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We propose a fine analysis of second order optimality conditions for theoptimal control of semi-linear parabolic equations with respect to the initialcondition. More precisely, we investigate the following problem: maximise withrespect 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 classicalboundary conditions, under constraints of the form $-\kappa_0\leq y\leq\kappa_1\text{ a.e.}\,, \int_\Omega y(t,\cdot)=V_0$. This class of problemsarises in several application fields. A challenging feature of these problemsis 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 aLaplace-type method to provide some answers to this question. This Laplace typemethod is of independent interest.

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