Controllability of the semilinear wave equation governed by a multiplicative control

TL;DR

This paper investigates the approximate and exact controllability of semilinear wave equations using a single multiplicative control, providing constructive methods and explicit controls, with applications to one-dimensional systems.

Contribution

It introduces new controllability results for semilinear wave equations with multiplicative control, including explicit steering controls and uniform exact controllability in undamped cases.

Findings

Approximate controllability achieved with a single multiplicative control.

Explicit steering controls are constructed for the system.

Exact controllability established for one-dimensional undamped wave equations.

Abstract

In this paper we establish several results on approximate controllability of a semilinear wave equation by making use of a single multiplicative control. These results are then applied to discuss the exact controllability properties for the one dimensional version of the system at hand. The proof relies on linear semigroup theory and the results on the additive controllability of the semilinear wave equation. The approaches are constructive and provide explicit steering controls. Moreover, in the context of undamped wave equation, the exact controllability is established for a time which is uniform for all initial states.

Equations

$$\left\{\begin{array}[]{lll}w_{tt}=&\Delta w+v(x,t)w+f(t,w,w_{t}),&\hbox{in}\,\;\;\Omega\\ w=&0,&\hbox{on}\ \partial\Omega\\ w(x,0)=&w_{{}_{1}},\,w_{t}(x,0)=w_{{}_{2}},&\hbox{in}\,\Omega\\ \end{array}\right.$$

$$\phi (t)\le a+b{\int }_0^t\phi (s)ds,\text{mbox}fora.e.t\in [0,T]$$

$$\phi (t)\le a{e}^{bt},\text{mbox}fora.e.t\in [0,T].$$

$$B_{n}(u)(t)=\sum_{k=0}^{n}\left(\begin{array}[]{c}n\\ k\\ \end{array}\right)t^{k}(1-t)^{n-k}u(\frac{k}{n}),\;\;n\geq 1.$$

$$B_{n}(u)^{\prime}(t)=n\sum_{k=0}^{n-1}(\begin{array}[]{c}n-1\\ k\\ \end{array})t^{k}(1-t)^{n-1-k}(u(\frac{k+1}{n})-u(\frac{k}{n})),$$

$$\phi(x)=\left\{\begin{array}[]{ll}c\>e^{\frac{1}{\|x\|^{2}-1}},&\hbox{if}\;\|x\|<1\\ 0,&\hbox{if}\;\|x\|\geq 1\end{array}\right.$$

$$\begin{array}[]{cccc}k_{r}(x)&=&c\>r^{-n}\displaystyle\int_{B(x,r)}h(s)e^{\frac{1}{\|\frac{x-s}{r}\|^{2}-1}}ds\\ \\ &\leq&\displaystyle c\>\|h\|_{L^{\infty}(R^{n})}\>\displaystyle\int_{B(O,1)}ds.\end{array}$$

$$\left\{\begin{array}[]{llll}y_{t}&=&Ay+v(t)By+F(t,y),&t\in(0,T_{0})\\ \\ y(0)&=&y_{0}=(w_{1},w_{2})&\end{array}\right.$$

$$\Vert w(T)-w_1{\Vert }_{H_0^1(\Omega )}<ϵ\text{mbox}and\Vert w_t(T)-{\theta }_2{\Vert }_{L^2(\Omega )}<ϵ.$$

$$v(x,t)=v_T(x):=\frac{a(x)}{T},t\in (0,T_0).$$

$$y(t)=S(t)y_{0}+\int_{0}^{t}S(t-s)\big{(}v_{T}(x)By(s)+F(s,y(s))\big{)}ds,$$

$$y(T)-y^{d}=\int_{0}^{T}e^{(T-s)v_{T}(x)B}\big{(}Ay(s)+F(s,y(s))\big{)}ds,$$

$$A_{\lambda }y(t)=S(t)A_{\lambda }{y}_0+{\int }_0^t{A}_{\lambda }S(t-s)(v_T(x)By(s)+F(s,y(s)))ds.$$

$$\begin{array}[]{lll}A_{\lambda}y(t)&=&S(t)A_{\lambda}y_{0}+\int_{0}^{t}A_{\lambda}S(t-s)(v_{T}(x)By(s))ds+\\ &&\int_{0}^{t}\lambda R(\lambda,A)S^{\prime}(t-s)F(s,y(s))ds\end{array}$$

$$\begin{array}[]{lll}\int_{0}^{t}\lambda R(\lambda,A)S^{\prime}(t-s)F(s,y(s))ds&=&-\int_{0}^{t}\frac{d}{ds}\big{(}\lambda R(\lambda;A)S(t-s)F(s,y(s))\big{)}ds-\\ &&\int_{0}^{t}\lambda R(\lambda;A)S(t-s)\big{(}\frac{\partial F}{\partial s}(s,y(s))+\frac{\partial F}{\partial y}(s,y(s))y_{s}(s)\big{)}ds,\end{array}$$

$$\begin{array}[]{lll}\int_{0}^{t}\lambda R(\lambda,A)S^{\prime}(t-s)F(s,y(s))ds&=&\lambda R(\lambda;A)\big{(}F(t,y(t))-S(t)F(0,y_{0})\big{)}-\\ &&\int_{0}^{t}\lambda R(\lambda;A)S(t-s)\big{(}\frac{\partial F}{\partial s}(s,y(s))+\frac{\partial F}{\partial y}(s,y(s))y_{s}(s)\big{)}ds.\end{array}$$

$$\begin{array}[]{lll}\|A_{\lambda}y(t)\|&\leq&\|Ay_{0}\|+\int_{0}^{t}\|A(v_{T}(x)By(s))\|ds+\|F(0,y_{0})\|+\|F(t,y(t))\|+\\ \\ &&\int_{0}^{t}\|\frac{\partial F}{\partial s}(s,y(s))+\frac{\partial F}{\partial y}(s,y(s))y_{s}(s)\|ds\cdot\end{array}$$

$$\Vert y(t)\Vert \le C\Vert y_0\Vert +C,\forall t\in [0,T]\subset [0,1],$$

$$\Vert F(t,y(t))\Vert \le C(1+\Vert y_0\Vert ),C=C(\Vert a{\Vert }_{L^{\infty }(\Omega )})>0,$$

$${\int }_0^t\Vert \frac{\partial F}{\partial s}(s,y(s))+\frac{\partial F}{\partial y}(s,y(s))y_s(s)\Vert ds\le LT+L{\int }_0^t\Vert y_s(s)\Vert ds.$$

$$\begin{array}[]{lll}\|Ay(t)\|&\leq&\|Ay_{0}\|+\int_{0}^{t}\|A(v_{T}(x)By(s))\|ds+\\ &&C(1+\|y_{0}\|_{\mathcal{D}(A)})+L\int_{0}^{t}\|y_{s}(s)\|ds\end{array}$$

$$\Delta (aw))=\Delta (a)w+2\nabla (a)\cdot \nabla (w)+a\Delta (w),\forall w\in H^2(\Omega ),$$

$$\Vert A(v_T(x)By(s))\Vert =\frac{1}{T}\Vert \Delta (aw(s)){\Vert }_{L^2(\Omega )}\le \frac{C}{T}\Vert y(s){\Vert }_{\mathcal{D}(A)},\forall s\in [0,T]$$

$${\int }_0^t\Vert A(v_T(x)By(s))\Vert ds\le \frac{C}{T}{\int }_0^t\Vert y(s){\Vert }_{\mathcal{D}(A)}ds,\forall t\in [0,T]\cdot$$

$$\begin{array}[]{ccc}\|y_{t}(t)\|&=&\|Ay(t)+v_{T}(x)By(t)+F(t,y(t))\|\\ &\leq&\frac{C}{T}\|y(t)\|_{\mathcal{D}(A)}+T+\|F(0,0)\|\end{array}$$

$$\Vert y(t){\Vert }_{D(A)}\le C\Vert y_0{\Vert }_{\mathcal{D}(A)}+C,\forall t\in [0,T],$$

$$y(t)=e^{tv_{T}(x)B}y_{0}+\int_{0}^{t}e^{(t-s)v_{T}(x)B}\big{(}Ay(s)+F(s,y(s))\big{)}ds,\;\forall t\in[0,T],$$

$$y(T)-y^{d}=\int_{0}^{T}e^{(T-s)v_{T}(x)B}\big{(}Ay(s)+F(s,y(s))\big{)}ds.$$

$$\Vert y(T)-y^d\Vert \le C_{\ast }T(\Vert y_0{\Vert }_{\mathcal{D}(A)}+1),C_{\ast }=C_{\ast }(\Vert a{\Vert }_{W^{2,\infty }(\Omega )}),$$

$$F_{n}(t)=\sum_{k=0}^{n}\left(\begin{array}[]{c}n\\ k\\ \end{array}\right)t^{k}(1-t)^{n-k}F(\frac{k}{n},y(\frac{k}{n})),\;t\in[0,1]\;n\geq 1.$$

Full text

Controllability of the semilinear wave equation governed by a multiplicative

control

M. Ouzahra

MASI Team, University of Sidi Mohamed Ben Abdellah

mohamed.ouzahra@usmba.ac.ma

I.   Introduction

In this paper, we study the controllability problem for a distributed parameter system governed by the following $n-$dimensional wave equation:

[TABLE]

where $\Omega$ is a bounded open set of $I\!\!\!R^{n},\;n\geq 1$ with a smooth boundary $\partial\Omega$. The real valued coefficient $v(x,t)$ is the multiplicative control and $f$ is the nonlinearity. Our goal is to identify a set of states $(w(\cdot,t),w_{t}(\cdot,t))$ that can be achieved by system (1) at a time $T>0$ using a suitable control $v(x,t)$. Such problems arise in various real situations (see [21] and the rich references therein). Research in the multiplicative controllability of distributed systems have been the subject of several works. The question of controllability of PDEs equations by multiplicative controls has attracted many researchers in the context of various type of equations, such as rod equation [3, 23], Beam equation [7], Schr$\ddot{o}$dinger equation [6, 23, 29], heat equation [10, 11, 15, 16, 18, 21, 25, 31]. Various approaches were used to tackle the question of multiplicative controllability of hyperbolic equations like (1). The homogeneous version of (1) (i.e, $f=0$) has been considered in [3, 8, 19, 21, 30]. The case of semilinear wave equation has been studied in [20] for equilibrium-like states of the form $(y^{d}_{1},0)$ using two controls, i.e. beside the control $v(x,t)$, a time-dependent control has been considered in the damped part. Furthermore, research in the controllability of the semilinear wave equation by additive controls have been the subject of several works (see [26, 28, 35, 36] and the references therein).

In this paper, we study the approximate and exact controllability for the system (1) by the means of a single multiplicative control, thus we will have a principal reduction in the means to control the system (1).

The paper is organized as follows: in the second section, we first consider the question of reaching approximately target states of the form $(w(0),\theta_{2})$ by applying a suitable time-independent control $v(x,t)=v_{T}(x)$ at a "short" time $T$. In the second part of the same section, we define a set of target states $(\theta_{1},\theta_{2})$ that can be approximately achieved by using a piecewise static control in "long" time. In Section 3, we apply the result of Section 2 to define a strategy of the controller $v(x,t)$ in order to get the exact achievement of a class of target states for both damped and undamped cases.

II.   Approximate controllability

i.   Preliminaries

The following lemmas will be used in several steps in the proof of our main results.

The next result concerns a Gronwall inequality regarding locally integrable functions.

Lemma 1

(see [14, 34]). Let $\varphi(t)$ be a nonnegative and locally integrable function on $[0,T],\;0\leq T<+\infty,$ such that the inequality

[TABLE]

holds for some nonnegative constants $a$ and $b$. Then

[TABLE]

Let us give the following lemma which concerns the uniform approximation of continuous functions using Bernstein polynomials.

Lemma 2

([13], pp. 108-113). Let $u:[0,1]\to X$ be a continuous function from $[0,1]$ to a Banach space $(X,\|\cdot\|_{X})$, and let $B_{n}(u)$ be the $n$th Bernstein polynomial for $u$:

[TABLE]

*Then the sequence $B_{n}(u)$ tends uniformly to $u,$ i.e., $\sup_{t\in[0,1]}\|B_{n}(u)(t)-u(t)\|_{X}\to 0,$ as $n\to+\infty.$

Furthermore, for all $n\geq 1,$ we have:*

[TABLE]

where $B_{n}(u)^{\prime}(t)$ is the derivative of $B_{n}(u)(t)$ with respect to $t.$

Let us show the following smoothness lemma:

Lemma 3

Let $\Omega$ be an open bounded set of $R^{n},\;n\geq 1$. For all $h\in L^{\infty}(\Omega)$ such that $h\geq 0$, a.e. in $\Omega,$ there exists $(h_{r})\subset C^{\infty}(R^{n})$ such that:

*(i) $(h_{r}|_{\Omega})$ is uniformly bounded with respect to $r$, (where $h_{r}|_{\Omega}$ designs the restriction of $h_{r}$ to $\Omega$),

(ii) for all $r>0;\;h_{r}>0,\;$ a.e in $\overline{\Omega},$

and

(iii) $h_{r}|_{\Omega}\to h$ in $L^{2}(\Omega),$ as $r\to 0^{+}$.*

Proof 1

*Let us extend $h$ by [math] to $R^{n}$ so that the obtained extension, still denoted by $h$, lies in $L^{2}(R^{n})\cap L^{\infty}(R^{n})$.

Let us introduce the following function:*

[TABLE]

where $c$ is a positive constant such that: $\int_{R^{n}}\phi=1.$ For all $r>0$, let $\phi_{r}(x)=r^{-n}\phi(\frac{x}{r}),\;$ a.e. $x\in R^{n}$ and let $k_{r}$ be the convolution of $h$ with: $\phi_{r};\,k_{r}=\phi_{r}\ast h.$ This directly yields $k_{r}\in{\mathcal{C}}^{\infty}(R^{n}),\,k_{r}\geq 0$ a.e. in $\overline{\Omega}$ and $k_{r}\to h$ in $L^{2}(\Omega),$ as $r\to 0^{+}$ (see [9], pp. 69-71). Moreover, for every $r>0$ and for a.e. $x\in\Omega$, we have:

[TABLE]

In other word, the sequence $(k_{r})$ is uniformly bounded with respect to $r$. We conclude that $h_{r}:=k_{r}+\frac{r}{r+1},r>0$ satisfies the claimed properties.

ii.   A partial approximate controllability result

Let us consider the system (1) evolving on a time-interval $(0,T_{0})$ with a nonlinear term $f:(0,T_{0})\times H_{0}^{1}(\Omega)\times L^{2}(\Omega)\to L^{2}(\Omega)$ which is globally Lipschitz.

Letting $y=(w,w_{t})\in H:=H_{0}^{1}(\Omega)\times L^{2}(\Omega),$ we obtain the following equivalent first order system:

[TABLE]

where v(t)=v(\cdot,t)\in\mathcal{U}:=L^{\infty}(\Omega),\;B=\left(\begin{array}[]{cc}0&0\\ I&0\\ \end{array}\right) and A=\left(\begin{array}[]{cc}0&I\\ \Delta&0\\ \end{array}\right) with domain \mathcal{D}(A)=\big{(}H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\big{)}\times H_{0}^{1}(\Omega) and where for all $t\in(0,T_{0})$ and $y=(w_{1},w_{2})\in H;\;F(t,y)=(0,f(t,w_{1},w_{2})).$ Here, the state space $H$ is endowed with the following inner product: $\langle(u_{1},u_{2}),(v_{1},v_{2})\rangle=\langle u_{1},v_{1}\rangle_{H_{0}^{1}(\Omega)}+\langle u_{2},v_{2}\rangle_{L^{2}(\Omega)}$ with corresponding norm $\|\cdot\|.$ With this Hilbert structure, the operator $A$ generates a semigroup of isometries $S(t)$.

For any $\xi\in L^{2}(\Omega)$ we set: $\Lambda(\xi):=\{x\in\Omega/\;\xi(x)\neq 0\}$ and ${\bf 1}_{\Lambda(\xi)}$ will denote the characteristic function of $\Lambda(\xi).$

Our first main result concerns the approximate controllability toward a target state $w(T)=w_{1}$, $\partial_{t}w(T)=\theta_{2}$ within an arbitrarily small time-interval $(0,T)$, which depends on the choice of the initial state $y_{0}=(w_{1},w_{2})$, the target state $y^{d}=(w_{1},\theta_{2})$ and the precision of steering. The main idea here consists on looking for a static control such that the respective solution to (3) is such that $y(T)-y^{d}\to 0,$ as $T\to 0^{+}.$ This idea was first used by Khapalov in [18] in the context of reaction-diffusion equation (see also [11]).

Theorem 1

Let $(w_{1},w_{2})\in H$ and $\theta_{2}\in L^{2}(\Omega)$ and let us set $a(x):=\frac{\theta_{2}-w_{2}}{w_{1}}{\bf 1}_{\Lambda(w_{1})}.$ Assume that: (i) $a\in L^{\infty}(\Omega)$ and (ii) for a.e., $x\in\Omega;\;w_{1}(x)=0\Rightarrow\theta_{2}(x)=w_{2}(x).$ Then for any $\epsilon>0,$ there are a time $T=T(w_{1},w_{2},\theta_{2},\epsilon)\in(0,T_{0})$ and a static control $v(\cdot,t)=v_{T}(\cdot)\in W^{2,\infty}(\Omega)$ such that for the respective solution to (1), the following inequalities hold:

[TABLE]

Proof 2

Let $\epsilon>0$, and let us consider the state $y^{d}=(w_{1},\theta_{2})$ to be achieved. For any time of steering $0<T<T_{0}$ we consider the control

[TABLE]

Since $a\in L^{\infty}(\Omega),$ there is a unique mild solution $y(t)$ to (3) (see [32], p. 184), which is given by the following variation of constants formula:

[TABLE]

for all $t$ in $[0,T_{0}].$ We aim to show that the control (4) guarantees the steering of system (3) to $y^{d}$ at any small time $T>0$, so we can assume in the sequel that $0<T<T_{0}:=1$.

Case 1.* $a(\cdot)\in W^{2,\infty}(\Omega)$ and $y_{0}\in\mathcal{D}(A).$

We will distinguish two subcases:*

Case 1.1.* Assume that the operator $F$ is $\mathcal{C}^{1}$ and globally Lipschitz from $(0,T_{0})\times H$ to $H$. Here, the mild solution is a classical one. In particular we have $y(t)\in\mathcal{D}(A)=H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\times H_{0}^{1}(\Omega),\;\forall t\in[0,T_{0}]$ (see [32], p. 187).

It comes from the assumption (i) and from (4) that: e^{Tv_{T}B}=\left(\begin{array}[]{cc}I&0\\ a&I\\ \end{array}\right), so the assumption (ii) leads to: $e^{Tv_{T}B}y_{0}=y^{d}.$

The idea of the proof will consist on proving the following formula:*

[TABLE]

*and showing that the term in the right-hand side of the relation (6) tends to zero as $T\to 0^{+}.$

In order for $y(t)$ to satisfy (6), it suffices to show that $Ay(\cdot)\in L^{1}(0,1)$ (see [4]). For this end, let us apply the bounded operator $A_{\lambda}=\lambda R(\lambda;A)A$ to (5), where $R(\lambda;A)$ is the resolvent of $A$. Thus*

[TABLE]

This gives

[TABLE]

where $S^{\prime}(t)$ is the derivative of $S(t)$ with respect to $t$. We have

[TABLE]

where $y_{s}$ refers to the derivative w.r.t "$s$". Thus

[TABLE]

Using the fact that $S(t)$ is a contraction semigroup, we deduce that:

[TABLE]

Moreover, using (5), we deduce via Gronwall’s inequality that:

[TABLE]

*for some positive constant $C=C(\|a\|_{L^{\infty}(\Omega)})>0$ which is independent of $T$.

Then using the fact that $F$ is Lipschitz we get:*

[TABLE]

and

[TABLE]

Then, letting $\lambda\to+\infty,$ we deduce that:

[TABLE]

*where $L$ is a Lipschitz constant of $F$ and the constant $C=C(\|a\|_{L^{\infty}(\Omega)})>0$ is independent of $T$.

In the sequel, the letter $C$ will be used to denote a generic positive constant which is independent of $T$.

Let us now study the terms of the right hand of inequality (10). We have $v_{T}(x)By(t)=(0,\frac{a(x)}{T}w(t)),$ thus since $a\in W^{2,\infty}(\Omega)$ it comes that $v_{T}(x)By(t)\in\mathcal{D}(A)$ for all $t\in[0,T].$ Moreover, we have the following second order Leibniz rule:*

[TABLE]

from which we get:

[TABLE]

where $C=C(\|a\|_{W^{2,\infty}(\Omega)})$ is independent of $T.$ It follows that:

[TABLE]

Since $y(t)$ is a classical solution, we have

[TABLE]

*for all $0<t\leq T,$ where $C=C(\|a\|_{L^{\infty}(\Omega)})$.

Reporting (11) and (12) in (10) and taking into account (8), we deduce via Gronwall’s inequality that:*

[TABLE]

where $C=C(\|a\|_{W^{2,\infty}(\Omega)})$ is independent of $T.$ Thus $Ay(\cdot)\in L^{1}(0,T)$, and hence the following variation of constants formula holds:

[TABLE]

from which it comes

[TABLE]

Based on (15) and using (13) and the fact that $F$ is Lipschitz, we deduce that:

[TABLE]

and hence $\|y(T)-y^{d}\|<\epsilon$, whenever $0<T<\inf({T_{0}=1,\frac{\epsilon}{C_{*}(1+\|y_{0}\|_{\mathcal{D}(A)})}})$.

*Case 1.2. *** Here, we only assume that the operator $F$ is globally Lipschitz from $(0,T_{0})\times H$ to $H$ (with a Lipschitz constant $L>0$), and let $y(t)$ be the mild solution of (3) corresponding to control $v_{T}(x)$ given by (4). Then we can approximate the function $t\to F(t,y(t))$ uniformly with $\mathcal{C}^{1}-$functions $(F_{n})$ in $[0,T_{0}]=[0,1]$. More precisely, according to Lemma 2, one can consider the following Bernstein polynomial:

[TABLE]

From (9) we get:

[TABLE]

Moreover, for all $n\geq 1$ we have:

[TABLE]

*where $F^{\prime}_{n}(t)$ is the derivative of $F_{n}(t)$.

Let us show that the sequence of derivative $(F^{\prime}_{n})$ is uniformly bounded in $[0,T]$.

For all $h,t\in[0,T]$ such that $t+h\in[0,T],$ we have:*

[TABLE]

from which, we derive:

[TABLE]

where $C=C(\|a\|_{L^{\infty}(\Omega)})$ is independent of $T$, which by Gronwall’s inequality gives the following estimate:

[TABLE]

*where $C=C(\|a\|_{L^{\infty}(\Omega)})$ is independent of $T$.

It follows from the expression of $F^{\prime}$ and the last inequality that:*

[TABLE]

*As a consequence, $F_{n}$ is $L_{T}-$Lipschitz on $[0,T]$.

In the sequel, we will apply the techniques of Case 1.1 to the following approached system:*

[TABLE]

*Let $y_{n}(t)$ denote the classical solution of the system (18).

Based on the variation of constants formula, we can show via the Gronwall’s inequality that there is $N=N(T,\epsilon)\in\mathbf{I\!N}$ such that:*

[TABLE]

Moreover, applying the relation (7) to $y_{n}(t)$ leads to:

[TABLE]

We have

[TABLE]

Then we deduce that:

[TABLE]

Letting $\lambda\to+\infty$, we get

[TABLE]

where $C=C(\|a\|_{L^{\infty}(\Omega)})$ is a positive constant which is independent of $T.$ Then by proceeding as in the Case 1.1., we get an estimate like (16), namely:

[TABLE]

*where $C=C(\|a\|_{W^{2,\infty}(\Omega)})>0$ is independent of $N.$

It follows that $\|y_{N}(T)-y^{d}\|<\epsilon/2,$ for some $T$ small enough, and hence*

[TABLE]

**Case 2. *** $a(\cdot)\in W^{2,\infty}(\Omega)$ and $y_{0}\in H.$

Let $T>0,$ and for all $\lambda>0$ we set $y_{0\lambda}:=\lambda R(\lambda;A)y_{0}\in\mathcal{D}(A).$ Let $y_{\lambda}$ be the mild solution of (3) corresponding to the initial state $y_{0\lambda}=(w_{1\lambda},w_{2\lambda})$ with the same control as in the Case 1., i.e., $v(x,t)=v_{T}(x)=\frac{a}{T},\;0<t<T_{0}$.

We have*

[TABLE]

It follows from the variation of constants formula that:

[TABLE]

Then, using the contraction property of the semigroup $S(t)$, it comes:

[TABLE]

Gronwall’s lemma yields

[TABLE]

It follows from $y^{d}=e^{aB}y_{0}$ that:

[TABLE]

We deduce that there is a $\lambda>0$, which is independent of $T\in(0,1),$ such that:

[TABLE]

*For such a $\lambda,$ we deduce from the same arguments as in the Case 1 that there exists $0<T<1$ such that: $\|y_{\lambda}(T)-e^{aB}y_{0\lambda}\|<\frac{\epsilon}{2}.$

We conclude that:*

[TABLE]

**Case 3: *** $a(\cdot)\in L^{\infty}(\Omega)$ and $y_{0}\in H.$

From Lemma 3, there is a sequence $(a_{k})\subset W^{2,\infty}(\Omega)$ which is uniformly bounded on $\Omega$ such that $a_{k}\to a$ in $L^{2}(\Omega),$ as $k\to+\infty.$ Here, we will consider the control: $v_{T}(x)=\displaystyle\frac{a_{k}}{T}$ for a suitably selected (large enough) $k\in\mathbf{N}$, and let $y(t)$ be the corresponding solution to (3) with the initial state $y(0)=y_{0}=(w_{1},w_{2}).$

Now, let $(w_{2l})\in L^{\infty}(\Omega)$ be such that $w_{2l}\to w_{2}$ in $L^{2}(\Omega),$ as $l\to+\infty,$ and let us consider the initial state $y_{0l}=(w_{1},w_{2l}).$

We have the following triangular inequality:*

[TABLE]

From the relation e^{a_{k}B}=\left(\begin{array}[]{cc}I&0\\ a_{k}&I\\ \end{array}\right), we deduce that:

[TABLE]

and $e^{a_{k}B}y_{0l}-e^{aB}y_{0l}=(0,(a_{k}-a)w_{2l}).$ Let $l\in\mathbf{N}$ be such that

[TABLE]

and for such value of $l,$ we consider a $k$ such that

[TABLE]

Then, for this value of $k$, it comes from the Case 2 that there exists $T>0$ such that:

[TABLE]

We conclude that

[TABLE]

Finally, since $e^{aB}y_{0}=y_{d},$ it comes

[TABLE]

Remark 1

For any initial state $(w_{1},w_{2}),$ the set of reachable states $\theta_{2}$ identified in the above theorem is convex.

iii.   Global approximate controllability

In this subsection, we will consider the following equation:

[TABLE]

where $T>0,\;h\in L^{\infty}(\Omega)$ and the nonlinear term $f:L^{2}(\Omega)\to L^{2}(\Omega)$ is a globally Lipschitz function. Here, we will study the approximate controllability problem for the system (22) toward a full state $(\theta_{1},\theta_{2})$ by using two static controls, applied subsequently in time.

For any $\zeta\in H^{2}(\Omega)$ we set $b_{\zeta}:=-\displaystyle\frac{\Delta\zeta+f(\zeta)}{\zeta}{\bf 1}_{\Lambda(\zeta)},$ and let us consider the following assumptions:

$(\mathcal{P}_{1})$: $b_{\zeta}\in L^{\infty}(\Omega),$

$(\mathcal{P}_{2})$: $h\geq 0,\;a.e.\;\Omega$. and there exist $\delta,\mathcal{T}>0$ such that:

[TABLE]

where $\varphi$ is the solution of

[TABLE]

$(\mathcal{P}_{3})$: $\langle f(y)+b_{\zeta}(x)y,y\rangle_{L^{2}(\Omega)}\leq 0,\;\forall y\in L^{2}(\Omega),$

$(\mathcal{P}_{4})$: for a.e. $x\in\Omega,$ we have: $\zeta(x)=0\Rightarrow f(\zeta)(x)=0.$

We have the following remarks regarding the estimate (23).

Remark 2

For $f=b_{\zeta}=0,$ the inequality (23) was established for $\mathcal{T}$ large enough provided there is a subset $O$ of the support of $h$ satisfying the following so-called geometrical control condition (GCC): "there exists $x_{0}\in\mathbf{R}^{n}$ such that $O$ is a neighborhood of the closure of the set $\Gamma(x_{0}):=\{x\in\partial\Omega/\;(x-x_{0})\nu(x)>0\}$", where $\nu(x)$ denotes the unit outward normal at $x\in\partial\Omega$ (see **[5]**). In particular, for $n=1$ and $g={\bf 1}_{\omega}$ the estimate (23) holds for $\omega=(a,b)\subset\Omega=(0,1)$ and $\mathcal{T}>2\inf(a,1-b)$ (see **[36]**). 2. 2.

Using robustness results on the observability property (see **[30]**), we can see that (23) also holds under the geometrical control condition for small Lipschitz constant $L_{\zeta}$ of the operator: $y\in L^{2}(\Omega)\mapsto f(y)+b_{\zeta}(x)y$. Indeed, let the above (GCC) hold, so that:

[TABLE]

(for some $\mathcal{T},\delta>0$), where $\phi$ is the solution of $\phi_{tt}=\Delta\phi,\phi(0)=\phi_{1}\in H_{0}^{1}(\Omega),\phi_{t}(0)=\phi_{2}\in L^{2}(\Omega).$ We can easily show that the solution $T(t)y_{0}=(\varphi(t),\varphi_{t}(t)),\;y_{0}=(\varphi_{1},\varphi_{2})$ of (24) verifies $\|T(t)y_{0}\|\leq e^{L_{\zeta}t}\|y_{0}\|,\;\forall t\geq 0.$ Then using the variation of constants formula, we get:

[TABLE]

where C=\left(\begin{array}[]{cc}0&0\\ 0&h(x)I\\ \end{array}\right). From this and (25) it comes:

[TABLE]

Hence the estimate (23) holds whenever $L_{\zeta}<\gamma^{-1}(\delta),$ where $\gamma^{-1}$ is the inverse function of $\gamma:s\mapsto\|h\|_{\infty}\mathcal{T}^{2}se^{s\mathcal{T}}(1+e^{s\mathcal{T}}).$ 3. 3.

An other situation in which (23) holds is the case of functions: $f(y)(x)=k(y(x)),$ where $k:R\to R$ is such that $k(0)=0$ and $sk(s)\leq-s^{2}\|b_{\zeta}\|_{L^{\infty}(\Omega)},\;\forall s\in R$ (see **[33]**).

The following result concerns the approximate controllability toward target states of the form $(\theta_{1},0)$.

Theorem 2

Let $\theta_{1}\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$ and let assumptions $(\mathcal{P}_{1})-(\mathcal{P}_{4})$ hold for $\zeta=\theta_{1}$. Then for every initial state $(w_{0},w_{1})\in H$ and for every $\epsilon>0,$ there are a time $T=T(w_{1},w_{2},\theta_{1},\epsilon)>0$ and a piecewise static control $v(\cdot,t)=v_{T}(\cdot)\in L^{\infty}(\Omega)$ such that the respective solution to (22) satisfies:

[TABLE]

Proof 3

Let $\epsilon>0$ be fixed. Let us consider the control:

[TABLE]

and let us set $z=w-\theta_{1}$. Thus the system (22) takes the form:

[TABLE]

*where $z_{1}=w_{1}-\theta_{1}$ and $z_{2}=w_{2}.$

We have: $\Delta\theta_{1}-(\Delta\theta_{1}/\theta_{1}){\bf 1}_{\Lambda(\theta_{1})}\theta_{1}=0$ (see [2, 30]). Then, using this and the fact that for almost every $x$ in $\Omega;\ :\theta_{1}(x)=0\Rightarrow f(\theta_{1})(x)=0,$ the system (28) (controlled with (27)) becomes:*

[TABLE]

*where $f_{\theta_{1}}(z)=f(z+{\theta_{1}})-f({\theta_{1}}),\;z\in L^{2}(\Omega)$.

Let $\vartheta$ be a solution of the system:*

[TABLE]

Then, remarking that $\varphi:=\vartheta+\theta_{1}$ satisfies the equation: $\varphi_{tt}=\Delta\varphi+b_{\theta_{1}}(x)\varphi+f(\varphi),$ we deduce by taking $\varphi(0)=\vartheta_{1}$ and $\varphi_{t}(0)=\vartheta_{2}$ in (24) that:

[TABLE]

Moreover, since $f_{\theta_{1}}$ is Lipschitz and satisfies $f_{\theta_{1}}(0)=0$ and $\langle f_{\theta_{1}}(z)+b_{\theta_{1}}(x)z,z\rangle\leq 0,$ for all $z\in L^{2}(\Omega),$ we deduce that the solution of (29) can be defined for all $t\geq 0$ and satisfies the following exponential decay (see [33]):

[TABLE]

*for some constants $M,\sigma>0$ which are independent of $T$.

We deduce that for $T>\frac{1}{\sigma}|\ln(\frac{M\|(w_{1}-\theta_{1},w_{2})\|}{\epsilon})|$, the solution of (22) satisfies the following estimate:*

[TABLE]

Our main result in this section concerns the case of a full target state $\theta=(\theta_{1},\theta_{2})$ and is stated as follows:

Theorem 3

Let $(\theta_{1},\theta_{2})\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\times L^{2}(\Omega)$ be such that: $d(x):=\frac{\theta_{2}}{\theta_{1}}{\bf 1}_{\Lambda(\theta_{1})}\in L^{\infty}(\Omega)$ and that for a.e. $x\in\Omega,$ we have $\theta_{1}(x)=0\Rightarrow\theta_{2}(x)=0.$ We further assume that assumptions $(\mathcal{P}_{1})-(\mathcal{P}_{4})$ hold for $\zeta=\theta_{1}$. Then for every initial state $(w_{1},w_{2})\in H$ and for every $\epsilon>0,$ there are a time $T=T(w_{1},w_{2},\theta_{1},\theta_{2},\epsilon)>0$ and a piecewise static control $v(\cdot,t)=v_{T}(\cdot)\in L^{\infty}(\Omega)$ such that the respective solution to (22) satisfies:

[TABLE]

Proof 4

From Theorem 2, we deduce that for any $\epsilon>0$ there is a time $T_{1}=T_{1}(w_{1},w_{2},\theta_{1})$ such that the control: $v(x,t)=v_{1}(x)=-\frac{\Delta\theta_{1}+f(\theta_{1})}{\theta_{1}}{\bf 1}_{\Lambda(\theta_{1})},\;t\in(0,T_{1})$ guarantees the following estimate for the corresponding solution of (22):

[TABLE]

We will continue to control our initial system (22) on $(T_{1},T)$ until the achievement of the full target state $\theta=(\theta_{1},\theta_{2})$, where $T>T_{1}$ is to be determined. Consider the following system:

[TABLE]

We will use Theorem 1 to reach $(w(T_{1}^{-}),\theta_{2})$ at a time $T>T_{1}$ which is close to $T_{1}.$ For this end, let us observe that by virtue of (34) the system (35) can be approximated by the following one:

[TABLE]

*Applying Theorem 1 to system (36), we deduce the existence of a static control $v(x,t)=v_{2}(x)\in L^{\infty}(\Omega)$ such that the corresponding state $(\tilde{w}(T),\tilde{w}_{t}(T))$ is close to $(\theta_{1},\theta_{2})$ at some $T=T(\theta_{1},T_{1})$ which is sufficiently close to $T_{1}^{+}$.

Using the same control $v(x,t)=v_{2}(x)$ for (35), we can see by Gronwall’s inequality and the variation of constants formula that:*

[TABLE]

where $C=L+\|v_{2}\|_{L^{\infty}(\Omega)}$ and $L$ is a Lipschitz constant of the function $F:H\rightarrow H;\;y=(w_{1},w_{2})\mapsto(0,f(w_{1})-hw_{2})$. Thus

[TABLE]

*whenever $0<T-T_{1}<1.$ We deduce that (33) holds.

We conclude that the initial system (22) can be approximately steered to $(\theta_{1},\theta_{2})$ at $T$ by using the control:*

[TABLE]

This completes the proof.

Remark 3

According to the proof of Theorem 1, we can see that in the case: $\frac{\theta_{2}}{\theta_{1}}{\bf 1}_{\Lambda(\theta_{1})}\in W^{2,\infty}(\Omega)$, one can take the control $v_{2}(x)=\frac{\theta_{2}}{(T-T_{1})\theta_{1}}{\bf 1}_{\Lambda(\theta_{1})}$ in the time-interval $(T_{1},T).$

III.   Exact controllability

In this section, we study the set of target states that can be exactly achieved at a finite time by the system (22) for $n=1$. The idea in this part consists first, thanks to the continuity of the Sobolev embedding $H^{1}(\Omega)\hookrightarrow\mathcal{C}^{0}(\overline{\Omega})$ for $n=1$, in applying the results of Section 2 in order to make the state closer to the desired one at a time $T_{1}$ with respect to $L^{\infty}-$norm. Then one can exploit the results of the exact additive controllability of semilinear wave equation to construct a time $T$ and a control $v(x,t)$ on $(T_{1},T)$ that guarantee the exact steering of the target state at $T$.

In this section, we take $n=1$ and $\Omega=(0,l),\;l>0$.

i.   Damped case

i.1 The case of homogeneous boundary conditions

In this part, we will study the exact controllability of the one dimensional version of the equation (22) evolving in a time-interval $(0,T)$.

For any $\zeta\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$ and $0<t_{0}<T$, we consider the following system:

[TABLE]

where $O$ is a sub-domain of $\Omega$, and let us consider the following property:

$(\mathcal{P}_{5}):$ For every $t_{0}>0$, the system (38) is exactly null controllable at some time $T>t_{0}$ with a control $u(x,t)$ satisfying

[TABLE]

where $c_{{}_{T-t_{0}}}>0$ is a constant depending on $T-t_{0}$.

We refer the reader to [17, 26, 28, 35, 36, 37, 38] for some results on the exact controllability problem for equations like (38).

We are ready to state our first main result of this section.

Theorem 4

*Let $n=1,\;(w_{1},w_{2})\in H_{0}^{1}(\Omega)\times L^{2}(\Omega)-\{(0,0)\}$ and let $\theta_{1}\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$ be such that: $\theta_{1}\neq 0,$ a.e in $\overline{O}$ for some open subset $O$ of $\Omega.$ Assume that assumptions $(\mathcal{P}_{1})-(\mathcal{P}_{5})$ hold for $\zeta=\theta_{1}$.

Then there exist $T=T(w_{1},w_{2},\theta_{1})>0$ and a control $v(\cdot,\cdot)\in L^{2}(0,T;L^{2}(\Omega))$ such that the respective solution to the system (22) satisfies $w(T)=\theta_{1}$ and $w_{t}(T)=0.$*

Proof 5

Let us set $z=w-\theta_{1}$ in the system (22). We have:

[TABLE]

Then for any fixed $0<\epsilon<1,$ it comes from Theorem 2 that there is a time $T_{1}>0$ (large enough) such that the control defined by $v(x,t)=b_{\theta_{1}}=-\frac{\Delta\theta_{1}+f(\theta_{1})}{\theta_{1}}{\bf 1}_{\Lambda(\theta_{1})}$ guarantees the following estimate:

[TABLE]

Let $T>T_{1}$, and let us consider the following system

[TABLE]

*where $u(x,t)$ is an additive control.

By assumption, there exists $T>T_{1}$ and $u(\cdot)\in L^{2}(T_{1},T;L^{2}(\Omega))$ satisfying (39) and such that the respective solution to system (42) satisfies: $(\psi(T),\psi_{t}(T))=(0,0)$. Then, in order to construct a control that steers (22) to $(\theta_{1},0)$, it suffices to look for a control $v(x,t)=v_{1}(x,t)+b_{\theta_{1}}(x)$ on $(T_{1},T)$ such that:*

[TABLE]

For this purpose, we will show that $\psi(x,t)+\theta_{1}(x)\neq 0,$ a.e. $x\in O\times(T_{1},T),$ and then take for $t\in(T_{1},T)$:

[TABLE]

The solution of (42) satisfies the following integral formula:

[TABLE]

Since $f_{\theta_{1}}$ is Lipschitz and vanishes at [math], we deduce from the formula (43) and by using (39) and (41) that:

[TABLE]

which by using the Gronwall’s inequality gives:

[TABLE]

and so

[TABLE]

This together with the continuity of the embedding for $n=1$ $H_{0}^{1}(\Omega)\hookrightarrow L^{\infty}(\Omega)$ (see e.g. [1]) gives:

[TABLE]

*Moreover, since $\theta_{1}\neq 0,$ a.e in $\overline{O},$ we deduce from the fact that the embedding $H^{1}(O)\hookrightarrow\mathcal{C}^{0}(\overline{O})$ is continuous (recall that $n=1$) that $|\theta_{1}|\geq\mu>0,$ a.e. in $O$.

Then, taking $0<\epsilon<\displaystyle\frac{\mu}{2C}$ in (44), we deduce that for all $t\in(T_{1},T)$ we have*

[TABLE]

Then, one can choose the control $v_{1}$ as follows:

[TABLE]

*From (45) and the fact that $u\in L^{2}(T_{1},T;L^{2}(\Omega))$, it comes that $v_{1}\in L^{2}(T_{1},T;L^{2}(\Omega))$.

With this control, the system (40) becomes:*

[TABLE]

*It is obvious that $\psi$ is a solution of (47). Let us show that this is the unique one.

Let $z\in H_{0}^{1}(\Omega)$ be a solution of (47). The H$\ddot{o}$lder’s inequality leads to:*

[TABLE]

[TABLE]

*for some constant $C>0.$

This together with (45) and the variation of constants formula enables us to establish the following inequality:

$\|(z,z_{t})(t)-(\psi,\psi_{t})(t)\|\leq$**

[TABLE]

*for some constants $C_{1},C_{2}>0.$ As a consequence we have $z(t)-\psi(t)=0$ for all $t\in[T_{1},T].$ Then solution of the system (40) is such that $z(T)=0$ and $z_{t}(T)=0$ and hence $w(T)=\theta_{1}$ and $w_{t}(T)=0$.

We conclude that the control defined by:*

[TABLE]

steers the system (22) from the initial state $(w_{{}_{1}},w_{{}_{2}})$ to the desired one $(\theta_{1},0)$ at $T.$

i.2 The case of nonhomogeneous boundary conditions

Here, we intend to study the possibility of achieving a full state $\theta=(\theta_{1},\theta_{2})$ for the following one dimensional system with nonhomogeneous Dirichlet boundary conditions:

[TABLE]

with the same assumptions as in (22), and $\sigma_{1},\sigma_{2}\in\mathbf{R}.$

For any $\zeta\in H^{2}(\Omega)$ satisfying the compatibility condition $\zeta|_{\partial\Omega}=\sigma:=(\sigma_{1},\sigma_{2})$, we consider the following system with additive globally distributed control:

[TABLE]

where $0<t_{0}<T$.

In the sequel, we will consider the case of exact steering of (49) from an initial state $\psi_{0}=(\psi_{1},\psi_{2})$ to a target state $\psi_{d}=(\psi_{1}^{d},\psi_{2}^{d})$ under a control $u\in L^{2}(t_{0},T;L^{2}(\Omega))$ that satisfies the following bound inequality with respect to initial and target states:

[TABLE]

where $c_{{}_{T-t_{0}}}>0$ is a bounded function of $T-t_{0}$ and C\big{(}\|\psi_{0}\|_{{H}},\|\psi_{d}\|_{{H}}\big{)}>0 is a function of $\|\psi_{0}\|_{{H}}$ and $\|\psi_{d}\|_{{H}}$. In the next theorem, we will consider the case where $t_{0}$ and $T$ are close to each other, which may be linked to the question of exact controllability in short time (see [12, 22, 27]). Note that, since the control acts in all of $\Omega$, the exact controllability of (49) holds in any time $T>0$. This may be deduced from the case of the linear version of (49) (i.e. $f_{\zeta}=0$) and the fact that, in the case of globally distributed control, the nonlinearity can be suppressed in a trivial way.

We will again proceed as in the case of homogeneous boundary conditions, but here we need to use an auxiliary $\zeta\in H^{2}(\Omega)$ such that $w-\zeta$ is the solution of a system like (49) with the condition that $\zeta\neq 0$ a.e. on $\overline{\Omega}.$ This is why we deal with nonhomogeneous boundary conditions. Moreover, unlike the case of homogeneous BC, here the estimate (50) involves the term $\|\psi_{d}\|_{{H}}$, so we require more than the null exact controllability of the auxiliary system (49).

For any $\zeta\in H_{\sigma}^{2}(\Omega):=\{\zeta\in H^{2}(\Omega):\;\zeta|_{\partial\Omega}=\sigma\}$, we consider the following assumption:

$(\mathcal{P}_{6}):\;$ The system (49) is exactly controllable at any $T>t_{0}>0$ large enough, with a control satisfying (50).

Let us also introduce the affine space $H_{\sigma}^{1}(\Omega):=\{\zeta\in H^{1}(\Omega):\;\zeta|_{\partial\Omega}=\sigma\}$.

We have:

Theorem 5

Let $n=1,\;(w_{1},w_{2})\in H_{\sigma}^{1}(\Omega)\times L^{2}(\Omega)$ and let $(\theta_{1},\theta_{2})\in H_{\sigma}^{1}(\Omega)\times L^{2}(\Omega).$ If assumptions $(\mathcal{P}_{1})-(\mathcal{P}_{4})$ and $(\mathcal{P}_{6})$ hold for some $\zeta\in H_{\sigma}^{2}(\Omega)$ such that $\zeta\neq 0,$ a.e in $\overline{\Omega}=[0,l],$ then there exist a time $T=T(w_{1},w_{2},\zeta)>0$ and a control $v(\cdot,\cdot)\in L^{2}(0,T;L^{2}(\Omega))$ such that the corresponding solution of the system (48) satisfies $(w(T),w_{t}(T))=(\theta_{1},\theta_{2}).$

Proof 6

Consider the following system:

[TABLE]

For any fixed $0<\epsilon<1,$ it comes from the proof of Theorem 2 that there is a time $T_{1}>0$ (large enough) such that the control defined by $v(x,t)=b_{\zeta}=-\frac{\Delta\zeta+f(\zeta)}{\zeta}{\bf 1}_{\Lambda(\zeta)}$ guarantees the following estimate:

[TABLE]

Let $T>T_{1}$, and let us consider the following additive-control system:

[TABLE]

*where $u(x,t)$ is the additive control.

By assumption, there exists $u(\cdot)\in L^{2}(T_{1},T;L^{2}(\Omega))$ satisfying (50) and is such that the respective solution to system (53) satisfies: $(\psi(T),\psi_{t}(T))=(\theta_{1}-\zeta,\theta_{2})$.

Then, in order to construct a control that steers (48) to $(\theta_{1},\theta_{2})$, it suffices to build a control $v(x,t)=v_{1}(x,t)+b_{\zeta}(x)$ on $(T_{1},T)$ such that:*

[TABLE]

which may be done as in the proof of Theorem 4 by observing (thanks to estimate (50)) that the additive steering control $u(x,t)$ satisfies:

[TABLE]

*whenever $T$ is sufficiently close to $T_{1}$.

Hence the respective solution to (53) satisfies the estimate:*

[TABLE]

This enables us to show that $\psi(t)+\zeta\neq 0$, a.e. in $\Omega,$ so that, one can consider the control $v_{1}$ defined by:

[TABLE]

which renders the system (51) equivalent to the following one:

[TABLE]

*whose unique solution is the same as the one of (53). In other words, $z=\psi$. Then the solution of the system (51) is such that $z(T)=\theta_{1}-\zeta$ and $z_{t}(T)=\theta_{2}.$

Let us now set: $w=z+\zeta.$ Then $w$ is the unique solution of the system (48) and we have $(w(T),w_{t}(T))=(\theta_{1},\theta_{2})$. We conclude that the control defined by:*

[TABLE]

guarantees the exact steering of the system (48) from the initial state $(w_{{}_{1}},w_{{}_{2}})$ to $(\theta_{1},\theta_{2})$ at $T.$

Remark 4

If the assumptions of Theorem 5 hold for $\zeta=\theta_{1},$ then the exact controllability required for the system (49) can be restricted to target states of the form: $\psi_{d}=(0,\psi_{2}^{d})$.

ii.   Undamped case

In this subsection, we will establish an exact controllability result for an uniform time $T$ when dealing with undamped equation. We consider the following one dimensional undamped equation:

[TABLE]

where $f$ is globally Lipschitz. In the context of additive controls, Zuazua [36] has considered the exact internal controllability of the one dimensional version of following semilinear system:

[TABLE]

The multidimensional case has been treated in [35].

For any $\zeta\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$ and $0<t_{0}<T$, we consider the following system:

[TABLE]

which we will assume to be exactly null controllable at $T>t_{0}$ for $t_{0}$ small enough and $T$ large enough (i.e. for $t_{0}<\alpha<T$ for some $\alpha>0$) with controls $u\in L^{2}(t_{0},T;L^{2}(O))$ such that:

[TABLE]

for some positive constant $C=C_{T}$ depending only on $T,$ for $t_{0}>0$ small enough.

The next theorem states our second main result of this section.

Theorem 6

Let $n=1,\;(w_{1},w_{2})\in H_{0}^{1}(\Omega)\times L^{2}(\Omega)-\{(0,0)\},$ and let $\theta_{1}\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$ be such that $\theta_{1}(x)\neq 0,$ a.e. $x\in\overline{O}$ for some open subset $O$ of $\Omega$ and $b_{\theta_{1}}\in L^{\infty}(\Omega)$.

If for some $\alpha>0,$ the target state $\theta:=(\theta_{1},0)$ is approximately reachable at a small time $0<T_{1}<\alpha$ with a control $v_{T_{1}}\in L^{2}(0,T_{1};L^{2}(\Omega))$, and if the system (59) is exact null controllable at $T>\alpha$ for $\zeta=\theta_{1}$ with a control $u$ satisfying (60), then there exists a control $v\in L^{2}(0,T;L^{2}(\Omega))$ such that the corresponding solution of (57) satisfies: $w(T)=\theta_{1}$ and $w_{t}(T)=0.$

Proof 7

Let $T>\alpha$, and let us set $z=w-\theta_{1}$ in the system (57). Then we have

[TABLE]

For any fixed $0<\epsilon<1,$ there are $T_{1}\in(0,\alpha)$ small enough and a control $v_{T_{1}}$ that provide the following estimate:

[TABLE]

Letting $v(x,t)=v_{1}(x,t)+b_{\theta_{1}}$, we deduce that $z$ satisfies the following homogeneous equation:

[TABLE]

Let us now consider the following system:

[TABLE]

*where $u(x,t)$ is an additive control.

By assumption, there exists a control $u(\cdot,\cdot)\in L^{2}(T_{1},T;L^{2}(\Omega))$ such that $\psi(T)=\psi_{t}(T)=0$ and*

[TABLE]

where the positive constant $C$ can be chosen independent of $T_{1}$. Then, in order to construct a control that steers (57) to $(\theta_{1},0)$ at $T$, it suffices to look for a control $v(x,t)$ on $(T_{1},T)$ such that:

[TABLE]

For the remainder part, it suffices to reproduce the corresponding part in the proof of Theorem 4 to deduce that the state $(\theta_{1},0)$ can be exactly achieved using the following control:

[TABLE]

Remark 5

The results of Theorems 4 $\&$ 6 can be extended to several dimension in hight energy spaces (see [30] for the bilinear case).

iii.   Example

Here, we will present an illustrating example. Let us consider the following semilinear and linear systems respectively with additive control:

[TABLE]

and

[TABLE]

where $O$ is an open subset of $\Omega,$ the functions $\mu$ and $h$ are such that $\mu,h\in L^{\infty}(\Omega),$ the nonlinear term $k:R\to R$ is Lipschitz, $u_{0}(x,t)$ and $u(x,t)$ are the additive controls and belong to $L^{2}(O\times(0,T))$.

Let us first examine the exact controllability of (66). For this end, we start with proving that under the assumption of exact controllability of the linear part (67), the semilinear system (66) is exactly controllable over the same time interval as the linear version (67).

The following elementary controllability result for the system (66) is sufficient for our purpose.

Lemma 4

*Assume that:

(i) supp$(h)\subset O,$

and

(ii) for all $y\in L^{2}(\Omega),\;$ we have supp$(k\circ y)\subset O$.

If the linear system (67) is null exactly controllable with a control satisfying (39), then so is the semilinear system (66).*

Proof 8

Let us consider the system (66) and the following one:

[TABLE]

*For any couple of control $u_{0}(x,t)$ and corresponding solution $\varphi$ of (68), we consider the control: $u(x,t)=u_{0}(x,t)+h(x)\varphi_{t}(x,t)-k(\varphi(x,t))$.

From the assumptions on $u_{0}(x,t),\;h$ and $k$ we can see that $u(x,t)$ satisfies (39) and ${\bf 1}_{O}u(x,t)={\bf 1}_{O}u_{0}(x,t)+h(x)\varphi_{t}(x,t)-k(\varphi(x,t)).$

Then with the control $u(x,t)$, the system (66) takes the form:*

[TABLE]

which admits $\varphi$ as a particular solution, and by uniqueness it comes $\psi=\varphi$. Hence the null exact controllability of the semilinear system (66) follows from the one of its linear part (68).

Let us now describe our illustrative example. Consider the system (22) with $h={\bf 1}_{O}$ for some proper open subset $O=(a,b)$ of $\Omega=(0,3)$ such that $[1,2]\subset O$ and let $f$ be such that $f(y)(x)=k(y(x)),$ where $k(s)=-s\alpha(s)\;$ and $\alpha(s)=-c\>(s-a)^{2}(s-b)^{2}{\bf 1}_{O},\>c>0.$ Here, the function $k:\mathbf{R}\to\mathbf{R}$ is $\mathcal{C}^{1}$ and supp$(k)\subset O$, so $k$ is Lipschitz.

Now in order to define our target state, consider the function defined by:

[TABLE]

and for each $\eta>0$ we set $\xi_{\eta}=\eta\xi_{1}$, and consider the target state $(\theta_{1},\theta_{2})=(\xi_{\eta},-\Delta\xi_{\eta}).$ We have (\theta_{1},\theta_{2})\in\big{(}H_{0}^{1}(0,3)\cap H^{2}(0,3)\big{)}\times L^{2}(0,3) and $\frac{\Delta\theta_{1}}{\theta_{1}}\in L^{\infty}(0,3)$. Moreover for $\eta>0$ small enough i.e. $\eta<\frac{a}{\|\xi_{1}\|_{L^{\infty}(\Omega)}},$ we have $k(\xi_{\eta}(x))=0,\;\forall x\in\Omega$ so that $b_{\theta_{1}}(x)=-\frac{\Delta\theta_{1}}{\theta_{1}}=\frac{\theta_{2}}{\theta_{1}}.$

Let us show that (23) holds. For this end, let us write: $b_{\theta_{1}}(x)y(x)+f(y)(x)=p(x)y(x)+q(y(x))$ with $p(x)=b_{\theta_{1}}(x)-\|b_{\theta_{1}}\|_{L^{\infty}(\Omega)}{\bf 1}_{(1,2)}\leq 0$ and $q(s)=k(s)+s\|b_{\theta_{1}}\|_{L^{\infty}(\Omega)}{\bf 1}_{(1,2)}.$

Observing that $sq(s)=sk(s)\leq 0$ for all $s\in O\setminus(1,2)$, we can see that $sq(s)\leq 0,\;\forall s\in\mathbf{R},$ whenever $c>\|b_{\xi_{1}}\|_{L^{\infty}(\Omega)}\;\displaystyle\max_{s\in(1,2)}\frac{1}{(s-a)^{2}(s-b)^{2}}.$

From [17, 33], we deduce that the estimate (23) holds. Moreover, we have $\langle b_{\theta_{1}}(x)y+f(y),y\rangle\leq 0,\;y\in L^{2}(\Omega).$ Then according to Theorem 3, one can approximately achieve $(\theta_{1},\theta_{2})$ (for $\eta<\frac{a}{\|\xi_{1}\|_{L^{\infty}(\Omega)}}$) using the control:

[TABLE]

for $T_{1}$ large enough ($T_{1}\to+\infty$), $T_{1}^{\prime}$ sufficiently close to $T_{1}$ and for $a,b$ and $c$ satisfying the above mentioned conditions.

Here, the control $v_{2}\in W^{2,\infty}(\Omega)$ is an approximation of $-\frac{\Delta\xi_{1}}{(T_{1}^{\prime}-T_{1})\xi_{1}}{\bf 1}_{O}$ in $L^{2}(\Omega)$.

Let us establish the null exact controllability of the additive-control system (42). From the definition of $\theta_{1},h$ and $f$, it is clear that the assumptions of Lemma 4 are satisfied for $\mu=b_{\theta_{1}}.$ Moreover, we know that (see [27, 35, 36]) there exist a $T>T^{\prime}_{1}$ and a control $u_{0}\in L^{2}(T^{\prime}_{1},T;L^{2}(O))$ satisfying (39) for $t_{0}=T_{1}^{\prime}$ that steers the linear system:

[TABLE]

to $(0,0)$ at $T$. Then it follows from Lemma 4 that the control $u(x,t)=u_{0}(x,t)+h(x)\varphi_{t}-f_{\theta_{1}}(\varphi)$ guarantees the null exact steering of system (42) to $(0,0)$ and satisfies (39). Then applying Theorem 4, we deduce that the control:

[TABLE]

guarantees the exact steering of system (22) to $(\theta_{1},0)$, where $\psi$ is the solution of system (42) corresponding to the control $u(x,t)$.