Differential Stability of Discrete Optimal Control Problems with Possibly Nondifferentiable Costs

In this paper, a family of discrete optimal control problems that depend on parameters is considered. The control problems are reformulated as parametric optimization problems. By establishing/exploiting abstract results on subdifferentials of optimal value functions of parametric optimization problems, we derive formulas for estimating/computing subdifferentials of optimal value functions of parametric discrete optimal control problems in both nonconvex and convex cases. Namely, for control problems with nonconvex costs, upper-evaluations on the regular subdifferential and the limiting (Mordukhovich) subdifferential of the optimal value function are obtained without using the (strict) differentiability of the costs. Meanwhile, for control problems with convex costs, besides results on estimating/computing the subdifferential (in the sense of convex analysis) of the optimal value function, it is worth pointing out that some properties of the optimal value function are first discussed in this paper.