Optimal control of a class of reaction–diffusion systems

Abstract

The optimal control of a system of nonlinear reaction–diffusion equations is considered that covers several important equations of mathematical physics. In particular equations are covered that develop traveling wave fronts, spiral waves, scroll rings, or propagating spot solutions. Well-posedness of the system and differentiability of the control-to-state mapping are proved. Associated optimal control problems with pointwise constraints on the control and the state are discussed. The existence of optimal controls is proved under weaker assumptions than usually expected. Moreover, necessary first-order optimality conditions are derived. Several challenging numerical examples are presented that include in particular an application of pointwise state constraints where the latter prevent a moving localized spot from hitting the domain boundary.

Appendix: Proof of Lemma 3.1

Let us consider sequences $\{f_k\}_{k=1}^\infty \subset L^2(Q)^n$, $\{g_k\}_{k=1}^\infty \subset L^2(\varSigma )^n$, and $\{h_k\}_{k = 1}^\infty \subset H^1(\varOmega )^n$ satisfying

$$\begin{aligned}&\Vert f_k\Vert _{L^1(Q)^n} \le \Vert \nabla _yg(\cdot ,\cdot ,{{\bar{y}}}){{\bar{\mu }}}_Q \Vert _{{\mathcal {M}}(Q)^n},\ f_k {\mathop {\rightharpoonup }\limits ^{*}} \nabla _yg(\cdot ,\cdot ,{{\bar{y}}}){{\bar{\mu }}}_Q \quad \text { in } {\mathcal {M}}(Q)^n, \end{aligned}$$

(5.1)

$$\begin{aligned}&\Vert g_k\Vert _{L^1(\varSigma )^n} \le \Vert \nabla _yg(\cdot ,\cdot ,{{\bar{y}}}){{\bar{\mu }}}_\varSigma \Vert _{{\mathcal {M}}(\varSigma )^n},\ g_k {\mathop {\rightharpoonup }\limits ^{*}} \nabla _yg(\cdot ,\cdot ,{{\bar{y}}}){{\bar{\mu }}}_\varSigma \quad \text { in } {\mathcal {M}}(\varSigma )^n, \end{aligned}$$

(5.2)

$$\begin{aligned}&\left\{ \begin{array}{l}\displaystyle \Vert h_k\Vert _{L^1(\varOmega )^n} \le \Vert \nabla _yg(\cdot ,T,{{\bar{y}}}(T)){{\bar{\mu }}}_\varOmega \Vert _{{\mathcal {M}}({{\bar{\varOmega }}} \times \{T\})^n}\\ h_k {\mathop {\rightharpoonup }\limits ^{*}} \nabla _yg(\cdot ,T,{{\bar{y}}}(T)) {{\bar{\mu }}}_\varOmega \text { in } {\mathcal {M}}({{\bar{\varOmega }}} \times \{T\})^n.\end{array}\right. \end{aligned}$$

(5.3)

We also consider a sequence $\{w_k\}_{k= 1}^\infty \subset H^1(\varOmega )^n$ such that

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle w_k \rightarrow C^\top _\varOmega [C_\varOmega {{\bar{y}}}(\cdot ,T) - y_\varOmega ] \ \text { strongly in } L^2(\varOmega )^n\\ \displaystyle \text {and } \Vert w_k\Vert _{L^2(\varOmega )^n} \le \Vert C^\top _\varOmega [C_\varOmega {{\bar{y}}}(\cdot ,T) - y_\varOmega ]\Vert _{L^2(\varOmega )^n}\quad \forall k. \end{array}\right. \end{aligned}$$

(5.4)

Now we consider the problem

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\frac{\partial \varphi _k}{\partial t} - \varDelta \varphi _k + \frac{\partial R}{\partial y}(\cdot ,\cdot ,{{\bar{y}}})\varphi _k + A^\top \varphi _k = C_Q^\top [C_Q{{\bar{y}}} - y_Q] + f_k&{}\quad \text {in }Q,\\ \displaystyle \partial _\nu \varphi _k = \displaystyle g_k&{}\quad \text {on } \varSigma , \\ \varphi _k(\cdot ,T) = w_k + h_k&{}\quad \text {in } {{\bar{\varOmega }}}. \end{array}\right. \end{aligned}$$

(5.5)

This is a standard linear problem for which existence and uniqueness of a solution $\varphi _k \in W(0,T)^n$ are well known. Moreover, since $w_k + h_k \in H^1(\varOmega )^n$, we also have that $\varphi _k \in H^1(Q)^n$. From Theorem 3.3 we infer that

$$\begin{aligned} \varphi _k \rightharpoonup {{\bar{\varphi }}} \ \text { in } L^r\left( 0,T;W_0^{1,s}(\varOmega )\right) ^n\quad \forall r, s \in [1, 2) \text { with } \frac{2}{r}+\frac{d}{s} > 1 + d. \end{aligned}$$

(5.6)

Moreover, the following strong convergence holds [7]

$$\begin{aligned} \lim _{k \rightarrow \infty }\Vert {{\bar{\varphi }}} - \varphi _k\Vert _{L^q(Q)^n} = 0\quad \forall 1\le q < \frac{d + 2}{d}. \end{aligned}$$

(5.7)

Now, we define $\varphi _{M,k}(x,t) = {\text {Proj}}_{[-M,+M]^n}({{\bar{\varphi }}}_k(x,t))$. Since $\varphi _k \in H^1(Q)^n$, then we also have that $\varphi _{M,k} \in H^1(Q)^n$. From the inequality $|\varphi _{M}(x,t) - \varphi _{M,k}(x,t)| \le |{{\bar{\varphi }}}(x,t) - \varphi _{k}(x,t)|$ and (5.7) we infer that $\varphi _{M,k} \rightarrow \varphi _M$ strongly in $L^q(Q)^n$ for every $1\le q < \frac{d + 2}{d}$.

If we prove that $\{\varphi _{M,k}\}_{k = 1}^\infty$ is bounded in $L^2(0,T;H^1(\varOmega ))^n$, then the convergence $\varphi _{M,k} \rightarrow \varphi _M$ in $L^q(Q)^n$ implies that $\varphi _M \in L^2(0,T;H^1(\varOmega ))^n$ as well. Taking $\eta = |C_R| + 1$, multiplying (5.5) by ${\mathrm{e}}^{2\eta t}\varphi _{M,k}$ and integrating in Q

$$\begin{aligned}&\int _Q-{\mathrm{e}}^{2\eta t}\frac{\partial \varphi _k}{\partial t}\cdot \varphi _{M,k}\, dx\, dt + \int _Q{\mathrm{e}}^{2\eta t}\nabla \varphi _k \cdot \nabla \varphi _{M,k}\, dx\, dt\nonumber \\&\quad + \int _Q{\mathrm{e}}^{2\eta t}\left( \frac{\partial R}{\partial y}(x,t,{{\bar{y}}})\varphi _k\right) \cdot \varphi _{M,k}\, dx\, dt + \int _Q{\mathrm{e}}^{2\eta t}\varphi _k^\top A(x,t)\varphi _{M,k}\, dx\, dt\nonumber \\&\qquad = \int _Q{\mathrm{e}}^{2\eta t}\left( C_Q^\top [C_Q{{\bar{y}}} - y_Q] + f_k\right) \cdot \varphi _{M,k}\, dx\, dt+ \int _\varSigma {\mathrm{e}}^{2\eta t} g_k\cdot \varphi _{M,k}\, dx\, dt. \end{aligned}$$

(5.8)

Now using that $\varphi _k\cdot \partial _t\varphi _{M,k} = \varphi _{M,k}\cdot \partial _t\varphi _{M,k} = \frac{1}{2}\partial _t|\varphi _{M,k}|^2$, we obtain

$$\begin{aligned} \int _Q-\text {e}^{2\eta t}\frac{\partial \varphi _k}{\partial t}\cdot \varphi _{M,k}\, dx\, dt =&-\int _0^T\frac{d}{dt}\int _\varOmega \text {e}^{2\eta t}\varphi _k\cdot \varphi _{M,k}\, dx\, dt\nonumber \\&+ 2\eta \int _Q\text {e}^{2\eta t}\varphi _k\cdot \varphi _{M,k}\, dx\, dt + \int _Q\text {e}^{2\eta t}\varphi _k\cdot \frac{\partial \varphi _{M,k}}{\partial t}\, dx\, dt\nonumber \\ =&-\int _\varOmega \text {e}^{2\eta T}\varphi _k(x,T)\cdot \varphi _{M,k}(x,T)\, dx \nonumber \\&+ \int _\varOmega \varphi _k(x,0)\cdot \varphi _{M,k}(x,0)\, dx\nonumber \\&+ 2\eta \int _Q\text {e}^{2\eta t}\varphi _k\cdot \varphi _{M,k}\, dx\, dt +\frac{1}{2}\int _Q\text {e}^{2\eta t}\partial _t|\varphi _{M,k}|^2\, dx\, dt. \end{aligned}$$

(5.9)

For the last term we have

$$\begin{aligned}&\frac{1}{2}\int _Q\text {e}^{2\eta t}\partial _t|\varphi _{M,k}|^2\, dx\, dt\nonumber \\&\quad = \frac{1}{2}\int _0^T\frac{d}{dt}\int _\varOmega \text {e}^{2\eta t}|\varphi _{M,k}|^2\, dx\, dt - \eta \int _0^T\int _\varOmega \text {e}^{2\eta t}|\varphi _{M,k}|^2\, dx\, dt\nonumber \\&\quad \ge \frac{1}{2}\int _\varOmega \text {e}^{2\eta T}|\varphi _{M,k}(x,T)|^2\, dx - \frac{1}{2}\int _\varOmega |\varphi _{M,k}(x,0)|^2\, dx \nonumber \\&\qquad - \eta \int _Q\text {e}^{2\eta t}\varphi _k\cdot \varphi _{M,k}\, dx\, dt. \end{aligned}$$

(5.10)

From (5.9) and (5.10) and using that $\varphi _k\cdot \varphi _{M,k} \ge |\varphi _{M,k}|^2$ we deduce

$$\begin{aligned} \int _Q-\text {e}^{2\eta t}\frac{\partial \varphi _k}{\partial t}\cdot \varphi _{M,k}\, dx\, dt \ge&-\text {e}^{2\eta T}\int _\varOmega (w_k + h_k)\cdot \varphi _{M,k}(x,T)\, dx\\&+ \frac{1}{2}\int _\varOmega |\varphi _{M,k}(x,0)|^2\, dx + \eta \int _Q\text {e}^{2\eta t}\varphi _k\cdot \varphi _{M,k}\,dx\, dt\\&+ \frac{\text {e}^{2\eta T}}{2}\int _\varOmega |\varphi _{M,k}(x,T)|^2\, dx\\ \ge&-\text {e}^{2\eta T}\int _\varOmega (w_k + h_k)\cdot \varphi _{M,k}(x,T)\, dx \\&+ \eta \int _Q\text {e}^{2\eta t}\varphi _k\cdot \varphi _{M,k}\, dx\, dt\\&+ \frac{1}{2}\int _\varOmega |\varphi _{M,k}(x,T)|^2\, dx. \end{aligned}$$

Inserting this inequality in (5.8) and taking into account that $\partial _{x_i}\varphi _k\cdot \partial _{x_i}\varphi _{M,k} = \partial _{x_i}\varphi _{M,k}\cdot \partial _{x_i}\varphi _{M,k}$, we obtain with Young’s inequality, (5.1)–(5.4), and the estimate for $\Vert \varphi _k\Vert _{L^1(Q)^n}$ derived from Theorem 3.3

$$\begin{aligned}&\int _Q|\nabla \varphi _{M,k}|^2\, dx\, dt + \int _Q|\varphi _{M,k}|^2\, dx\, dt + \frac{1}{2}\int _\varOmega |\varphi _{M,k}(x,T)|^2\, dx\\&\quad \le \int _Q|\nabla \varphi _{M,k}|^2 dx dt + \int _Q\text {e}^{2\eta t}(\eta -|C_R|)\varphi _k\cdot \varphi _{M,k}\, dx dt+\frac{1}{2}\int _\varOmega |\varphi _{M,k}(\cdot ,T)|^2 dx \\&\quad \le \int _0^{T}-{\mathrm{e}}^{2\eta t}\frac{\partial \varphi _M}{\partial t}\cdot \varphi _{M,k}\, dx\, dt + \int _Q\text {e}^{2\eta t}\nabla \varphi _M \cdot \nabla \varphi _{M,k}\, dx\, dt\\&\qquad + \int _Q\text {e}^{2\eta t}\Big (\frac{\partial R}{\partial y}(x,t,{{\bar{y}}})\varphi _k\Big )\cdot \varphi _{M,k}\, dx\, dt + \text {e}^{2\eta T}\int _\varOmega (w_k + h_k)\cdot \varphi _{M,k}(x,T)\, dx\\&\quad = \int _Q\text {e}^{2\eta t}\Big (C_Q^\top [C_Q{{\bar{y}}} - y_Q] + f_k\Big )\cdot \varphi _{M,k}\, dx\, dt + \int _\varSigma {\mathrm{e}}^{2\eta t} g_k\cdot \varphi _{M,k}\, dx\, dt\\&\qquad + \text {e}^{2\eta T}\int _\varOmega (w_k + h_k)\cdot \varphi _{M,k}(T)\, dx - \int _Q{\mathrm{e}}^{2\eta t}\varphi _k^\top A(x,t)\varphi _{M,k}\, dx\, dt\\&\quad \le \text {e}^{2\eta T}\Big [\Vert C_Q^\top [C_Q{{\bar{y}}} - y_Q]\Vert _{L^2(Q)^n}\Vert \varphi _{M,k}\Vert _{L^2(Q)^n} + \Vert w_k\Vert _{L^2(\varOmega )^n}\Vert \varphi _{M,k}(T)\Vert _{L^2(\varOmega )^n}\\&\qquad + M\big (\Vert f_k\Vert _{L^1(Q)^n} + \Vert g_k\Vert _{L^1(\varSigma )^n} + \Vert h_k\Vert _{L^1(\varOmega )^n} + \Vert A\Vert _{L^\infty (Q,{\mathbb {R}}^{n \times n})}\Vert \varphi _k\Vert _{L^1(Q)^n)}\big )\Big ]\\&\quad \le C\Big [\Big \Vert C_Q^\top [C_Q{{\bar{y}}} - y_Q]\Big \Vert ^2_{L^2(Q)} + \Vert C^\top _\varOmega [C_\varOmega {{\bar{y}}}(\cdot ,T) - y_\varOmega ]\Vert ^2_{L^2(\varOmega )^n}\\&\qquad + M\Vert \nabla _yg(\cdot ,\cdot ,{{\bar{y}}})\Vert _{C(K)^n}\Vert {{\bar{\mu }}}_Q\Vert _{{\mathcal {M}}({{\bar{Q}}})}\Big ] + \frac{1}{2}\int _Q|\varphi _{M,k}|^2 dx dt + \frac{1}{4}\Vert \varphi _{M,k}(T)\Vert ^2_{L^2(\varOmega )^n}. \end{aligned}$$

Finally, we get from the above inequality with (5.4) that each $\varphi _{M,k}$ satisfies (3.15). Hence, $\varphi _M$ also does it. $\square$