Generalized Gradient Elements for Nonsmooth Optimal Control Problems

Recent advances in nonsmooth sensitivity analysis are extended to describe particular elements of Clarke's generalized gradient for the nonsmooth objective function of a nonsmooth optimal control problem, in terms of states of an auxiliary dynamic system. The considered optimal control problem is a generic nonlinear open-loop problem, in which the cost function and the right-hand side function describing the system dynamics may each be nonsmooth. The desired generalized gradient elements are obtained under two parametric discretizations of the control function: a representation as a linear combination of basis functions, and a piecewise constant representation. If the objective function under either discretization is convex, then the corresponding generalized gradient elements are subgradients, without requiring any convexity assumptions on the system dynamics.