Narrow Results

A planar interception problem of a maneuverable target is considered using the linearized kinematic model with variable velocities and first-order dynamics of the interceptor and target. The maneuverabilities of the interceptor and target are assumed to be variable. By using an auxiliary zero-sum linear-quadratic differential game with cheap controls of both players, an interception guidance law, linear with respect to the state vector, is derived. An analytical description of the set of initial positions (capture set) is obtained, from which this guidance law provides zero miss distance, subject to given maneuverabilities of the interceptor and target. A numerical example illustrating the analytical results is presented.

A linear-quadratic zero-sum singular differential game, where the cost functional does not contain the minimizer's control cost, is considered. Due to the singularity, the game cannot be solved either by applying the MinMax principle of Isaacs, or by using the Bellman–Isaacs equation method. In this paper, the solution of the singular game is obtained by using an auxiliary differential game with the same equation of dynamics and with a similar cost functional augmented by an integral of the square of the minimizer's control multiplied by a small positive weighting coefficient. This auxiliary game is a regular cheap control zero-sum differential game. For the analysis of such a cheap control differential game, in the present paper a singular perturbation technique is applied. Based on this analysis, the minimizing control sequence and the maximizer's optimal strategy in the original (singular) game are derived. Moreover, the existence of the value of the original game is established and its expression is derived. The solution is illustrated by an interception example.