Exact controllability of mild solutions for a class of semilinear differential equations

Abstract

In this paper, by using the noncompact measure, we study the exact controllability of mild solutions for semilinear differential equation when the nonlocal term is Lipschitz continuous in general Banach space. Here, the regularity conditions of noncompact measure assumption in previous literature are not used, but instead we adopt the estimate of noncompact measure in different topological spaces, give the proper control function by the property of quotient space, and obtain the exact controllability of the differential equation. Therefore, our results can be regarded as a Generalization of the controllability field.

Keywords

Exact controllability / noncompact measure / semilinear differential equation / nonlocal condition

1 Introduction

Byszewski [2] first developed classical differential equations into nonlocal differential equations. Due to the fact that nonlocal differential equations can better and more accurately describe physical phenomena than classical Cauchy problem, many scholars have conducted detailed research in this field under different conditions for linear operator A and nonlinear term f.

In this paper, we mainly discuss the exact controllability of mild solutions to nonlocal differential equations as follows:

(1.1)\{\begin{array}{l}{y}^{\prime }\left(t\right)=Ay\left(t\right)+f(t,y(t))+Bu\left(t\right),t\in [0,h],\\ y\left(0\right)=y_0-g\left(y\right),\end{array}

where y(\cdot ) takes value in Banach space Y, control function u\in L^2([0,h],U), and U is Hilbert space. A is an infinitesimal generator of strongly continuous semigroup \left\{T\right(t),t\ge 0\}, and there exists a constant Q>0 such that \underset{t\in [0,h]}{\mathrm{s}\mathrm{u}\mathrm{p}} \Vert T\left(t\right)\Vert \le Q. B:U\to Y is a bounded linear operator and g:C([0,h],Y)\to Y is a nonlocal term.

In 1960, Kalman first proposed the concept of controllability of linear systems in finite dimensional spaces. Controllability is a core issue in control theory, which has attracted widespread attention from scholars. Some scholars have also extended this concept to the case of nonlinear systems in infinite dimensional spaces. In recent years, scholars have studied the exact controllability of infinite dimensional semilinear evolution systems using different methods (see [3, 6-15]). In the study of exact controllability problems, the property of operator W is crucial in constructing control functions. However, in infinite dimensional spaces, it is generally not possible to achieve exact controllability when the relevant operator is compact (see [4]). In references [3, 8, 12, 15], the authors used Hausdorff noncompact measures to discuss the exact controllability of the system under the condition of noncompact semigroup. In order to overcome the difficulty of losing compactness, the author assumes the following regularity condition of W^{-1} for non-compact measures:

(1.2){\gamma }_U(W^{-1}(\mathrm{\Omega })(t))\le \kappa (t){\gamma }_Y(\mathrm{\Omega }),

where \kappa (\cdot )\in L_{+}^1([0,h\left]\right), {\gamma }_U is a Hausdorff noncompact measure defined on space U. Based on this, this paper has two purposes: the first is to provide estimates of noncompact measures in different topological spaces, directly removing the regularity condition assumed in the previous literature (1.2); the second is to use the properties of the quotient space to provide a suitable control function u. Therefore, our results generalize some conclusions known in this field.

2 Preliminaries

Definition 2.1　Function y\in C([0,h],Y). If for any t\in [0,h], there exists

y\left(t\right)=T\left(t\right)[y_0-g(y)]+{\int }_0^t T(t-s)f(s,y(s))\mathrm{d}s+{\int }_0^{t}T(t-s)Bu\left(s\right)\mathrm{d}s,

then y is the mild solution of Equation (1.1).

Define the Hausdorff noncompact measure as follows:

\gamma \left(\mathrm{\Omega }\right)=\mathrm{i}\mathrm{n}\mathrm{f}\{\epsilon >0:\mathrm{\Omega }\text{has finite}\epsilon \text{grid in}Y\},

where Ω is a bounded set in Y.

Hausdorff noncompact measure γ has the following properties.

Lemma 2.1 [1]　 Assume B_1, B_2\subseteq Y is a bounded set in Y, then the following conclusions are established:

(1) B₁ is relatively compact if and only if \gamma \left(B_1\right)=0;

(2) \gamma \left(B_1\right)\le \gamma \left(B_2\right), when B_1\subseteq B_2;

(3) \gamma (B_1+B_2)\le \gamma \left(B_1\right)+\gamma \left(B_2\right), where B_1+B_2=\{u_1+u_2;u_1\in B_1,u_2\in B_2\};

(4) \gamma \left(\lambda B_1\right)=\left|\lambda \right|\gamma \left(B_1\right), where λ∈R;

(5) If \Gamma :D\left(\Gamma \right)\subseteq Y\to Z is Lipschitz continuous, and Lipschitz constant is k, then for any bounded set V\subseteq D\left(\Gamma \right), there exists {\gamma }_Z\left(\Gamma V\right)\le k\gamma \left(V\right), where Z is a Banach space.

Noncompact measure γ₀ induced by γ is defined as

{\gamma }_0\left(V\right)=\mathrm{s}\mathrm{u}\mathrm{p}\left\{\gamma \right(\{y_n:n\ge 1\}):\{y_n{\}}_{n=1}^{+\mathrm{\infty }}\text{is a sequence in}V\}.

Obviously,

(2.1){\gamma }_0\left(V\right)\le \gamma \left(V\right)\le 2{\gamma }_0\left(V\right).

In particular, if Y is a divisible Banach space, then {\gamma }_0\left(V\right)=\gamma \left(V\right).

Mapping \mathrm{\Gamma }:V\subseteq Y\to Y. For any bounded set V_1\subseteq V, there exists positive constant such that \gamma \left(\mathrm{\Gamma }{V}_1\right)\le \mu \gamma \left(V_1\right), then we say Γ is \gamma-contracted.

Definition 2.2　Equation (1.1) is exactly controllable on the interval [0,h]. If for any y_0,y_1\in Y, there exists the control function u\in L^2([0,h],U), such that the mild solution y(\cdot ) of Equation (1.1) satisfies y\left(h\right)=y_1.

We make the following assumptions:

Assumption (W): define linear operator W:L^2([0,h],U)\to Y as

Wu={\int }_0^h T(h-s)Bu\left(s\right)\mathrm{d}s,

satisfying

(i) W is surjective, namely R\left(W\right)=Y.

Obviously, L^2([0,h],U)/\mathrm{k}\mathrm{e}\mathrm{r}\left(W\right) is a Banach space, assigning the norm

\Vert [u]\Vert =\underset{u\in \left[u\right]}{\mathrm{i}\mathrm{n}\mathrm{f}} \Vert u{\Vert }_{L^2([0,h],U)}=\underset{W\hat{u}=0}{\mathrm{i}\mathrm{n}\mathrm{f}} \Vert u+\hat{u}{\Vert }_{L^2([0,h],U)},

where \left[u\right] is the equivalence class of u. Define \tilde{W}:L^2([0,h],U)/\mathrm{k}\mathrm{e}\mathrm{r}\left(W\right)\to Y as

\tilde{W}\left[u\right]=Wu,u\in \left[u\right].

Then \tilde{W} is injective, and satisfies

\Vert \tilde{W}\left[u\right]{\Vert }_Y\le 2\Vert W\Vert \Vert \left[u\right]{\Vert }_{L^2([0,h],U)/\mathrm{k}\mathrm{e}\mathrm{r}\left(W\right)}.

Since R\left(W\right)=Y, with the Banach inverse operator theorem, there exist

{\tilde{W}}^{-1}:Y\to L^2([0,h],U)/\mathrm{k}\mathrm{e}\mathrm{r}\left(W\right),

and

{\tilde{W}}^{-1}\in \mathcal{L}(Y,L^2([0,h],U)/\mathrm{k}\mathrm{e}\mathrm{r}(W)).

Define operator P:L^2([0,h],U)/\mathrm{k}\mathrm{e}\mathrm{r}\left(W\right)\to L^2([0,h],U) as

P\left[u\right]=u_1,u_1\in \left[u\right],

which satisfies

\Vert P([u\left]\right)\Vert \le 2\Vert \left[u\right]\Vert ,u\in L^2([0,h],U).

(ii) There exists positive constant Q₁, Q₂, such that \Vert B\Vert \le Q_1,\Vert {\tilde{W}}^{-1}\Vert \le Q_2.

Assume (f): mapping f:[0,h]\times Y\to Y satisfies

(1) for any y\in Y, f(\cdot ,y):[0,h]\to Y is measurable; for almost everywhere t\in [0,h], f(t,\cdot ):Y\to Y is continuous;

(2) for every y\in C([0,h],Y) and \Vert y\Vert \le r,r>0, there exists

\Vert f(t,y(t))\Vert \le {\rho }_r\left(t\right),\text{a.e.}t\in [0,h],

where {\rho }_r\in L^1([0,h],{\mathbb{R}}^{+}), and

(iii) There exists k\in L^1([0,h],{\mathbb{R}}^{+}), such that for any bounded set V\subset Y,

\gamma \left(f\right(t,V))\le k\left(t\right)\cdot \gamma \left(V\right)

holds for almost everywhere.

Assumption (g): nonlocal term g:C([0,h],Y)\to Y is Lipschitz continuous, that is, there exists constant b\in R⁺, such that for any y_1,y_2\in C([0,h],Y), there is \Vert g\left(y_1\right)-g\left(y_2\right)\Vert \le b\Vert y_1-y_2\Vert, and .

3 Main results

To prove the exact controllability of Equation (1.1), we assume is P a linear operator, and for any y\in C([0,h],Y), define the control function

u_y\left(t\right)=P{\tilde{W}}^{-1}\{y_1-T(h)[y_0-g\left(y\right)]-{\int }_0^h T(h-\tau \left)f\right(\tau ,y\left(\tau \right)\left)\mathrm{d}\tau \right\}\left(t\right),t\in [0,h].

Under this control function, define the operator \mathrm{\Gamma }:C([0,h],Y)\to C([0,h],Y) as

\left(\mathrm{\Gamma }y\right)\left(t\right)=\left({\mathrm{\Gamma }}_{1}y\right)\left(t\right)+\left({\mathrm{\Gamma }}_{2}y\right)\left(t\right)+\left({\mathrm{\Gamma }}_{3}y\right)\left(t\right),

where

\begin{array}{rl}& \left({\mathrm{\Gamma }}_{1}y\right)\left(t\right)=T\left(t\right)[y_0-g(y)],\\ & \left({\mathrm{\Gamma }}_{2}y\right)\left(t\right)={\int }_0^t T(t-s)f(s,y(s))\mathrm{d}s,\\ & \left({\mathrm{\Gamma }}_{3}y\right)\left(t\right)={\int }_0^t T(t-s)B{u}_y\left(s\right)\mathrm{d}s.\end{array}

Lemma 3.1 [1]　 Assume bounded set V\subseteq C([0,h],Y). For any t\in [0,h],

\gamma \left(V\right(t))\le {\gamma }_c\left(V\right),

where V\left(t\right)=\left\{y\right(t):y\in V\}\subseteq Y, and {\gamma }_c is the Hausdorff noncompact measure defined on C([0,h],Y). If V is continuous, then \gamma \left(V\right(t)) is continuous on the interval [0,h]. Specifically,

{\gamma }_c\left(V\right)=\mathrm{s}\mathrm{u}\mathrm{p}\left\{\gamma \right(V\left(t\right)):t\in [0,h\left]\right\}.

Lemma 3.2 [1]　 If operator semigroup \left\{T\right(t),t\ge 0\} is equicontinuous, and \omega \in L^1([0,h],{\mathbb{R}}^{+}), then set

\left\{{\int }_0^t T\left(t-s\right)f\left(s\right)\mathrm{d}s:\Vert f\left(s\right)\Vert \le \omega \left(s\right)a.e.s\in [0,h]\right\}

is equicontinuous on the interval [0,h].

Next, we will introduce the main fixed point theorems in this paper.

Lemma 3.3 [5]　 Assume W is a nonempty bounded closed convex set in Banach space Y, if continuous mapping \mathrm{\Gamma }:W\to W is γ-contracted, then Γ contains at least one fixed point.

Lemma 3.4　 If assumptions (W), (f(i)‒f(ii)), and (g) hold, then there exists positive constant r, such that \mathrm{\Gamma }\left(B_r\right)\subset B_r, where B_r=\{y\in C([0,h],Y):\Vert y\Vert \le r\}.

Proof　Use proof by contradiction. If for any r, there is always y_r\in B_r satisfying \mathrm{\Gamma }\left(y_r\right)\notin B_r, and

\begin{array}{rl}\mathrm{\Gamma }\left(y_r\right)\left(t\right)=& T\left(t\right)[y_0-g(y_r)]+{\int }_0^t T(t-s)f(s,y_r(s))\mathrm{d}s+{\int }_0^t T(t-s)BP{\tilde{W}}^{-1}\\ & \times \left\{y_1-T\left(h\right)[y_0-g(y_r)]-{\int }_0^h T(h-\tau )f(\tau ,y_r(\tau ))\mathrm{d}\tau \right\}\left(s\right)\mathrm{d}s,\end{array}

it can be obtained with assumptions (W), (f(ii)) and (g) that for every t\in [0,h], there is

\begin{array}{rl}& \Vert \mathrm{\Gamma }\left(y_r\right)\left(t\right)\Vert \le Q\Vert y_0-g\left(y_r\right)\Vert +Q{\int }_0^t \Vert f(s,y_r(s))\Vert \mathrm{d}s\\ & +Q{Q}_1{\int }_0^t\Vert P{\tilde{W}}^{-1}\left(y_1-T(h)[y_0-g\left(y_r\right)]-{\int }_0^h T(h-\tau )f(\tau ,y_r(\tau ))\mathrm{d}\tau \right)\Vert \mathrm{d}s\\ & \le Q((b+1)\Vert y_0\Vert +br+\Vert g(y_0)\Vert )+Q{\int }_0^h {\rho }_r\left(s\right)\mathrm{d}s+Q{Q}_1\sqrt{h}\\ & \times {\Vert P{\tilde{W}}^{-1}\left(y_1-T\left(h\right)[y_0-g(y_r)]-{\int }_0^h T(h-\tau )f(\tau ,y_r(\tau ))\mathrm{d}\tau \right)\Vert }_{L^2([0,h],U)}\\ & \le Q((b+1)\Vert y_0\Vert +br+\Vert g(y_0)\Vert )+Q{\int }_0^h {\rho }_r\left(s\right)\mathrm{d}s+2Q{Q}_2{Q}_1\sqrt{h}\\ & \times \left[\Vert y_1\Vert +Q((b+1)\Vert y_0\Vert +br+\Vert g(y_0)\Vert )+Q{\int }_0^h {\rho }_r\left(s\right)\mathrm{d}s\right].\end{array}

However,

When the two sides of the equation are divided by r and obtain the limit r\to \mathrm{\infty }, we have

(Q\alpha +Qb)(1+2\sqrt{h}{Q}_1{Q}_{2}Q)\ge 1,

which contradicts with the Assumption (g). Hence, there must exist r>0 such that \mathrm{\Gamma }\left(B_r\right)\subset B_r. □

Lemma 3.5　 Assume assumptions (W), (f(iii)), and (g) hold. If semigroup \left\{T\right(t):t\in [0,h\left]\right\} is equicontinuous, and

(3.1)

then Γ is γ-contracted on B_r.

Proof　Assume W is a bounded set in B_r, it can be obtained with semigroup equicontinuity and Lemma 3.2 that Γ(W) is equicontinuous. From Lemma 2.1, Lemma 3.1, and Assumption (g),

\begin{array}{rl}\gamma \left(\mathrm{\Gamma }W\right)& \le \gamma \left({\mathrm{\Gamma }}_{1}W\right)+\gamma \left({\mathrm{\Gamma }}_{2}W\right)+\gamma \left({\mathrm{\Gamma }}_{3}W\right)\\ & =\underset{t\in [0,h]}{\mathrm{s}\mathrm{u}\mathrm{p}}\{\gamma (\{T\left(t\right)[y_0-g(y)]:y\in W\})\\ & +\gamma \left(\left\{{\int }_0^t T(t-s)\left[f\right(s,y\left(s\right))+B{u}_y(s)]\mathrm{d}s:y\in W\right\}\right)\}\\ & \le Qb{\gamma }_c\left(W\right)+\underset{t\in [0,h]}{\mathrm{s}\mathrm{u}\mathrm{p}} \gamma \left(\left\{{\int }_0^t T(t-s)f(s,y(s))\mathrm{d}s:y\in W\right\}\right)\\ & +\underset{t\in [0,h]}{\mathrm{s}\mathrm{u}\mathrm{p}} \gamma \left(\left\{{\int }_0^t T(t-s)B{u}_y\left(s\right)\mathrm{d}s:y\in W\right\}\right).\end{array}

By using Formula (2.1), for any \epsilon >0, there exists sequences \{y_k{\}}_{k=1}^{\mathrm{\infty }}\subset W and \left\{f\right(s,y_k\left(s\right)){\}}_{k=1}^{\mathrm{\infty }}\subset L^1([0,h],Y) satisfying

\begin{array}{rl}& \gamma \left(\left\{{\int }_0^t T(t-s)f(s,y(s))\mathrm{d}s:y\in W\right\}\right)\\ & \le 2\gamma \left(\left\{{\int }_0^t T(t-s)f(s,y_k(s))\mathrm{d}s:k\ge 1\right\}\right)+\epsilon \\ & \le \underset{t\in [0,h]}{\mathrm{s}\mathrm{u}\mathrm{p}} 2\gamma \left(\left\{{\int }_0^t T(t-s)f(s,y_k(s))\mathrm{d}s:k\ge 1\right\}\right)+\epsilon .\end{array}

Hence,

\gamma \left(\left\{{\int }_0^t T(t-s)f(s,y_k(s))\mathrm{d}s:k\ge 1\right\}\right)\le 2Q{\int }_0^t k\left(s\right)\gamma (\{y_k\left(s\right):k\ge 1\left\}\right)\mathrm{d}s,

(3.2)\begin{array}{rl}\gamma \left(\left\{{\int }_0^t T(t-s)f(s,y(s))\mathrm{d}s:y\in W\right\}\right)& \le \underset{t\in [0,h]}{\mathrm{s}\mathrm{u}\mathrm{p}} 4Q{\int }_0^t k\left(s\right)\gamma (\{y_k\left(s\right):k\ge 1\left\}\right)\mathrm{d}s+\epsilon \\ & \le \underset{t\in [0,h]}{\mathrm{s}\mathrm{u}\mathrm{p}} 4Q{\int }_0^t k\left(s\right)\gamma (W(s))\mathrm{d}s+\epsilon \\ & \le 4Q\Vert k{\Vert }_{L^1}{\gamma }_c\left(W\right)+\epsilon .\end{array}

Next, we estimate

\gamma \left(\left\{{\int }_0^t T(t-s)B{u}_y\left(s\right)\mathrm{d}s:y\in W\right\}\right).

By using Equation (2.1), there exists u_W^k\left(s\right)\in L^2([0,h],U) such that

\gamma \left(\left\{{\int }_0^t T(t-s)B{u}_y\left(s\right)\mathrm{d}s:y\in W\right\}\right)\le 2\gamma \left(\left\{{\int }_0^t T(t-s)B{u}_W^k\left(s\right)\mathrm{d}s:k\ge 1\right\}\right)+\epsilon .

Let

\xi :={\gamma }_{L^2}\left(\left\{u_W^k:k\ge 1\right\}\right).

Then for any {\xi }^{\prime }>\xi, there exists finite point \{v_1,v_2,\dots ,v_j\}\subset L^2([0,h],U), such that for any u_W^k\in L^2([0,h],U),

\Vert u_W^k-v_i{\Vert }_{L^2([0,h],U)}\le {\xi }^{\prime },i=1,2,\dots ,j\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}.

Hence,

\begin{array}{rl}\Vert {\int }_0^t T(t-s)B{u}_W^k\left(s\right)\mathrm{d}s-{\int }_0^t T(t-s)B{v}_i\left(s\right)\mathrm{d}s\Vert & \le {\int }_0^h Q{Q}_1\Vert u_W^k\left(s\right)-v_i\left(s\right)\Vert \mathrm{d}s\\ & \le \sqrt{h}Q{Q}_1\Vert v_i-u_W^k{\Vert }_{L^2([0,h],U)}\\ & \le \sqrt{h}Q{Q}_1{\xi }^{\prime }.\end{array}

According to the properties of noncompact meassures,

\begin{array}{rl}\gamma \left(\left\{{\int }_0^t T(t-s)B{u}_W^k\left(s\right)\mathrm{d}s:k\ge 1\right\}\right)& \le \sqrt{h}Q{Q}_1{\gamma }_{L^2}(\{u_W^k:k\ge 1\left\}\right)\\ & \le \sqrt{h}Q{Q}_1{\gamma }_{L^2}(\{u_y:y\in W\left\}\right).\end{array}

Given

u_y=P{\tilde{W}}^{-1}\{y_1-T(h)[y_0-g\left(y\right)]-{\int }_0^h T(h-\tau \left)f\right(\tau ,y\left(\tau \right)\left)\mathrm{d}\tau \right\},y\in W,

\begin{array}{rl}{\gamma }_{L^2}(u_y:y\in W)& \le 2Q_2\gamma \left(\left\{T\left(h\right)[x_0-g(y)]+{\int }_0^h T(h-s)f(s,y(s))\mathrm{d}s:y\in W\right\}\right)\\ & \le 2Q_{2}Qb{\gamma }_c\left(W\right)+4Q_{2}Q{\int }_0^h k\left(s\right)\gamma (\{y\left(s\right):y\in W\left\}\right)\mathrm{d}s\\ & \le 2Q_{2}Qb{\gamma }_c\left(W\right)+4Q_{2}Q\Vert k{\Vert }_{L^1}{\gamma }_c\left(W\right).\end{array}

So,

\gamma \left(\left\{{\int }_0^t T\left(t-s\right)B{u}_y^k\left(s\right)\mathrm{d}s:k\ge 1\right\}\right)\le 2\sqrt{h}{Q}_1{Q}_2{Q}^2\left(b+2\Vert k{\Vert }_{L^1}\right){\gamma }_c\left(W\right).

Hence,

(3.3)\gamma \left(\left\{{\int }_0^t T(t-s)B{u}_y\left(s\right)\mathrm{d}s:y\in W\right\}\right)\le 4\sqrt{h}{Q}_1{Q}_2{Q}^2(b+2\Vert k{\Vert }_{L^1}){\gamma }_c\left(W\right)+\epsilon .

By combining (3.2) and (3.3),

\begin{array}{rl}{\gamma }_c\left(\mathrm{\Gamma }W\right)& \le {\gamma }_c\left({\mathrm{\Gamma }}_{1}W\right)+{\gamma }_c\left({\mathrm{\Gamma }}_{2}W\right)+{\gamma }_c\left({\mathrm{\Gamma }}_{3}W\right)\\ & \le (Qb+4Q\Vert k{\Vert }_{L^1}+4\sqrt{h}{Q}_1{Q}_2{Q}^2(b+2\Vert k{\Vert }_{L^1})){\gamma }_c\left(W\right)+2\epsilon .\end{array}

It can be obtained with the arbitrariness of ε

{\gamma }_c\left(\mathrm{\Gamma }W\right)\le (Qb+4Q\Vert k{\Vert }_{L^1}+4\sqrt{h}{Q}_1{Q}_2{Q}^2(b+2\Vert k{\Vert }_{L^1})){\gamma }_c\left(W\right).

From Equation (3.1), it can be seen that Γ is strictly γ-contracted. □

Theorem 3.1　 Assume assumptions (W), (f), and (g) hold. If semigroup \left\{T\right(t):t\in [0,h\left]\right\} is equicontinuous, and Equation (3.1) is established, then Equation (1.1) is exactly controllable on the interval [0,h].

Proof　Obviously, the fixed point of operator Γ is the mild solution of Equation (1.1). Next, we will use Lemma 3.3 to prove the existence of fixed point to operator Γ. Firstly, by combining the conclusions of Lemma 3.4 and Lemma 3.5, it can be seen that B_r\to B_r is strictly γ-contracted.

Secondly, we prove is Γ a continuous operator. Let \{y_n{\}}_{n=1}^{+\mathrm{\infty }} be a sequence in C([0,h],Y) and \underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} y_n=y. It can be obtained from assumptions (f(i)) that for almost everywhere s\in [0,h], f(s,y_n(s)) converges to f(s,y(s)). Hence

\begin{array}{rl}& \Vert \left(\mathrm{\Gamma }{y}_n\right)-\mathrm{\Gamma }\left(y\right)\Vert \\ & \le Q\Vert g\left(y_n\right)-g\left(y\right)\Vert +Q{\int }_0^h \Vert f(s,y_n(s))-f(s,y(s))\Vert \mathrm{d}s+Q{Q}_1{\int }_0^h \Vert u_{y_n}\left(s\right)\\ & -u_y\left(s\right)\Vert \mathrm{d}s\\ & \le Q\Vert g\left(y_n\right)-g\left(y\right)\Vert +Q{\int }_0^h \Vert f(s,y_n(s))-f(s,y(s))\Vert \mathrm{d}s+Q{Q}_1\sqrt{h}\Vert u_{y_n}-u_y{\Vert }_{L^2}\\ & \le Q\Vert g\left(y_n\right)-g\left(y\right)\Vert +Q{\int }_0^h \Vert f(s,y_n(s))-f(s,y(s))\Vert \mathrm{d}s+2Q_1{Q}_2{Q}^2\sqrt{h}\\ & \times \left[\Vert g\left(y_n\right)-g\left(y\right)\Vert +{\int }_0^h \Vert f(s,y_n(s))-f(s,y(s))\Vert \mathrm{d}s\right].\end{array}

We can get with Assumption (g) and controlled convergence theorem that \underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} \mathrm{\Gamma }{y}_n=\mathrm{\Gamma }y in C([0,h],Y). Based on the above and Lemma 3.3, Γ has at least one fixed point y\in C([0,h],Y) on B_r, that is

\begin{array}{rl}y\left(t\right)=& T\left(t\right)[y_0-g(y)]+{\int }_0^t T(t-s)f(s,y(s))\mathrm{d}s+{\int }_0^t T(t-s)BP{\tilde{W}}^{-1}\\ & \times \left\{y_1-T\left(h\right)[y_0-g(y)]-{\int }_0^h T(h-\tau )f(\tau ,y(\tau ))\mathrm{d}\tau \right\}\left(s\right)\mathrm{d}s,t\in [0,h].\end{array}

To prove the exact controllability of Equation (1.1), according to Definition 2.2, we only need to prove y\left(h\right)=y_1 for the above u_y\in L^2([0,h],U). In fact, let

e:=y_1-T\left(h\right)[y_0-g(y)]-{\int }_0^h T(h-\tau )f(\tau ,y(\tau ))\mathrm{d}\tau ,\left[u_y\right]={\tilde{W}}^{-1}e.

Obviously

\tilde{W}\left[u_y\right]=e=W{u}_y={\int }_0^h T(h-s)B{u}_y\left(s\right)\mathrm{d}s,u_y\in \left[u_y\right].

Furthermore,

\begin{array}{rl}y\left(h\right)& =T\left(h\right)[y_0-g(y)]+{\int }_0^h T(h-s)f(s,y(s))\mathrm{d}s+{\int }_0^h T(h-s)B{u}_y\left(s\right)\mathrm{d}s\\ & =T\left(h\right)[y_0-g(y)]+{\int }_0^h T(h-s)f(s,y(s))\mathrm{d}s+y_1-T\left(h\right)[y_0-g(y)]\\ & -{\int }_0^h T(h-\tau )f(\tau ,y(\tau ))\mathrm{d}\tau \\ & =y_1,\end{array}

which indicates the exact controllability of Equation (1.1) on the interval[0,h]. □

References

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