Existence of Optimal Shapes in Parabolic Bilinear Optimal Control Problems

The aim of this paper is to prove the existence of optimal shapes in bilinear parabolic optimal control problems. We consider a parabolic equation \(\partial _tu_m-\Delta u_m=f(t,x,u_m)+mu_m\) ${\partial }_t{u}_m-\mathrm{\Delta }{u}_m=f(t,x,u_m)+m{u}_m$ . The set of admissible controls is given by \(A=\{m\in L^\infty \,, m_-\leqq m\leqq m_+{\text { almost everywhere, }}\int _\Omega m(t,\cdot )=V_1(t)\}\) $A=\{m\in L^{\infty },m_{-}\leqq m\leqq m_{+}\text{almost everywhere,}{\int }_{\mathrm{\Omega }}m(t,·)=V_1(t)\}$ , where \(m_\pm =m_\pm (t,x)\) $m_{\pm }=m_{\pm }(t,x)$ are two reference functions in \(L^\infty ({(0,T)\times {\Omega }})\) $L^{\infty }((0,T)\times \mathrm{\Omega })$ , and where \(V_1=V_1(t)\) $V_1=V_1(t)$ is a reference integral constraint. The functional to optimise is \(J:m\mapsto \iint _{(0,T)\times {\Omega }} j_1(u_m)+\int _{\Omega }j_2(u_m(T))\) $J:m\mapsto {\iint }_{(0,T)\times \mathrm{\Omega }}{j}_1(u_m)+{\int }_{\mathrm{\Omega }}{j}_2(u_m(T))$ . Roughly speaking, we prove that, if \(j_1\) $j_1$ and \(j_2\) $j_2$ are non-decreasing and if one is increasing, then any solution of \(\max _A J\) ${max}_{A}J$ is bang-bang: any optimal \(m^*\) $m^{\ast }$ writes \(m^*=\mathbb {1}_E m_-+\mathbb {1}_{E^c}m_+\) $m^{\ast }={\mathbb{1}}_E{m}_{-}+{\mathbb{1}}_{E^c}{m}_{+}$ for some \(E\subset {(0,T)\times {\Omega }}\) $E\subset (0,T)\times \mathrm{\Omega }$ . From the point of view of shape optimization, this is a parabolic analog of the Buttazzo-Dal Maso theorem in shape optimisation. The proof is based on second-order criteria and on an approximation-localisation procedure for admissible perturbations. This last part uses the theory of parabolic equations with measure data.