Linear-Quadratic Systems

Abstract

Linear systems with a quadratic performance criterion attract the attention of many investigations for the following reasons: they describe many actual phenomena adequately enough; on the other hand, the corresponding optimal control problems, as a rule, can be completely solved analytically. By way of an example let us specify works [4, 21, 74, 120, 124, 142, 149, 173, 211, 218]. Another version of linear-quadratic systems (with the continuous time, with the incomplete information, deterministic, etc.) were investigated in [1, 16, 60, 90, 91, 101, 131, 136]. Stochastic models with functional constraints were studied quite rarely [69, 175, 177, 180, 181, 183]. In the present chapter a complete analytical investigation is presented for the three versions of the linear-quadratic system with constraints: the finite horizon case, the discounted homogeneous model, and the homogeneous model with average losses. The results of Section 3.1 cannot be applied because the spaces X and A are not compact. The solution is of the Markov (stationary) selector type which depends on the initial probability distribution in the case of a finite horizon and in the discounted model.