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<p>It is known in control theory that the problem how an interconnection of ISS (abbreviation of input-to-state stability) subsystems remains ISS leads to a cycle condition, which is a set of inequalities involving the composition of unbounded Kamke functions. In this paper we convert the problem on those inequalities to functional equations and obtain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {K}_\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">K</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> solutions by finding fixed points in a complete metric space or a locally convex topological linear space. We also give <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {K}_\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">K</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> solutions by piecewise construction. Our results can be applied to discussing ISS of interconnection.</p>

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Kamke functions for stability of interconnected systems

  • 期刊论文

  • 发表日期：

    2026年4月29日

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  • Xiaobing Gong,
  • Xiao Tang,
  • Weinian Zhang

摘要

It is known in control theory that the problem how an interconnection of ISS (abbreviation of input-to-state stability) subsystems remains ISS leads to a cycle condition, which is a set of inequalities involving the composition of unbounded Kamke functions. In this paper we convert the problem on those inequalities to functional equations and obtain \(\mathcal {K}_\infty \) K ∞ solutions by finding fixed points in a complete metric space or a locally convex topological linear space. We also give \(\mathcal {K}_\infty \) K ∞ solutions by piecewise construction. Our results can be applied to discussing ISS of interconnection.