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

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 $y\in L^\infty(\OT)$ the cost functional $J(y)=\iint_\OT j_1(t,x,u)+\int_\O j_2(x,u(T,\cdot))$ 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 $-\kappa_0\leq y\leq \kappa_1\text{ a.e.}\,, \int_\O y(t,\cdot)=V_0$. 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^*<\kappa_1\}$ where $y^*$ is an optimiser. This set is in general non-empty and it is important (for instance for numerical applications) to understand the behaviour of $y^*$ in this set: which values can $ y^*$ 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