Differential Stability of Convex Discrete Optimal Control Problems

TL;DR

This paper investigates the differential stability of convex discrete optimal control problems in Banach spaces, providing subdifferential estimates for the optimal value function, especially under nondifferentiability conditions.

Contribution

It extends existing results by deriving subdifferential estimates for the optimal value function in convex discrete optimal control, including cases with nondifferentiable objectives.

Findings

Upper estimate for subdifferential of the optimal value function.

Equality of the estimate when the objective is differentiable.

Singular subdifferential always contains the origin.

Abstract

Differential stability of convex discrete optimal control problems in Banach spaces is studied in this paper. By using some recent results of An and Yen [Appl. Anal. 94, 108--128 (2015)] on differential stability of parametric convex optimization problems under inclusion constraints, we obtain an upper estimate for the subdifferential of the optimal value function of a parametric convex discrete optimal control problem, where the objective function may be nondifferentiable. If the objective function is differentiable, the obtained upper estimate becomes an equality. It is shown that the singular subdifferential of the just mentioned optimal value function always consists of the origin of the dual space.