Sensitivity Analysis for Parametric Optimal Control of Semilinear Parabolic Equations

Parametric optimal control problems for semilinear parabolic equations are considered. Using recent Lipschitz stability results for solutions of such problems, it is shown that, under standard coercivity conditions, the solutions are Bouligand differentiable (in $L^p$, p finite) functions of the parameter. The differentials are characterized as the solutions of accessory linear-quadratic problems. A uniform second order expansion of the optimal value function is obtained, as a corollary.