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<p>We derive converse Lyapunov theorems for input-to-state stability (ISS) of linear infinite-dimensional analytic systems. While we show that ISS in general does not imply the existence of a coercive quadratic ISS Lyapunov function, even if the input operator is bounded, we prove that indeed quadratic ISS Lyapunov functions always exist for <i>p</i>-admissible input operators with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p&lt;2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>&lt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, provided the semigroup is similar to a contraction on a Hilbert space. The constructions are semi-explicit and rely on classical results on analytic semigroups and similarity to contractive ones. In the case of self-adjoint generators, they coincide with the canonical Lyapunov function being the norm squared.</p>

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Quadratic ISS Lyapunov functions for linear analytic systems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月8日

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  • Andrii Mironchenko,
  • Felix L. Schwenninger

摘要

We derive converse Lyapunov theorems for input-to-state stability (ISS) of linear infinite-dimensional analytic systems. While we show that ISS in general does not imply the existence of a coercive quadratic ISS Lyapunov function, even if the input operator is bounded, we prove that indeed quadratic ISS Lyapunov functions always exist for p-admissible input operators with \(p<2\) p < 2 , provided the semigroup is similar to a contraction on a Hilbert space. The constructions are semi-explicit and rely on classical results on analytic semigroups and similarity to contractive ones. In the case of self-adjoint generators, they coincide with the canonical Lyapunov function being the norm squared.