Inverse Optimal Control with Convex Features and Box Constraints

Abstract

We study an inverse optimal-control problem for identifying nonnegative weights in a convex-feature lower-level objective. The resulting model is an optimistic bilevel optimal-control problem whose lower level is a strongly convex, linearly constrained control problem with box constraints in \(L^2\). For the reduced response map \(x\mapsto (y_x,u_x)\), we prove global single-valuedness and an explicit Lipschitz estimate, without assuming differentiability of active sets. Exploiting the affine dependence of the lower-level functional on \(x\), we show that the value function is concave and Lipschitz and derive an asymptotically exact optimal-value relaxation. We further establish local identifiability and \(O(\delta)\) noise stability by a gradient-free argument based on second-order growth. Finally, a KKT reformulation yields a function-space MPCC; Scholtes relaxation gives C-stationarity under uniform multiplier boundedness. A decoupled scalar example provides an explicit strict-complementarity certificate via finiteness of threshold contacts.