Stationary Hamilton-Jacobi Equations for Convex Control Problems: Uniqueness and Duality of Solutions

Abstract

This note summarizes some of the results of the author obtained in [7], with examples coming also from [6] and [8]. We focus on Hamilton-Jacobi and convex duality characterizations of the value function \(V : {\rm I\!R}^ n \mapsto {\rm I\!R}\) associated with a convex generalized problem of Bolza on an infinite time interval:

\(\displaystyle V ( \zeta ) = \inf \left\{\int^{ + \infty}_0 L ( x ( t ) , \dot x ( t )) dt | x ( 0 ) = \zeta \right\}\).     (1)

The minimization is carried out over all locally absolutely continuous arcs \(x : [0 , + \infty ) \mapsto {\rm I\!R}^n\) (that is, \(\dot x ( \cdot )\) is integrable on bounded intervals), subject to the initial condition x ( 0 ) = ζ. The Lagrangian \(L : {\rm I\!R}^{ 2 n} \mapsto \overline{\rm I\!R}\) is allowed to be nonsmooth and infinite-valued; the key assumption on it is the full convexity:

L ( x, v ) is convex in (x, v ).

Such problems of Bolza can model control problems with explicit linear dynamics, convex running costs, and control constraints; see Section 3 for references and examples.