An Optimal Control Problem with Terminal Stochastic Linear Complementarity Constraints

TL;DR

This paper studies an optimal control problem with stochastic linear complementarity constraints, proposing discrete approximation methods, proving convergence, and providing error bounds, supported by a numerical example.

Contribution

It introduces a discrete approximation framework for stochastic linear complementarity constrained control problems, proving convergence and error estimates.

Findings

Feasible and optimal solutions exist under certain matrix conditions.

Discrete solutions converge to the original problem's solution with probability 1.

Error bounds and asymptotic behavior of the approximation methods are established.

Abstract

In this paper, we investigate an optimal control problem with terminal stochastic linear complementarity constraints (SLCC), 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 conditions that the expectation of the stochastic matrix in the SLCC 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 as $\epsilon \downarrow 0$, $\nu\to \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.