Linear differential games with terminal payoff

A linear differential game whose payoff is determined by a convex function of the state vector at a given terminal time is investigated. An explicit expression for the greatest lower bound of the value of the game is obtained. A simple condition which guarantees the existence of the value of the game is derived by a geometric approach. Under this condition, instead of solving the Bellman equation, the value of the game can be calculated by maximizing a function on the unit sphere of the output space whose dimension is usually much less than that of the state space. The method of synthesizing the optimal strategies is also derived. Another game called a minimax energy game whose payoff is given by the difference between the consumed energies of both players is treated briefly, and an extension of the concept of controllability is discussed.