same argument shows that

$$x_n\to x\mathsf{\text{in}}{L}^2\left(T_1,min\left\{2T_1,T\right\};V\right)\cap W^{1,2}\left(T_1,min\left\{2T_1,T\right\};V^{*}\right).$$

Repeating this process, we conclude that

$$x_n\to x\mathsf{\text{in}}{L}^2\left(-h,T;V\right)\cap W^{1,2}\left(0,T;V^{*}\right).$$

Remark 2.2. For x ∈ L ²(0, T; V ), we set

$$G\left(t,x\right)={\int }_0^{t}k\left(t-s\right)g\left(s,x\left(s\right)\right)ds$$

where k belongs to L ²(0, T) and g : [0, T] × V → H be a nonlinear mapping satisfying

$$\mid g\left(t,x\right)-g\left(t,y\right)\mid \le L\parallel x-y\parallel$$

for a positive constant L. Let x ∈ L ²(0, T; V ), T > 0. Then, G(·, x) ∈ L ²(0, T; H) and

$$\parallel G\left(\cdot ,x\right){\parallel }_{L^2\left(0,T;H\right)}\le L\parallel k{\parallel }_{L^2\left(0,T\right)}\sqrt{T}\parallel x{\parallel }_{L^2\left(0,T;V\right)}.$$

Moreover, if x ₁ , x ₂ ∈ L ²(0, T; V ), then

$$\parallel G\left(\cdot ,x_1\right)-G\left(\cdot ,x_2\right){\parallel }_{L^2\left(0,T;H\right)}\le L\parallel k\parallel \sqrt{T}\parallel x_1-x_2{\parallel }_{L^2\left(0,T;V\right)}.$$

Then, with the condition that

$${\omega }_1^{-1}{e}^{{\omega }_2{T}_0}L\parallel k\parallel \sqrt{T_0}<1$$

in place of the condition (2.9), we can obtain the results of Theorem 2.1.

3 Approximate controllability

In what follows we assume that the embedding V ⊂ H is compact, and A is a continuous operator from V to V* satisfying (A1) and (A2). For h ∈ L ²(0, T; H) and let x _h be the solution of the following equation with B = I:

$$\left\{\begin{array}{c}\hfill x^{\prime }\left(t\right)+Ax\left(t\right)=G\left(t,x\right)+h\left(t\right),0<t,\hfill \\ \hfill x\left(0\right)=0x\left(s\right)=0-h\le s\le 0,\hfill \end{array}\right.$$

(3.1)

where

$$G\left(t,x\right)=g\left(t,x_t,{\int }_0^{t}k\left(t,s,x_s\right)ds\right).$$

We define the solution mapping S from L ²(0, T; V*) to L ²(0, T; V ) by

$$\begin{array}{cc}\hfill \left(Sh\right)\left(t\right)=x_h\left(t\right),\hfill & \hfill h\in L^2\left(0,T;V^{*}\right)\hfill \end{array}.$$

(3.2)

Let $\mathcal{A}$ and $\mathcal{G}$ be the Nemitsky operators corresponding to the maps A and G, which are defined by $\mathcal{A}(x)(\cdot )=Ax(\cdot )$ and $\mathcal{G}(h)(\cdot )=G(\cdot ,x_h)$, respectively. Then, since the solution x belongs to L ²(-h, T; V ) ∩ W ^1,2(0, T; V*) ⊂ C([0, T]; H), it is represented by

$$x_h(t)={\displaystyle {\int }_0^t(}(I+\mathcal{G}-\mathcal{A}S)h)(s)ds,$$

(3.3)

and with aid of Lemma 2.2 and Proposition 2.1

$$\begin{array}{l}\parallel Sh{\parallel }_{L^2(0,T;V)\cap W^{1,2}(0,T;V^{*})}=\parallel x_h\parallel \le C_1\parallel G(\eta ,x_h)+h{\parallel }_{L^2(0,T;V^{*})}\\ \le C_1\{L_0\sqrt{T}+L_2{K}_0{T}^{3/2}/\sqrt{3}+(L_1\sqrt{T}+L_2{K}_1{T}^{3/2}/\sqrt{2})\parallel x{\parallel }_{L^2(0,T;V)}\\ +\parallel h{\parallel }_{L^2(0,T;V^{*})}\}\\ \le C_1\{L_0\sqrt{T}+L_2{K}_0{T}^{3/2}/\sqrt{3}\\ +(L_1\sqrt{T}+L_2{K}_1{T}^{3/2}/\sqrt{2})(1+\parallel h{\parallel }_{L^2(0,T;V^{*})})+\parallel h{\parallel }_{L^2(0,T;V^{*})}\}.\end{array}$$

(3.4)

Hence, if h is bounded in L ² (0, T; V*), then so is x _h in L ²(0, T; V)∩W ^1,2(0, T; V*). Since V is compactly embedded in H by assumption, the embedding L ²(0, T; V) ∩ W ^1,2 (0, T; V*) ⊂ L ² (0, T; H) is compact in view of Theorem 2 of Aubin [14]. Hence, the mapping h ↦ Sh = x _h is compact from L²(0, T; V*) to L ²(0, T; H). Therefore, $\mathcal{G}$ is a compact mapping from L ²(0, T; V*) to L ²(0, T; H) and so is $\mathcal{A}S$ from L ²(0, T; V*) to itself. The solution of (SE) is denoted by x(T; g, u) associated with the nonlinear term g and control u at time T.

Definition 3.1. The system (SE) is said to be approximately controllable at time T if Cl{x(T; g, u): u ∈ L ²(0, T; U)} = V* where Cl denotes the closure in V*.

We assume

(T) $1-{\omega }_1^{-1}{\omega }_3{e}^{{\omega }_{2}T}>0$

(B) Cl{y : y(t) = (Bu)(t), a.e. u ∈ L ²(0, T; U)} = L ²(0, T; U)}. Here Cl is the closure in L ²(0, T; H).

Theorem 3.1. Let the assumptions (T) and (B) be satisfied. Then,

$$Cl\{(I-\mathcal{A}S)h:h\in L^2(0,T;V^{*})\}=L^2(0,T;V^{*}).$$

(3.5)

Therefore, the following nonlinear differential control system

$$\left\{\begin{array}{c}\frac{dx\left(t\right)}{dt}+Ax\left(t\right)=\left(Bu\right)\left(t\right),0<t\le T,\\ x\left(0\right)=x_0\end{array}\right.$$

(3.6)

is approximately controllable at time T.

Proof. Let z ∈ L ²(0, T; V*) and r be a constant such that

$$z\in U_r=\left\{x\in L^2\left(0,T;V^{*}\right):\parallel x{\parallel }_{L^2\left(0,T;V^{*}\right)}<r\right\}.$$

Take a constant d > 0 such that

$$(r+{\omega }_3+N_2\mid x_0\mid ){(1-N_2)}^{-1}<d,$$

(3.7)

where

$$N_2={\omega }_1^{-1}{\omega }_3{e}^{{\omega }_{2}T}.$$

Taking scalar product on both sides of (3.1) with G = 0 by x(t)

$$\begin{array}{c}\frac{1}{2}\frac{d}{dt}\mid x\left(t\right){\mid }^2+{\omega }_1\parallel x\left(t\right){\parallel }^2\le {\omega }_2\mid x\left(t\right){\mid }^2+\parallel h\left(t\right){\parallel }_{*}\parallel x\left(t\right)\parallel \\ \le {\omega }_2\mid x\left(t\right){\mid }^2+\frac{1}{2c}\parallel h\left(t\right){\parallel }_{*}^2+\frac{c}{2}\parallel x\left(t\right){\parallel }^2\end{array}$$

where c is a positive constant satisfying 2ω₁ - c > 0. Integrating on [0, t], we get

$$\begin{array}{c}\frac{1}{2}\mid x\left(t\right){\mid }^2+{\omega }_1{\int }_0^t\parallel x\left(s\right){\parallel }^{2}ds\le \frac{1}{2}\mid x_0{\mid }^2+\frac{1}{2c}{\int }_0^t\parallel h\left(s\right){\parallel }_{*}^{2}ds\\ +\frac{c}{2}{\int }_0^t\parallel x\left(s\right){\parallel }^{2}ds+{\omega }_2{\int }_0^t\mid x\left(s\right){\mid }^{2}ds,\end{array}$$

and hence,

$$\begin{array}{cc}\hfill \mid x\left(t\right){\mid }^2+\left(2{\omega }_1-c\right){\int }_0^t\parallel x\left(s\right){\parallel }^{2}ds& \le \mid x_0{\mid }^2+\frac{1}{c}{\int }_0^t\parallel h\left(s\right){\parallel }_{*}^{2}ds\hfill \\ +2{\omega }_2{\int }_0^t\mid x\left(s\right){\mid }^{2}ds.\hfill \end{array}$$

By using Gronwall's inequality, it follows that

$$\mid x\left(T\right){\mid }^2+\left(2{\omega }_1-c\right){\int }_0^T\parallel x\left(s\right){\parallel }^{2}ds\le e^{2{\omega }_{2}T}\left(\mid x_0{\mid }^2+\frac{1}{c}{\int }_0^T\parallel h\left(s\right){\parallel }_{*}^{2}ds\right),$$

that is,

$$\begin{array}{c}\parallel Sh{\parallel }_{L^2\left(0,T;V\right)}=\parallel x{\parallel }_{L^2\left(0,T;V\right)}\\ \le e^{{\omega }_{2}T}{\left(2{\omega }_1-c\right)}^{-1∕2}\left({\mid x}_0\mid +c^{-1∕2}\parallel h{\parallel }_{L^2\left(0,T;V^{*}\right)}\right).\end{array}$$

(3.8)

Let us consider the equation

$$z=(I-\mathcal{A}S)w.$$

(3.9)

Let w be the solution of (3.9). Then z ∈ U _d and taking c = ω ₁, from (3.7), (3.8)

$$\begin{array}{c}\parallel w{\parallel }_{L^2(0,T;V^{*})}\le \parallel z{\parallel }_{L^2(0,T;V^{*})}+\parallel \mathcal{A}Sw{\parallel }_{L^2(0,T;V^{*})}\\ \le r+{\omega }_3(\parallel Sw{\parallel }_{L^2(0,T;V^{*})}+1)\\ \le r+{\omega }_3\{{\omega }_1^{-1/2}{e}^{{\omega }_{2}T}(\mid x_0\mid +{\omega }_1^{-1/2}\parallel w\parallel )+1\},\end{array}$$

and hence

$$\parallel w\parallel \le (r+{\omega }_3+N_2\mid x_0\mid ){(1-N_2)}^{-1}<d$$

it follows that w ∉ ∂U _d where ∂U _d stands for the boundary of U _d . Thus, the homotopy property of topological degree theory there exists w ∈ L ²(0, T; V*) such that the equation (3.9) holds. Based on the assumption (B), there exists a sequence {u _n } ∈ L ²(0, T; U) such that Bu _n ↦ w in L ²(0, T; V*). Then, by the last paragraph of Theorem 2.1, we have that x(·; g, u _n ) ↦ x _w in L ²(0, T; V ) ∩ W ^1,2(0, T; V*) ⊂ C([0, T]; H). Hence, we have proved (3.5). Let y ∈ V*. Then, there exists an element u ∈ L ²(0, T; U) such that

$$\parallel \frac{y}{T}-(I-\mathcal{A}S)Bu{\parallel }_{L^2(0,T;V^{*})}<\frac{\epsilon }{\sqrt{T}}.$$

Thus

$$\begin{array}{c}\parallel y-x(T){\parallel }_{*}=\parallel y-{\displaystyle {\int }_0^T(}(I-\mathcal{A}S)Bu)(s)ds{\parallel }_{*}\\ \le {\displaystyle {\int }_0^T\parallel }\frac{y}{T}-((I-\mathcal{A}S)Bu)(s){\parallel }_{*}ds\\ \le \sqrt{T}\parallel \frac{y}{T}-(I-\mathcal{A}S)Bu{\parallel }_{L^2(0,T;V^{*})}<\epsilon .\end{array}$$

Therefore, the system (3.6) is approximately controllable at time T.

In order to investigate the controllability of the nonlinear control system, we need to impose the following condition.

(F) g is uniformly bounded: there exists a constant M _g such that

$$\mid g\left(t,x,y\right)\mid \le M_g,$$

for all x, y ∈ V.

By (F) it holds that

$$\parallel G\left(\cdot ,x\right){\parallel }_{L^2\left(0,T;H\right)}\le M_g\sqrt{T},$$

and for every h ∈ L ²(0, T; V*)

$$\parallel \mathcal{G}(h){\parallel }_{L^2(0,T;H)}\le M_g\sqrt{T}$$

(3.10)

Theorem 3.2. Let the assumptions (T), (B), and (F) be satisfied. Then, we have

$$Cl\{(\mathcal{G}+I-\mathcal{A}S)h:h\in L^2(0,T;V^{*})\}=L^2(0,T;V^{*}).$$

(3.11)

Thus, the system (SE) is approximately controllable at time T.

Proof. Let U _r be the ball with radius r in L ²(0, T; V*) and z ∈ U _r . To prove (3.11), we will also use the degree theory for the equation

$$z=\lambda (\mathcal{G}-\mathcal{A}S)w+w,0\le \lambda \le 1$$

(3.12)

in open ball U _d where the constant d satisfies

$$(r+{\omega }_3+N_2\mid x_0\mid +M_g\sqrt{T}){(1-N_2)}^{-1}<d$$

(3.13)

where the constant N ₂ is in Theorem 3.1. If w is the solution of (3.12), then z ∈ U _d and from Lemma 2.1

$$\begin{array}{l}\parallel w{\parallel }_{L^2(0,T;V^{*})}\le \parallel z\parallel +\parallel \mathcal{A}Sw\parallel +\parallel \mathcal{G}w\parallel \\ \le r+{\omega }_3(\parallel Sw\parallel +1)+M_g\sqrt{T})\\ r+{\omega }_3\{{\omega }_1^{-1/2}{e}^{{\omega }_{2}T}(\mid x_0\mid +{\omega }_1^{-1/2}\parallel w\parallel )+1\}+M_g\sqrt{T},\end{array}$$

and hence

$$\parallel w\parallel \le (r+{\omega }_3+N_2\mid x_0\mid +M_g\sqrt{T}){(1-N_2)}^{-1}<d$$

it follows that w ∉ ∂U _d . Hence, there exists w ∈ L ²(0, T; V*) such that the equation (3.12) holds. Using the similar way to the last part of Theorem 3.1 and the assumption (B), there exists a sequence {un} ∈ L ²(0, T; U) such that Bu _n ↦ w in L ²(0, T; V*) and x(·, g, u _n ) ↦x _w in L ²(0, T; V ) ∩ W ^1,2(0, T; V*) ⊂ C([0, T]; H). Thus, we have proved (3.11), and the system (1.1) is approximately controllable at time T.

4 Example

Let -A be an operator associated with a bounded sesquilinear form a(u, v) defined in V × V and satisfying Gårding inequality

$$Rea\left(u,v\right)\ge c_0\parallel u{\parallel }^2-c_1\mid u{\mid }^2,c_0>0,c_1\ge 0$$

for any u ∈ V. It is known that A generates an analytic semigroup in both H and V*. By virtue of the Riesz-Schauder theorem, if the embedding V ⊂ H is compact, then the operator A has discrete spectrum:

$$\sigma \left(A\right)=\left\{{\mu }_n:n=1,2,...\right\}$$

which has no point of accumulation except possibly when μ = ∞. Let μ _n be a pole of the resolvent of A of order k _n and P _n the spectral projection associated with μ _n

$$P_n=\frac{1}{2\pi i}{\int }_{{\Gamma }_n}{\left(\mu -A\right)}^{-1}d\mu ,$$

where Γ _n is a small circle centered at μ _n such that it surrounds no point of σ(A) except μ _n . Then, the generalized eigenspace corresponding to μ _n is given by

$$H_n=P_{n}H=\left\{P_{n}u:u\in H\right\},$$

and we have that from $P_n^2=P_n$ and H _n ⊂ V ; it follows that

$$P_{n}V=\left\{P_{n}u:u\in V\right\}=H_n.$$

Definition 4.1. The system of the generalized eigenspaces of A is complete in H if Cl {span{H _n : n = 1, 2,...}} = H where Cl denotes the closure in H.

We need the following hypotheses:

(B1) The system of the generalized eigenspaces of A is complete.

(B2) There exists a constant d > 0 such that

$$\parallel \upsilon \parallel \le d\parallel B\upsilon \parallel ,\upsilon \in L^2\left(0,T;U\right).$$

We can see many examples which satisfy (B2)(cf. [8, 11]).

Consider about the intercept controller B define d by

$$\left(Bu\right)\left(t\right)=\sum _{n=1}^{\infty }{u}_n\left(t\right),$$

(4.1)

where

$$u_n=\left\{\begin{array}{cc}\hfill 0,\hfill & \hfill 0\le t\le \frac{T}{n}\hfill \\ \hfill P_{n}u\left(t\right),\hfill & \hfill \frac{T}{n}<t\le T.\hfill \end{array}\right.$$

Hence, we see that u ₁(t) ≡ 0 and u _n (t) ∈ Im P _n .

First of all, for the meaning of the condition (B) in section 3, we need to show the existence of controller satisfying Cl{Bu : u ∈ L ²(0, T; U)} ≠ L ²(0, T; H). In fact, by completion of the generalized eigenspaces of A, we may write that $f\left(t\right)={\sum }_{n=1}^{\infty }{P}_{n}f\left(t\right)$ for ∈L ²(0, T; H). Let us choose f ∈L ²(0, T; H) satisfying

$${\int }_0^T\parallel P_{1}f\left(t\right){\parallel }^{2}dt>0.$$

Then, since

$$\begin{array}{c}{\int }_0^T\parallel f\left(t\right)-Bu\left(t\right){\parallel }^{2}dt={\int }_0^T\sum _{n=1}^{\infty }\parallel P_n\left(f\left(t\right)-Bu\left(t\right)\right){\parallel }^{2}dt\\ \ge {\int }_0^T\parallel P_1\left(f\left(t\right)-Bu\left(t\right)\right){\parallel }^{2}dt={\int }_0^T\parallel P_{1}f\left(t\right){\parallel }^{2}dt>0,\end{array}$$

the statement mentioned above is reasonable.

Let f ∈ L ²(0, T; H) and α = T/(T - T/n). Then we know

$$f\left(\cdot \right)\equiv \alpha K_{\left[T,T∕n\right]}f\left(\alpha \left(\cdot -T∕n\right)\right)\mathsf{\text{in}}{L}^2\left(0,T;H\right),$$

where K _[T,T/n]is the characteristic of [T,T/n]. Define

$$w\left(s\right)=\sum _{n=1}^{\infty }{w}_n\left(s\right),w_n\left(s\right)=\alpha K_{\left[T,T∕n\right]}{B}^{-1}{P}_{n}f\left(\alpha \left(s-T∕n\right)\right).$$

Thus $\left(Bw\right)\left(t\right)={\sum }_{n=1}^{\infty }{P}_{n}f\left(s\right),\mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.}}$ Since the system of the generalized eigenspaces of A is complete, it holds that for every f ∈ L ²(0, T; H) and ∈ > 0

$$\parallel f\left(\cdot \right)-\sum _{n=1}^{\infty }{P}_{n}f\left(\cdot \right){\parallel }_{L^2\left(0,T;H\right)}=\parallel f\left(\cdot \right)-Bw{\parallel }_{L^2\left(0,T;H\right)}<\epsilon .$$

Thus, the intercept controller B define d by (4.1) satisfies the condition (B).