Localising optimality conditions for the linear optimal control of semilinear equations \emph{via} concentration results for oscillating solutions of linear parabolic equations

Abstract

We propose a fine analysis of second order optimality conditions for the optimal control of semi-linear parabolic equations with respect to the initial condition. More precisely, we investigate the following problem: maximise with respect to $y\in L^{\infty }((0;T)\times \Omega )$ the cost functional $J(y)={\iint }_{(0;T)\times \Omega }{j}_1(t,x,u)+{\int }_{\Omega }{j}_2(x,u(T,\cdot ))$ where ${\partial }_{t}u-\Delta u=f(t,x,u)+y,u(0,\cdot )=u_0$ with some classical boundary conditions, under constraints of the form $-{\kappa }_0\le y\le {\kappa }_1\text{ a.e.},{\int }_{\Omega }y(t,\cdot )=V_0$. This class of problems arises in several application fields. A challenging feature of these problems is the study of the so-called abnormal set $\{-{\kappa }_0<y^{\ast }<{\kappa }_1\}$ where $y^{\ast }$ is an optimiser. This set is in general non-empty and it is important (for instance for numerical applications) to understand the behaviour of $y^{\ast }$ in this set: which values can $y^{\ast }$ take? In this paper, we introduce a Laplace-type method to provide some answers to this question. This Laplace type method is of independent interest.