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

Abstract

In this paper, a general principle for coercivity in optimal control problems is investigated. Utilizing regularity properties of solutions to second-order elliptic PDEs on Lipschitz domains, we provide quantitative sufficient second-order optimality condition for control-constrained semilinear elliptic optimal control problems. The main property of the investigated optimal control problems is the absence of a Tikhonov regularization term on the objective function. For these problems, the second variation is a quadratic form in the linearized state, and its curvature function may vanish or change sign. The quantitative condition we introduce implies coercivity with respect to the $L^2$-norm of the linearized-states. As a consequence, the stability of the sufficient second-order condition under perturbations of the states and the tracking data is shown.