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<p>Smoothness of the boundary of reachable sets of a&#xa0;linear control system with constraints on a&#xa0;control in the form of a&#xa0;ball in the space <i>L</i><sub><i>p</i></sub> for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p &gt; 1\)</EquationSource> </InlineEquation> is studied. It is shown that, under the condition of controllability of the system, the boundary of its reachable set is a&#xa0;<i>C</i><sup>1</sup>-smooth submanifold of dimension <i>n</i> − 1 in the state space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb R^n\)</EquationSource> </InlineEquation>. The proof relies on the properties of support functions and duality relations for convex subsets of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb R^n\)</EquationSource> </InlineEquation>.</p>

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Smoothness of the boundary of reachable sets under control constraints in Lp and duality relations

  • 期刊论文

  • 发表日期：

    2026年1月21日

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  • Mikhail Gusev

摘要

Smoothness of the boundary of reachable sets of a linear control system with constraints on a control in the form of a ball in the space Lp for \(p > 1\) is studied. It is shown that, under the condition of controllability of the system, the boundary of its reachable set is a C1-smooth submanifold of dimension n − 1 in the state space \(\mathbb R^n\) . The proof relies on the properties of support functions and duality relations for convex subsets of \(\mathbb R^n\) .