A Maximum Principle for One Infinite Horizon Impulsive Control Problem

Abstract

Abstract The paper is concerned with a nonlinear impulsive control system with trajectories of bounded variation. Necessary conditions of optimality in a form of the Maximum Principle are derived for a class of infinite horizon impulsive optimal control problems. For the overtaking optimality criterion under the assumption that all gradients of the payoff function are bounded, we construct a transversality condition for the adjoint variable in terms of limit points of the gradient of the payoff function. In the case when this limit point is unique, this condition supplements the system of the Maximum Principle and determines a unique solution of the adjoint system. This solution can be written explicitly with the use of the (Cauchy type) formula proposed earlier by S.M. Aseev and A.V. Kryazhimskii. The key idea of the proof is the application of the convergence of subdifferentials within Halkin’s scheme.