How to check numerically the sufficient optimality conditions for infinite-dimensional optimization problems

We consider general non-convex optimal control problems. Many results for such problems rely on second-order sufficient optimality conditions. We propose a method to verify whether the second-order sufficient optimality conditions hold in a neighborhood of a numerical solution. This method is then applied to abstract optimal control problems. Finally, we consider an optimal control problem subject to a semi-linear elliptic equation that appears to have multiple local minima. Mathematics Subject Classification (2000). Primary 49J20; Secondary 49K20, 65N15.