An Optimal Control Problem with Terminal Stochastic Linear Complementarity Constraints

In this paper, we investigate an optimal control problem with a crucial ODE constraint involving a terminal stochastic LCP and its discrete approximation using the relaxation, the sample average approximation (SAA), and the implicit Euler time-stepping scheme. We show the existence of feasible solutions and optimal solutions to the optimal control problem and its discrete approximation under the condition that the expectation of the stochastic matrix in the stochastic LCP is a Z-matrix or an adequate matrix. Moreover, we prove that the solution sequence generated by the discrete approximation converges to a solution of the original optimal control problem with probability 1 by the repeated limits in the order of \epsilon \downarrow 0, \nu \rightarrow \infty, and h \downarrow 0, where \epsilon is the relaxation parameter, \nu is the sample size, and h is the mesh size. We also provide asymptotics of the SAA optimal value and error bounds of the time-stepping method. A numerical example is used to illustrate the existence of optimal solutions, the discretization scheme, and error estimation.