Solution of a zero-sum linear quadratic differential game with singular control cost of minimiser

Abstract

We consider a finite horizon, zero-sum linear quadratic differential game. The feature of this game is that a weight matrix of the minimiser’s control cost in the cost functional is singular. Due to this singularity, the game can be solved neither by applying the Isaacs MinMax principle nor using the Bellman–Isaacs equation approach, i.e. this game is singular. A previous paper of one of the authors analysed such a game in the case where the cost functional does not contain the minimiser’s control cost at all, i.e. the weight matrix of this cost equals zero. In this case, all coordinates of the minimiser’s control are singular. In the present paper, we study the general case where the weight matrix of the minimiser’s control cost, being singular, is not, in general, zero. This means that only a part of the coordinates of the minimiser’s control is singular, while others are regular. The considered game is treated by a regularisation, i.e. by its approximate conversion to an auxiliary regular game. The latter has the same equation of dynamics and a similar cost functional augmented by an integral of the squares of the singular control coordinates with a small positive weight. Thus, the auxiliary game is a partial cheap control differential game. Based on a singular perturbation’s asymptotic analysis of this auxiliary game, the existence of the value of the original (singular) game is established, and its expression is obtained. The maximiser’s optimal state feedback strategy and the minimising control sequence in the original game are designed. It is shown that the coordinates of the minimising control sequence, corresponding to the regular coordinates of the minimiser’s control, are point-wise convergent in the class of regular functions. The optimal trajectory sequence and the optimal trajectory in the considered singular game also are obtained. An illustrative example is presented.