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Abstract <p> An algorithm is proposed for constructing an asymptotic approximation to the solution ofa discrete-time weakly controllable linear-quadratic optimal control problem with small step sizein the critical case. The asymptotic expansion consists of a sum of a regular series and twoboundary-layer series containing boundary functions in neighborhoods of two fixed endpoints. Theconstruction of the asymptotic expansion is based on the decomposition of the state space intoorthogonal sums of subspaces and the use of the corresponding orthogonal projectors. Explicitrelations for determining the terms of the asymptotic expansion of any order are provided. Anexample illustrating the proposed method is presented.</p>

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Asymptotics of the Solution of Discrete-Time Linear-Quadratic Optimal Control Problems with Small Step and Weak Control in the Critical Case

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • G. A. Kurina,
  • N. T. Hoai

摘要

Abstract

An algorithm is proposed for constructing an asymptotic approximation to the solution ofa discrete-time weakly controllable linear-quadratic optimal control problem with small step sizein the critical case. The asymptotic expansion consists of a sum of a regular series and twoboundary-layer series containing boundary functions in neighborhoods of two fixed endpoints. Theconstruction of the asymptotic expansion is based on the decomposition of the state space intoorthogonal sums of subspaces and the use of the corresponding orthogonal projectors. Explicitrelations for determining the terms of the asymptotic expansion of any order are provided. Anexample illustrating the proposed method is presented.