A terminal evasion problem is considered for discrete-time systems with an initially linear structure, where controls $u$, $U$ and $v$, $V$ operate; here $u$ and $v$ are included additively, and $U$ and $V$ are included in the system matrix. The presence of $U$ and $V$ leads to nonlinear systems of bilinear type. The aim of $v$ and $V$ is to avoid the trajectory hitting a given terminal set at a given final instant for any admissible $u$ and $U$. It is assumed that the terminal set is a parallelepiped, the controls $u$ and $v$ are constrained by parallelotope-valued constraints, $U$ and $V$ are constrained by interval constraints. A polyhedral method for synthesizing controls $v$ and $V$ is developed using polyhedral (parallelepiped-valued and parallelotope-valued) tubes, which can be found from recurrence relations using explicit formulas. To construct a solution to the problem under study, an auxiliary one-step polyhedral evasion problem with four controls $u$, $v$ and $U$, $V$ is first considered. Several solutions are found and compared for it. The paper generalizes the author previous results to systems in which the matrix controls $V$ may be present, and also strengthens them for the case without $V$ (in the sense of weakening assumptions and the possibility of constructing polyhedral tubes smaller in inclusion). Furthermore, guaranteed lower bounds are found for the deviation of the trajectory from the cross-sections of polyhedral solvability tubes.
Keywords: conflict-controlled systems, evasion problem, bilinearity, control synthesis, polyhedral methods, parallelepipeds, parallelotopes
Received April 23, 2026
Revised May 12, 2026
Accepted May 17, 2026
Elena Kirillovna Kostousova, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: kek@imm.uran.ru
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Cite this article as: E.K. Kostousova. On a polyhedral method of synthesis of additive and matrix controls for a terminal evasion problem in discrete-time systems. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol 32, no. 3, pp. 122–138.
