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We study a minimax optimal control problem with finite horizonand additive final cost. After introducing an auxiliary problem,we analyze the dynamical programming principle (DPP) and wepresent a Hamilton-Jacobi-Bellman (HJB) system. We prove theexistence and uniqueness of a viscosity solution for this system.This solution is the cost function of the auxiliary problem andit is possible to get the solution of the original problem interms of this solution

On a system of Hamilton-Jacobi-Bellman inequalities associated to a minimax problem with additive final cost