W. E. Schmitendorf

On a class of nonzero-sum, linear-quadratic differential games

A class of two-player, nonzero-sum, linear-quadratic differential games is investigated for Nash equilibrium solutions when both players use closed-loop control and when one or both of the players are required to use open-loop control. For three formulations of the game, necessary and sufficient conditions are obtained for a particular strategy set to be a Nash equilibrium strategy set. For a fourth formulation of the game, where both players use open-loop control, necessary and sufficient conditions for the existence of a Nash equilibrium strategy set are developed. Several examples are presented in order to illustrate the differences between this class of differential games and its zero-sum analog.

Existence of optimal open-loop strategies for a class of differential games

In this paper, the class of differential games with linear system equations and a quadratic performance index is investigated for saddlepoint solutions when one or both of the players use open-loop control. For each formulation of the game, a necessary and sufficient condition is obtained for the existence of an optimal strategy pair that generates a regular optimal path. For those cases where a solution exists, the unique saddle-point solution is presented. Also, relationships are established between the time intervals of existence of solutions for the various formulations of the game.

Conjugate-point conditions for variational problems with delayed argument

A variational problem with delayed argument is investigated. The existing necessary conditions are first reviewed. New results, two conjugate-point conditions, are then derived for this problem. The method of proof is similar to that used by Bliss for the classical problem. An example shows that the two conditions are not equivalent, and that the first-order necessary conditions, the strengthened Legendre conditions, and the conjugate-point conditions do not in general constitute a set of sufficient conditions for the delay problem. It is shown that a special case, referred to as the separated-integrand problem, leads to considerable simplification of the results for the general problem.

A sufficiency condition for coalitive pareto-optimal solutions

In a k-player, nonzero-sum differential game there exists the possibility that a group of players will form a coalition and work together. If all k players form the coalition, the criterion usually chosen is Pareto optimality, whereas if the coalition consists of only one player, a minmax or Nash equilibrium solution is sought. In this paper, games with coalitions of more than one but less than k players are considered. Coalitive Pareto optimality is chosen as the criterion. Sufficient conditions are presented for coalitive Paretooptimal solutions and the results are illustrated with an example.

Differential Games Without Pure Strategy Saddle-Point Solutions

In some two-player, zero-sum differential games, pure strategy saddle-point solutions do not exist. For such games, the concept of a minmax strategy is examined, and sufficient conditions for a control to be a minmax control are presented. Both the open-loop and the closed-loop cases are considered.

The associated disturbance-free system: A means for investigating the controllability of a disturbed system

This paper addresses the problem of state controllability in the presence of additive disturbances. In contrast to the stochastic controllability problem, the formulation given here does not require a probabilistic description of the uncertainty. Instead, the objective is to steer the state to the target in a so-called guaranteed sense. That is, one only assumes the availability of ana priori bounding setQ for the values of the uncertainty. Within this context, the goal is to decide whether one can find a control, having values restricted to a set Ω, which guarantees the transfer of the state to a prespecified target. Hence, we are led to define the notion of (Ω,Q)-controllability.The main result of this paper is given in Theorem 4.1. Loosely speaking, this theorem gives criteria for (Ω,Q)-controllability, which involve looking at the convergence properties of certain scalar-valued time functions which are created from the known data. In order to achieve this result, it is first shown that the given system (Sx with its targetX(·) can be associated with another system (SY) having targetY(·). Although (Sx) has disturbances in its dynamical description, (SY) is disturbance-free. Moreover, it is shown that (Sx) is (Ω,Q)-controllable toX(·) if and only if (SY) is Ω-controllable toY(·). After transforming the pair [(Sx),X(·)] into [(SY,Y(·)], one can use the known results (e.g., Ref. 1) on Ω-controllability of deterministic systems to determine the controllability properties of the system with disturbances. This line of thought motivates calling (SY) the associated disturbance-free system. Finally, it is also shown how the required calculations can be greatly simplified for the special case of a polyhedral target.

Optimal control of systems with multiple criteria when disturbances are present

Optimal control problems with a vector performance index and uncertainty in the state equations are investigated. Nature chooses the uncertainty, subject to magnitude bounds. For these problems, a definition of optimality is presented. This definition reduces to that of a minimax control in the case of a scalar cost and to Pareto optimality when there is no uncertainty or disturbance present. Sufficient conditions for a control to satisfy this definition of optimality are derived. These conditions are in terms of a related two-player zero-sum differential game and suggest a technique for determining the optimal control. The results are illustrated with an example.

A conjugate-point condition for a class of differential games.

A conjugate-point necessary condition is derived for a class of differential games. This is done by considering the conjugate-point condition for the minimum problem and maximum problem associated with a differential game. Two definitions of a conjugate point and two conjugate-point necessary conditions result. These two definitions and necessary conditions are shown to be equivalent and are combined into one definition and one necessary condition.

Pontryagin's principle for problems with isoperimetric constraints and problems with inequality terminal constraints: Reply

A reply is made to the comment of Forster and Long.

Global reachability results for systems with constrained controllers

The global reachability problem is to determine if, given a specified initial state, the state of a system can be steered via an admissible control to every point in the state space. This paper addresses the problem of global reachability when there are magnitude constraints on the controls. Necessary conditions and sufficient conditions for a system to be globally reachable are presented. The results are compared with those available for the global controllability problem.