Stability Analysis of Optimal Control Problems with a Second-Order State Constraint - Hermant, Audrey (SIAM journal on optimization, 2009)
This paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second-order. The strengthened Legendre-Clebsch condition is assumed to hold, and no assumption on the structure of the contact set is made. Under a weak secondorder sufficient condition (taking into account the active constraints), we show that the solutions are Lipschitz continuous w.r.t. the perturbation parameter in the L^sup 2^ norm, and Holder continuous in the L∞ norm. We use a generalized implicit function theorem in metric spaces by Dontchev and Hager [SIAM J. Control Optim., 36 (1998), pp. 698-718]. The difficulty is that multipliers associated with second-order state constraints have a low regularity (they are only bounded measures). We obtain Lipschitz stability of a "primitive" of the state constraint multiplier. [PUBLICATION ABSTRACT]