Leonard D. Berkovitz

Optimal feedback controls

Optimal control problems governed by ordinary differential equations with control constraints that are not necessarily compact are considered. Conditions imposed on the data and on the structure of the terminal sets imply that the minimum is attained and that the value function is locally Lipschitz. A necessary condition in terms of lower directional Dini derivates of the value function is given. The condition reduces to the Bellman–Hamilton–Jacobi (BHJ) condition at points of differentiability of the value, and for a subclass of the problems considered implies that the value is a viscosity solution of the BHJ equation. A strengthened version of the necessary condition gives an optimal feedback control and a procedure for approximating optimal controls.

The Existence of Value and Saddle Point in Games of Fixed Duration

Differential games of fixed duration are defined. The definition of strategy follows that of Friedman, while the definition of payoff follows that of Krasovskii and Subbotin. It is shown by relatively elementary methods that games of fixed duration which satisfy the Isaacs condition have values and saddle points. It is also shown under appropriate hypotheses on the data of the problem that if the Isaacs condition holds, then the value is uniformly Lipschitz continuous on bounded sets and satisfies the Isaacs equation at all points of differentiability. The relationship of the value as defined here to other values is studied.

Characterizations of the Values of Differential Games

The upper and lower directional derivates of the value function of a differential game are shown to satisfy certain relationships. These relationships imply that the value is a viscosity solution of the Isaacs equation.

The Equivalence of Some Necessary Conditions for Optimal Control in Problems with Bounded State Variables.

Abstract : The problem of optimal control is considered as earlier studied separately by Gamkrelidze, Berkovitz, and Dreyfus --wherein a constraint is placed on the state variables. The three have studied the problem from different viewpoints: Gamkrelidze accounts for the constraint by modifying Pontryagins maximum principle arguments; Berkovitz applies directly the classical theory developed for the problem of Bolza; and Dreyfus utilizes a dynamic programming approach. The results deduced by the latter approach, in some respects, appear to be unrelated to the Gamkrelidze and Berkovitz results, which are in agreement. It is demonstrated that the two sets of results are actually related. (Author)

A Theory of Differential Games

The differential games that we shall consider can be formulated intuitively as follows. The state of the game at time t is given by a vector x(t) in R n and is determined by a system of differential equations

Two Person Zero Sum Differential Games: An Overview

{{dx} \over {dt}} = f\left( {t,x,u(t),v(t)} \right)\,x({t_0}) = {x_0},

Addenda to “A variational problem related to an optimal filter problem with self-correlated noise”

(1.1) where u(t) is chosen by Player I at each time t and v(t) is chosen by Player II at each time t. The choices are constrained by the conditions u(t) ∊ Y and v(t)∊ Z, where Y and Z are preassigned sets in euclidean spaces. The choice of u(t) is governed by a set of rules or “strategy” U selected by Player I prior to the start of play and the choice of v(t) is governed by a “strategy” V selected by Player II prior to the start of play. Play proceeds from the initial point (t0,x0) until the point (t, ϕ(t)), where ϕ is the solution of (1.1), reaches some preassigned terminal set T. The point at which (t, ϕ(t))reaches T is called the terminal point and is denoted by (t f ,ϕ(t f )),or (t f ‚x f ). The payoff is