Appl. Math. Control Sci., 2021, Issue 3, Pages 7–34(Mi pstu49)
Analogue of the discrete maximum principle and the necessary optimality condition of singular controls in one two-parametric discrete optimal control problem
Abstract:
A two-stage (stepwise) optimal control problem for linear two-parameter systems with distributed control functions is considered. The aim of the work is to establish the necessary optimality condition under the assumption that the convexity of the set of admissible controls is satisfied and the connection condition is nonlinear. Using increments of the quality functional in the form of two-dimensional linear inhomogeneous systems of difference equations, a formula is obtained that allows one to obtain both a discrete analogue of the Pontryagin maximum principle and to study the case of its degeneration. A theorem is formulated that is an analogue of the discrete Pontryagin maximum principle for the problem under consideration. In the case of special controls, the discrete maximum principle degenerates and, therefore, becomes ineffective, including in the verification sense. Therefore, it is necessary to have new necessary conditions for optimality. A special, in the sense of the Pontryagin maximum principle, case of a discrete maximum condition, under which admissible controls are considered special, is studied. A necessary condition for optimality of singular controls is established.
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