Asymptotic analysis for optimal control of the Cattaneo model (Q6111113)

The Cattaneo model consists in a damped wave equation: \(\tau y_{r}^{\prime \prime }+y_{r}^{\prime }-\Delta y_{r}=u_{r}\), posed in \((0,T)\times \Omega \) , where \(\Omega \subset \mathbb{R}^{d}\) is a bounded and Lipschitz domain with boundary \(\partial \Omega \) and \(T>0\), \(y_{r}\) denoting the temperature, \(u_{r}\) the heat source and \(\tau >0\) a delay parameter. Homogeneous Dirichlet boundary conditions are imposed \(y_{r}=0\) on \( (0,T)\times \partial \Omega \), together with initial conditions \( y_{r}(0,\cdot )=y_{0}\in H_{0}^{1}(\Omega )\), \(y_{r}^{\prime }(0,\cdot )=y_{1}\in L^{2}(\Omega )\). The authors recall the existence of a unique weak solution \(y_{r}\in Y(0,T)=\{v\in L^{2}(0,T;H_{0}^{1}(\Omega ))\mid v^{\prime }\in L^{2}(0,T;L^{2}(\Omega ))\) and \(v^{\prime \prime }\in L^{2}(0,T;H^{-1}(\Omega ))\}\) to the Cattaneo problem, which depends continuously on the data. They consider the optimal control problem: \( \min_{(y_{r},u_{r})}J_{\tau ,\nu }(y_{r},u_{r})=\frac{1}{2}\left\Vert y_{r}-y_{d}\right\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}+\frac{\nu \tau }{2} \left\Vert y_{r}(T)-y_{d}(T)\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\lambda }{2}\left\Vert u_{r}\right\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}\), such that \(y_{r}\) is the solution to the Cattaneo problem with \(u_{r}\in U_{ad}\), the set of admissible controls, where \(\lambda \geq 0\) and \(\nu \in \{0,1\}\). Assuming that \(y_{0}\in H^{3}(\Omega )\cap H_{0}^{1}(\Omega )\), \( y_{1}=\Delta y_{0}+\overline{u}\in H_{0}^{1}(\Omega )\), \(y_{r}\in H^{1}(0,T;H_{0}^{1}(\Omega ))\), and the set \(U_{ad}\) of admissible controls is a weakly closed and convex subset of \(\widehat{U}=\{u\in H^{1}(0,T;H^{1}(\Omega ))\mid \left\Vert u\right\Vert _{H^{1}(0,T;H^{1}(\Omega ))}\leq M\), \(u(0)=\overline{u}\}\), the authors apply standard techniques for linear-quadratic optimal control problems to prove the existence of a unique optimal control \(u_{r}^{\ast }\in U_{ad}\). The authors are also interested in the case where \(\tau =0\), which leads to the heat equation \(y^{\prime }-\Delta y=u\) in \((0,T)\times \Omega \), with the boundary condition \(y=0\) on \((0,T)\times \partial \Omega \) and the initial condition \(y(0,\cdot )=y_{0}\). The optimal control problem becomes: \( \min_{(y,u)}J(y,u)=\frac{1}{2}\left\Vert y-y_{d}\right\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}+\frac{\lambda }{2}\left\Vert u\right\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}\), such that \(y\) is the solution to the heat problem with \(u\in U_{ad}\). The authors recall the existence of a unique weak solution \(y\in W(0,T)=\{v\in L^{2}(0,T;H_{0}^{1}(\Omega ))\mid v^{\prime }\in L^{2}(0,T;L^{2}(\Omega ))\}\) to the heat problem, which depends continuously on the data, and under the same hypotheses than above the existence of a unique optimal control \(u^{\ast }\in U_{ad}\). In both cases, they write the first-order optimality conditions. The authors prove the convergence of \((y_{r}^{\ast },u_{r}^{\ast })\) to \((y^{\ast },u^{\ast })\) in the weak topology of \(L^{2}(0,T;H^{1}(\Omega ))\times L^{2}(0,T;L^{2}(\Omega ))\) when \(\tau \rightarrow 0\) and they prove some rates of convergence. In the last part of their paper, the authors present the results of numerical simulations for different values of the parameters \( \lambda \) and \(\tau \).