2014/10/08 by B. S. Mordukhovich, Yuan Tian, Mordukhovich, B. S. +1
Mathematics · #FOS: Mathematics #Optimization and Control (math.OC) #math.OC
paper · pdf · doi:10.48550/arxiv.1410.2207
arxiv created 2015/06/01 · arxiv updated 2015/06/02
This paper concerns the study of the generalized Bolza problem governed by differential inclusions satisfying the so-called "relaxed one-sided Lipschitzian" (ROSL) condition with respect to the state variables subject to various types of nonsmooth endpoint constraints. We construct discrete approximations of differential inclusions with ROSL right-hand sides by using the implicit Euler scheme for approximating time derivatives, and then we justify an appropriate well-posedness of such approximations. Our principal result establishes the strong approximation (in the sense of the W1,2 norm convergence) of an "intermediate" (between strong and weak minimizers) local optimal solution of the continuous-time Bolza problem under the ROSL assumption by optimal solutions of the implicitly discretized finite-difference systems. Finally, we derive necessary optimality conditions for the discretized Bolza problems via suitable generalized differential constructions of variational analysis. The obtained results on the well-posedness of discrete approximations and necessary optimality conditions allow us to justify a numerical approach to solve the generalized Bolza problem for one-sided Lipschitzian differential inclusions by using discrete approximations constructed via the implicit Euler scheme.