Shadow Prices and Duality for a Class of Optimal Control Problems

Abstract

A class of optimal control problems is considered in which the cost functional is locally Lipschitz (not necessarily convex or differentiable) and the dynamics linear and/or convex. By using generalized gradients and duality methods of functional analysis, necessary conditions are obtained in which the dual variables admit interpretation as shadow prices (or rates of change of the value function). Applications are presented in three settings: infinite-horizon optimal control, optimal control of partial differential equations, and a variational problem with unilateral state constraints. A theorem is proved which characterizes the generalized gradients of integral functionals on $L^p $.