Emmanuel N. Barron

Optimal control problems with no turning back

Abstract In this paper we consider the maximization of a payoff functional subject to a differential equality contraint over the class of monotonically increasing functions with values in [0, 1]. We will show that an optimal control exists, derive the system of inequalities (similar to a quasi-variational inequality) that the value function satisfies and derive various properties of the value function sufficient to characterize it. Furthermore, we derive a perturbation result using the theory of Lipschitz controls. Finally, we also consider the case when the control functions are of bounded total variation and relate the problems considered herein to the impulse control problem of Bensoussan-Lions.

Differential Games with Lipschitz Control Functions and Fixed Initial Control Positions

Abstract Differential games in which one or both players are restricted to choosing control functions which are uniformly Lipschitz continuous and which start at fixed initial conditions always have a value. We derive the Hamilton-Jacobi equation which this value satisfies a.e. as a function of the initial time t , the initial state x , and the initial control positions. We also show that a “Lipschitz Game” has an approximate saddle point in pure strategies. The approach of Friedman to differential games is used.

Functions Which Are Quasiconvex under Linear Perturbations

A quasiconvex function is a function which has convex sublevel sets. This paper studies robustly quasiconvex functions, that is, quasiconvex functions which remain quasiconvex under small linear perturbations. Relations to pseudoconvexity and other generalized convexity concepts and necessary and sufficient first-order conditions for robust quasiconvexity of smooth functions are presented. Convex-analytic properties and convexification of robustly quasiconvex functions are studied. Supporting robustly quasiconvex functions by simpler functions is discussed, with duality theory as motivation. Through the use of a second-order condition for robust quasiconvexity of nonsmooth functions, nontrivial examples of such functions are given.

Control problems for parabolic equations on controlled domains

Given the one-dimensional heat equation vt = vxx on the controlled domain Q(y) = {(t, x); 0 < x < y(t), 0 < t < T} subject to some initial-boundary conditions, we study the problem of optimally selecting y(·) from some admissible class so as to maximize a given payoff of fixed duration. Q(y) is thus a controlled domain. We also study the problem in which the heat equation holds in Q(y, z) = {z(t) < x < y(t), 0 < t < T}; z minimizing, y maximizing, i.e., the differential game. The principle techniques involved are (i) transforming the controlled domain to an uncontrolled domain and then (ii) using the method of lines for parabolic equations to enable us to use known results for control systems governed by ordinary differential equations. Sufficient conditions for existence in an admissible class is given and the method of lines allows numerical techniques to be applied to determine the optimal control in our class.

Remarks on the existence of value in differential games

Abstract We give a new criterion for the existence of value in differential games. The method of proof involves Lipschitz differential games and hence extends to games with more general dynamics. The connection between using measurable control functions or simply constants is clarified.