The construction of the guaranteeing control in the time-optimization problem for linear discrete-time systems with summary control constraints

Context and relevance. The time-optimization problem is solved for a linear stationary system with discrete-time and summary first-order constraints on control. Objective. Demonstrate the possibility of constructing the guaranteeing control in the time-optimization problem. Hypothesis. The guaranteed solution found by applying the polyhedral approximation algorithm will converge to the optimal one. Methods and materials. This task has a number of features that complicate its solution using standard methods such as dynamic programming and the maximum principle. For this reason, it is proposed to use a geometric approach and an apparatus of null-controllable sets to solve the time-optimization problem. Results. For the case when the summary constraints are piecewise-linear, it is proved that all null-controllable sets are polyhedrons, which makes it possible to reduce the initial control problem to solving a number of linear programming problems. For arbitrary convex summary constraints, the possibility of constructing a guaranteeing solution in terms of time-optimization problem based on polyhedral approximation methods is shown. Conclusions. The convergence of the guaranteeing solution to the optimal one is investigated: it is proved that it will be completed in a finite number of iterations if the polyhedral approximation algorithm used guarantees convergence in the sense of the Hausdorff metric. The developed theoretical results are demonstrated using a numerical example.

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