Numerical Control of a Semilinear Wave Equation on an Interval

Abstract

We are concerned with the numerical exact controllability of the semilinear wave equation on the interval (0, 1). We introduce a Picard iterative scheme yielding a sequence of approximated solutions which converges towards a solution of the null controllability problem, provided that the initial data are small enough. The boundary control, which is applied at the endpoint $x=1$, is taken in the space $H^1_0(0,T)$ for $T=2$. For the linear part, the control input is obtained by imposing a transparent boundary condition at $x=1$. Next, we provide several simulations to show the efficiency of the algorithm, using collocation pseudospectral methods on Chebychev grids to discretize the second order derivative in space in the wave equation.

The authors are partially supported by CAPES: $\text {n}^\text {o}$ 88881.520205/2020-01 and MATH AMSUD: 21-MATH-03.

Appendix

1.1 Proof of Lemma 1

The estimate for $h\equiv 0$ follows from classical semigroup theory, $\big ( S(t) \big )_{t\in {\mathbb R}}$ being a unitary group on $\mathcal H_0$. Assume now that $h\ne 0$ and that $(z^0,z^1,g)=(0,0,0)$. Since $||(z ,\tilde{z})||_\mathcal H$ and $||z||_{H^1_x} + ||\tilde{z}||_{L^2_x}$ are equivalent norms on $\mathcal H$, it is sufficient to prove

$$ ||z||_{ C ([0,T],H^1_x) } + || z_t ||_{ C( [0,T],L^2_x)} \le C || h ||_{H^1_t}. $$

(Here, C denotes some constant that may vary from line to line.) It is well-known that

$$\begin{aligned} ||z||_{C([0,T],L^2_x)} +||z_t ||_{C([0,T],H^{-1}_x)} \le C \left( ||z^0||_{L^2_x} + ||z^1||_{H^{-1}_x} +||h||_{L^2_t} \right) \end{aligned}$$

(81)

where z denotes the solution by transposition of (39)–(41) with $(z^0,z^1,g)=(0,0,0)$. Introduce $w:=z_t$. Then w solves the system

$$\begin{aligned}&w_{tt}-w_{xx} = 0, \qquad (t,x)\in (0,T)\times (0,1)\\&w(t,0)=0,\ w(t,1)=h'(t), \qquad t \in (0,T)\\&(w(0,.),w_t(0,.))= (z^1 , z^0_{xx}) =(0,0). \end{aligned}$$

Since $h\in H^1_t$, we infer from (81) that

$$\begin{aligned} ||w||_{C([0,T],L^2_x)} + ||w_t ||_{ C( [0,T],H^{-1}_x)} \le C ||h||_{H^1_t} , \end{aligned}$$

(82)

so that the estimate for $||z_t ||_{C([0,T],L^2_x) }$ is established. For $||z||_{C([0,T],H^1_x) }$, it is sufficient to notice that $z_{xx}=z_{tt}=w_t$ and to combine

$$ ||z_{xx}||_{C([0,T], H^{-1}_x )} \le C ||h||_{H^1_t} $$

with the elliptic estimate

$$ ||z (t,.) ||_{H^1_x} \le C \left( ||z_{xx} (t,.) ||_{H^{-1}_x} + |z(t,0)| + |z(t,1)| \right) \le C ( ||z_{xx} (t,.) ||_{H^{-1}_x} + | h(t) | ) . $$

1.2 Proof of Lemma 2

For $(z,\tilde{z} ) \in C([0,T], \mathcal H)$, let

$$ ||| (z,\tilde{z} ) |||= \max _{t\in [0,T]} ||(z(t),\tilde{z}(t ) )||_{\mathcal H} =\max _{t\in [0,T] } \big ( ||z_x (t) ||^2_{L^2_x} + ||\tilde{z}(t) ||^2_{L^2_x} \big ) ^{\frac{1}{2}} . $$

We shall apply the contraction mapping principle in the ball $B(0,R)\subset C([0,T], \mathcal H)$, where $R>0$ will be chosen later on. Note that

$$ ||z||_{L^\infty _x(0,1)} \le ||z_x||_{L^2_x} $$

if $z\in H^1_x$ with $z(0)=0$. Assume that $z^0 \in H^1_0(0,1)$, and that

$$\begin{aligned} ||( z^0,z^1 )||_\mathcal H+ C_1 || h ||_{H^1_t} =: \delta\le & {} \delta _0. \end{aligned}$$

(83)

$$\begin{aligned} R< & {} \delta _0. \end{aligned}$$

(84)

Note that, for $(z,\tilde{z})\in B(0,R)$,

$$ ||z (t) ||_{L^\infty _x (0,1) } \le || z_x (t) ||_{L^2_x} \le ||| (z,\tilde{z}) ||| <\delta _0 \qquad \forall t\in [0,T] . $$

Pick two pairs $(z_1,\tilde{z}_1)$ and $(z_2,\tilde{z}_2)$ in B(0, R). Then by Lemma 1

$$\begin{aligned} |||\Gamma (z_1,\tilde{z}_1) -\Gamma (z_2,\tilde{z}_2)|||= & {} |||\int _0^t S(t-s) (0, f(z_1(s)) - f(z_2(s)) ) ds ||| \nonumber \\\le & {} || f(z_1) - f(z_2) ||_{L^1(0,T,L^2_x)} \nonumber \\\le & {} \eta ||z_1-z_2 ||_{L^1(0,T,L^2_x)} \nonumber \\\le & {} T\eta |||(z_1-z_2,\tilde{z}_1-\tilde{z}_2)|||. \end{aligned}$$

(85)

Note that $\Gamma$ contracts in B(0, R), for $T=2$ and $\eta$ as in (4). On the other hand, for any $(z,\tilde{z}) \in B(0,R)$, we have by Lemma 1 and (85)

$$\begin{aligned} |||\Gamma (z,\tilde{z}) |||\le & {} ||| S(t) (z^0,z^1) + W(h) ||| + ||| \int _0^t S(t-s) (0,f(z(s))) ds |||\nonumber \\\le & {} \delta + 2 \eta ||| (z, \tilde{z}) ) |||. \end{aligned}$$

(86)

Thus $\Gamma (B(0,R))\subset B(0,R)$ if

$$ \delta + 2\eta R \le R. $$

Pick

$$ R = \frac{\delta }{1-2\eta }\cdot $$

with

$$ 0<\delta< \delta _1 = (1-2\eta ) \delta _0 < \delta _0 . $$

Then $\Gamma$ has a unique fixed point $(z,\tilde{z})$ in B(0, R) by the contraction mapping principle. Clearly, $\tilde{z} = z_t$ and (48) holds with $C_2=(1-2\eta )^{-1}.$