On quantitative sufficient second-order optimality conditions for elliptic optimal control problems

In this paper, a quantitative condition for optimality for distributed optimal control problems with box-constraints that are subject to a semilinear elliptic equation is considered. An important property of the investigated optimal control problems is the absence of a Tikhonov regularization. It is well known that at a given control, the second variation is a quadratic form in the linearized state, and its curvature coefficient function may vanish or change sign. We present a quantitative condition that implies coercivity with respect to the $L^2$-norm of the linearized-states. As a consequence, the stability of the second-order condition under perturbations of the states and the tracking data is shown.