This paper is devoted to the study of discrete optimal control problems. We aim to obtain more constructive optimality conditions under weakened convexity assumptions. Based on a new approach introduced in this work, an optimality condition with respect to every component is obtained in the form of a global maximum principle. In addition, an optimality condition with respect to one of the components of a control in the form of the global maximum principle and with respect to another component of a control in the form of the linearized maximum principle are obtained. Furthermore, various second-order optimality conditions in terms of singular and quasi-singular controls with respect to the components are obtained on the fly.
In the present paper, a discrete optimization problem with rather general input data (without assumptions of convexity and smoothness) is considered. Taking into account the specific character of the discrete system, a necessary optimality condition which is not formulated in terms of the Hamilton-Pontryagin function is obtained.
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