The Effect of the Cost Functional on Asymptotic Solution to One Class of Zero-Sum Linear-Quadratic Cheap Control Differential Games

A finite-horizon zero-sum linear-quadratic differential game with non-homogeneous dynamics is considered. The key feature of this game is as follows. The cost of the control of the minimizing player (the minimizer) in the game’s cost functional is much smaller than the cost of the control of the maximizing player (the maximizer) and the cost of the state variable. This smallness is due to a positive small multiplier (a small parameter) for the quadratic form of the minimizer’s control in the integrand of the cost functional. Two cases of the game’s cost functional are studied: (i) the current state cost in the integrand of the cost functional is a positive definite quadratic form; (ii) the current state cost in the integrand of the cost functional is a positive semi-definite (but non-zero) quadratic form. The latter case has not yet been considered in the literature devoted to the analysis of cheap control differential games. For each of the aforementioned cases, an asymptotic approximation (by the small parameter) of the solution to the considered game is derived. It is established that the property of the aforementioned state cost (positive definiteness/positive semi-definiteness) has an essential effect on the asymptotic analysis and solution of the differential equations (Riccati-type, linear, and trivial), appearing in the solvability conditions of the considered game. The cases (i) and (ii) require considerably different approaches to the derivation of the asymptotic solutions to these differential equations. Moreover, the case (ii) requires developing a significantly novel approach. The asymptotic solutions of the aforementioned differential equations considerably differ from each other in cases (i) and (ii). This difference yields essentially different asymptotic solutions (saddle point and value) of the considered game in these cases, meaning it is of crucial importance to distinguish cases (i) and (ii) in the study of various theoretical and real-life cheap control zero-sum linear-quadratic differential games. The asymptotic solutions of the considered game in cases (i) and (ii) are compared with each other. An academic illustrative example is presented.