The paper studies discrete/finite-difference approximations of optimal control problems governed by continuous-time dynamical systems with endpoint constraints. Finite-difference systems, considered as parametric control problems with the decreasing step of discretization, occupy an intermediate position between continuous-time and discrete-time (with fixed steps) control processes and play a significant role in both qualitative and numerical aspects of optimal control. In this paper we derive an enhanced version of the Approximate Maximum Principle for finite-difference control systems, which is new even for problems with smooth endpoint constraints on trajectories and occurs to be the first result in the literature that holds for nonsmooth objectives and endpoint constraints. The results obtained establish necessary optimality conditions for constrained nonconvex finite-difference control systems and justify stability of the Pontryagin Maximum Principle for continuous-time systems under discrete approximations.
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