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 Ural Math. J., 2020, Volume 6, Issue 1, Pages 71–83 (Mi umj112)

The limits of applicability of the linearization method in calculating small-time reachable sets

Mikhail I. Gusev

Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: The reachable sets of nonlinear systems are usually quite complicated. They, as a rule, are non-convex and arranged to have rather complex behavior. In this paper, the asymptotic behavior of reachable sets of nonlinear control-affine systems on small time intervals is studied. We assume that the initial state of the system is fixed, and the control is bounded in the $\mathbb{L}_2$-norm. The subject of the study is the applicability of the linearization method for a sufficiently small length of the time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is defined in terms of the Banach-Mazur metric in the space of sets. The conditions depend on the behavior of the controllability Gramian of the linearized system — the smallest eigenvalue of the Gramian should not tend to zero too quickly when the length of the time interval tends to zero. The indicated asymptotic behavior occurs for a reasonably wide class of second-order nonlinear control systems but can be violated for systems of higher dimension. The results of numerical simulation illustrate the theoretical conclusions of the paper.

Keywords: Nonlinear control systems, Small-time reachable sets, Asymptotics, Integral constraints, Linearization.

 Funding Agency Grant Number Ural Mathematical Center This work is supported by the Research and Education Center of IMM UB of RAS in the framework of the Ural Mathematical Center (project “Set-Valued Dynamics in Control and Estimation Problems for Dynamical Systems with Uncertainty”).

DOI: https://doi.org/10.15826/umj.2020.1.006

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Citation: Mikhail I. Gusev, “The limits of applicability of the linearization method in calculating small-time reachable sets”, Ural Math. J., 6:1 (2020), 71–83

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\paper The limits of applicability of the linearization method in calculating small-time reachable sets
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\pages 71--83
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