Local Null-Controllability of a Nonlocal Semilinear Heat Equation

Abstract

This paper deals with the problem of internal null-controllability of a heat equation posed on a bounded domain with Dirichlet boundary conditions and perturbed by a semilinear nonlocal term. We prove the small-time local null-controllability of the equation. The proof relies on two main arguments. First, we establish the small-time local null-controllability of a $2 \times 2$ reaction-diffusion system, where the second equation is governed by the parabolic operator $\tau \partial _t - \sigma \varDelta$, $\tau , \sigma > 0$. More precisely, this controllability result is obtained uniformly with respect to the parameters $(\tau , \sigma ) \in (0,1) \times (1, + \infty )$. Secondly, we observe that the semilinear nonlocal heat equation is actually the asymptotic derivation of the reaction-diffusion system in the limit $(\tau ,\sigma ) \rightarrow (0,+\infty )$. Finally, we illustrate these results by numerical simulations.

Appendices

Energy Estimates for the Reaction-Diffusion System

In this section, we recall some classical energy estimates for the system (9), assuming $(a,b,c,d) \in \mathbb {R}^2 \times \mathbb {R}^{*} \times (-\infty ,0)$. More precisely, we consider for $F \in L^2((0,T)\times \varOmega )$,

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \partial _t u- \varDelta u = a u +b v + F&{}\mathrm {in}\ (0,T)\times \varOmega ,\\ \tau \partial _t v - \sigma \varDelta v = u-v &{}\mathrm {in}\ (0,T)\times \varOmega ,\\ \displaystyle u = \frac{\partial v}{\partial n}= 0,\ &{}\mathrm {on}\ (0,T)\times \partial \varOmega ,\\ (u,v)(0,\cdot )=(u_0,v_0)&{} \mathrm {in}\ \varOmega . \end{array}\right. } \end{aligned}$$

(118)

We have the following well-posedness result in $L^2$.

Proposition 7

There exists a positive constant $C = C(\varOmega ,T)=\exp (C(\varOmega )T)>0$ such that for every $(u_0,v_0) \in L^2(\varOmega )^2$, $F \in L^2((0,T)\times \varOmega )$, the solution (u, v) to (118) satisfies

$$\begin{aligned}&\left\Vert u\right\Vert _{C([0,T];L^2(\varOmega ))} + \left\Vert u\right\Vert _{L^2(0,T;H_0^1(\varOmega ))} + \left\Vert \partial _t u\right\Vert _{L^2(0,T;H^{-1}(\varOmega ))} \nonumber \\&\qquad + \sqrt{\tau } \left\Vert v\right\Vert _{C([0,T];L^2(\varOmega ))} + \left\Vert v\right\Vert _{L^2(0,T;H^1(\varOmega ))} + \sqrt{\tau } \left\Vert \partial _t v\right\Vert _{L^2(0,T;H^1(\varOmega )')}\nonumber \\&\quad \le C\left( \left\Vert u_0\right\Vert _{L^2(\varOmega )} + \sqrt{\tau } \left\Vert v_0\right\Vert _{L^2(\varOmega )} + \left\Vert F\right\Vert _{L^2((0,T)\times \varOmega )}\right) . \end{aligned}$$

(119)

Proof

We just give the sketch of the proof because it is standard, see [11, Sect. 7.1.2] for the details. We only give a priori estimates. We multiply the first equation of (118) by u and the second equation of (118) by v, then integrate in $Q_t=(0,t)\times \varOmega$,

$$\begin{aligned} \frac{1}{2} \int _{\varOmega } u(t)^2 + \int _{Q_t} |\nabla u|^2&= \frac{1}{2} \int _{\varOmega } u_0^2 + \int _{Q_t} F u + \int _{Q_t} a u^2 + \int _{Q_t} b uv,\\ \frac{\tau }{2} \int _{\varOmega } v(t)^2 + \sigma \int _{Q_t} |\nabla v|^2 + \int _{Q_t} v^2&= \frac{\tau }{2} \int _{\varOmega } v_0^2 + \int _{Q_t} uv. \end{aligned}$$

We use Young’s inequalities in the previous equations to obtain

$$\begin{aligned} \frac{1}{2} \int _{\varOmega } u(t)^2 + \int _{Q_t} |\nabla u|^2&\le C \left( \int _{\varOmega } u_0^2 + \int _{Q_t} F^2 + \int _{Q_t} u^2 + \int _{Q_t} v^2 \right) , \end{aligned}$$

(120)

$$\begin{aligned} \frac{\tau }{2} \int _{\varOmega } v(t)^2 + \sigma \int _{Q_t} |\nabla v|^2 + \frac{1}{2} \int _{Q_t} v^2&\le \frac{\tau }{2} \int _{\varOmega } v_0^2 +\,\frac{1}{2} \int _{Q_t} u^2 . \end{aligned}$$

(121)

We put (121) in (120) and use Gronwall’s estimate

$$\begin{aligned} \int _{\varOmega } u(t)^2 + \int _{Q_t} |\nabla u|^2 \le C \left( \int _{\varOmega } u_0^2 + \tau \int _{\varOmega } v_0^2 + \int _{Q_t} F^2\right) . \end{aligned}$$

(122)

Then, we use this previous bound in (121) to get

$$\begin{aligned} \tau \int _{\varOmega } v(t)^2 + \sigma \int _{Q_t} |\nabla v|^2 + \int _{Q_t} v^2 \le C \left( \int _{\varOmega } u_0^2 + \tau \int _{\varOmega } v_0^2 + \int _{Q_t} F^2\right) . \end{aligned}$$

(123)

By taking the supremum for $t \in [0,T]$ in (122) and (123), we obtain the conclusion of the proof. $\square$

We have the following maximal regularity estimate in $L^2$.

Proposition 8

There exists a positive constant $C = C(\varOmega ,T)=\exp (C(\varOmega )T)>0$ such that for every $(u_0,v_0) \in H_0^1(\varOmega )\times H^1(\varOmega )$, $F \in L^2((0,T)\times \varOmega )$, the solution (u, v) to (118) satisfies

$$\begin{aligned}&\left\Vert u\right\Vert _{C([0,T];H_0^1(\varOmega ))} + \left\Vert u\right\Vert _{L^2(0,T;H^2(\varOmega ))} + \left\Vert \partial _t u\right\Vert _{L^2(0,T;L^2(\varOmega ))} \nonumber \\&\quad + \sqrt{\tau } \left\Vert v\right\Vert _{C([0,T];H^1(\varOmega ))} + \left\Vert v\right\Vert _{L^2(0,T;H^2(\varOmega ))} + \sqrt{\tau } \left\Vert \partial _t v\right\Vert _{L^2(0,T;L^2(\varOmega ))} \nonumber \\&\qquad \le C\left( \left\Vert u_0\right\Vert _{H^1(\varOmega )} + \sqrt{\tau } \left\Vert v_0\right\Vert _{H^1(\varOmega )}\right) . \end{aligned}$$

(124)

Proof

It is a straightforward adaptation of the proof of [11, Sect. 7.1.3, Theorem 5], just by multiplying the first equation by $-\varDelta u$ and the second equation by $-\varDelta v$. $\square$

Proof of the Source Term Method

In this section, we give the proof of Proposition 3.

Proof

For $k \ge 0$, we define $T_k := T(1-q^{-k})$ where $q \in (1, \sqrt{2})$. On the one hand, let $a_0 := (u_0,\sqrt{\tau } v_0)$ and, for $k \ge 0$, we define $a_{k+1} := (u_S,\sqrt{\tau } v_S)(T_{k+1}^{-},.)$ where $(u_S,v_S)$ is the solution to

$$\begin{aligned} \left\{ \begin{array}{l l} \partial _t u_{S} - \varDelta u_{S} = a u_S + b v_S + S&{}\mathrm {in}\ (T_k,T_{k+1})\times \varOmega ,\\ \tau \partial _t v_{S} - \sigma \varDelta v_{S} = c u_S + d v_S&{}\mathrm {in}\ (T_k,T_{k+1})\times \varOmega ,\\ u_{S}= \frac{\partial v_{S}}{\partial n } = 0 &{}\mathrm {on}\ (T_k,T_{k+1})\times \partial \varOmega ,\\ (u,v)_{S}(T_k^{+},.)=0 &{}\mathrm {in}\ \varOmega . \end{array} \right. \end{aligned}$$

From classical energy estimates, see Proposition 7, we have

$$\begin{aligned} \left\Vert a_{k+1}\right\Vert _{L^{2}(\varOmega )^2} \le \left\Vert (u_S, \sqrt{\tau } v_S)\right\Vert _{C([T_k,T_{k+1}];L^{2}(\varOmega )^2)} \le C\left\Vert S\right\Vert _{L^{2}((T_k,T_{k+1});L^{2}(\varOmega ))}. \end{aligned}$$

(125)

On the other hand, for $k \ge 0$, we also consider the control systems

$$\begin{aligned} \left\{ \begin{array}{l l} \partial _t u_{h} - \varDelta u_{h} = a u_h + b v_h + h 1_{\omega } &{}\mathrm {in}\ (T_k,T_{k+1})\times \varOmega ,\\ \tau \partial _t v_{h} - \sigma \varDelta v_{h} = c u_h + d v_h&{}\mathrm {in}\ (T_k,T_{k+1})\times \varOmega ,\\ u_{S}= \frac{\partial v_{h}}{\partial n } = 0 &{}\mathrm {on}\ (T_k,T_{k+1})\times \partial \varOmega ,\\ (u_h,\sqrt{\tau } v_h)(T_k^{+},.)= a_k &{}\mathrm {in}\ \varOmega . \end{array} \right. \end{aligned}$$

From the null-controllability result, we deduce that we can define $h_k \in L^2((T_k, T_{k+1})\times \varOmega )$ such that $(u_S,v_S)(T_{k+1}^{-},\cdot ) = 0$ and thanks to the (precise) cost estimate,

$$\begin{aligned} \left\Vert h_k\right\Vert _{ L^{2}((T_k,T_{k+1})\times \varOmega )} \le M e^{\frac{M}{T_{k+1}-T_{k}}} \left\Vert a_k\right\Vert _{L^2(\varOmega )^2}. \end{aligned}$$

(126)

In particular, for $k=0$, we have

$$\begin{aligned} \left\Vert h_0\right\Vert _{ L^{2}((T_0,T_{1})\times \varOmega )} \le M e^{\frac{qM}{T(q-1)}} \left\Vert a_0\right\Vert _{L^2(\varOmega )^2}. \end{aligned}$$

And, since $\rho _0$ is decreasing

$$\begin{aligned} \left\Vert h_0/\rho _0\right\Vert _{L^{2}((T_0,T_{1})\times \varOmega )} \le \rho _0^{-1}(T_1) M e^{\frac{qM}{T(q-1)}} \left\Vert a_0\right\Vert _{L^2(\varOmega )^2}. \end{aligned}$$

(127)

For $k \ge 0$, since $\rho _{{\mathcal {S}}}$ is decreasing, combining (125) and (126) yields

$$\begin{aligned} \left\Vert h_{k+1}\right\Vert _{L^{2}((T_{k+1},T_{k+2})\times \varOmega )} \le C M e^{\frac{M}{T_{k+2}-T_{k+1}}} \rho _{{\mathcal {S}}}(T_k) \left\Vert S/\rho _{{\mathcal {S}}}\right\Vert _{L^{2}((T_{k},T_{k+1})\times \varOmega )}. \end{aligned}$$

(128)

In particular, by using $M e^{\frac{M}{T_{k+2}-T_{k+1}}} \rho _{{\mathcal {S}}}(T_k) = \rho _0(T_{k+2})$ coming from the definitions (74) and (75), we have

$$\begin{aligned} \left\Vert h_{k+1}\right\Vert _{L^{2}((T_{k+1},T_{k+2})\times \varOmega )}&\le C \rho _0(T_{k+2}) \left\Vert S/\rho _{{\mathcal {S}}}\right\Vert _{L^{2}((T_{k},T_{k+1})\times \varOmega )}. \end{aligned}$$

(129)

Then, from (129), by using the fact that $\rho _0$ is decreasing,

$$\begin{aligned} \left\Vert h_{k+1}/\rho _0\right\Vert _{L^{2}((T_{k+1},T_{k+2})\times \varOmega )} \le C \left\Vert S/\rho _{{\mathcal {S}}}\right\Vert _{L^{2}((T_{k},T_{k+1})\times \varOmega )}. \end{aligned}$$

(130)

As in the original proof, we can paste the controls $h_{k}$ for $k \ge 0$ together by defining

$$\begin{aligned} h := \sum \limits _{k \ge 0} h_k 1_{(T_k,T_{k+1})}. \end{aligned}$$

We have the estimate from (127) and (130)

$$\begin{aligned} \left\Vert h\right\Vert _{{\mathcal {H}}} \le C \left\Vert S\right\Vert _{{\mathcal {S}}} + C \rho _0^{-1}(T_1) M e^{\frac{qM}{T(q-1)}} \left\Vert a_0\right\Vert _{L^2(\varOmega )^2}. \end{aligned}$$

The state (u, v) can also be reconstructed by concatenation of $(u_S,v_S)$ + $(u_h,v_h)$, which are continuous at each junction $T_k$ thanks to the construction. Then, we estimate the state. We use the energy estimate on each time interval $(T_k, T_{k+1})$:

$$\begin{aligned} \left\Vert (u_S, \sqrt{\tau } v_S)\right\Vert _{L^{\infty }(T_k,T_{k+1};L^2(\varOmega )^2)} \le C \left\Vert S\right\Vert _{L^2((T_k,T_{k+1})\times \varOmega )}, \end{aligned}$$

and

$$\begin{aligned} \left\Vert (u_h,\sqrt{\tau } v_h)\right\Vert _{L^{\infty }(T_k,T_{k+1};L^2(\varOmega )^2)} \le C\left( \left\Vert a_k\right\Vert _{L^{2}(\varOmega )} + \left\Vert h\right\Vert _{L^{2}((T_k,T_{k+1})\times \varOmega )}\right) . \end{aligned}$$

Proceeding similarly as for the estimate on the control, we obtain respectively

$$\begin{aligned} \left\Vert (u_S, \sqrt{\tau } v_S)/\rho _0\right\Vert _{L^{\infty }(T_k,T_{k+1};L^2(\varOmega )^2)} \le C M^{-1} \left\Vert S\right\Vert _{{\mathcal {S}}}, \end{aligned}$$

and

$$\begin{aligned}&\left\Vert (u_h, \sqrt{\tau } v_h)/\rho _0\right\Vert _{L^{\infty }(T_k,T_{k+1};L^2(\varOmega )^2)} \\&\quad \le C M^{-1} \left\Vert S\right\Vert _{{\mathcal {S}}} + C \rho _0^{-1}(T_1) M e^{\frac{qM}{T(q-1)}} \left\Vert (u_0, \sqrt{\tau } v_0)\right\Vert _{L^{2}(\varOmega )^2}. \end{aligned}$$

Therefore, for an appropriate choice of constant $C >0$, (u, v) and h satisfy (80). This concludes the proof of Proposition 3. $\square$