We study a minimax optimal control problem with finite horizon and additive final cost. After introducing an auxiliary problem, we analyze the dynamical programming principle (DPP) and we present a Hamilton-Jacobi-Bellman (HJB) system. We prove the existence and uniqueness of a viscosity solution for this system. This solution is the cost function of the auxiliary problem and it is possible to get the solution of the original problem in terms of this solution.2000 Mathematics Subject Classification: 49L20, 49L25, 49K35, 49K15.
Introduction.The optimization of dynamic systems where the criterion is the maximum value of a function is a frequent problem in technology, economics, and industry. This problem appears, for example, when we attempt to minimize the maximum deviation of controlled trajectories with respect to a given "model" trajectory. Minimax problems differ from those usually considered in the optimal control literature where a cumulative cost is minimized. Since in some cases, minimax problems describe more appropriately decision problems arisen in controlled systems whose performance is evaluated with a unique scalar parameter, the minimax optimization has received considerable attention in recent publications (see, e.g., [4,5,6,7,8,9,10,11,12,13,15,16,17,18,19,22]). Furthermore, minimax problems are related to the design of robust controllers (see [14]).In addition, from the academic point of view, the minimax optimal control problem is of interest in the area of game theory because minimax problems can be seen as a game (see [18]) where a player applies ordinary controls and the other one-using complete and privileged information-chooses a stopping time. Problems of this type lead to the treatment of nonlinear partial differential inequalities akin to those appearing in the obstacle problem (with obstacle given in explicit or implicit form, see [10]). To find solutions of these systems, we must consider generalized solutions-even discontinuous solutions-since commonly there do not exist classical solutions of such systems (see [2,3]). The treatment of the infinite horizon problem also presents great analytical difficulties because the optimal cost is neither necessarily lower semicontinuous nor upper semicontinuous. Moreover, the