Coercive quadratic converse ISS Lyapunov theorems for linear analytic systems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915499693047808 |
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| author | Mironchenko, Andrii Schwenninger, Felix |
| author_facet | Mironchenko, Andrii Schwenninger, Felix |
| contents | We derive converse Lyapunov theorems for input-to-state stability (ISS) of linear infinite-dimensional analytic systems. We show that input-to-state stability of a linear system does not imply existence of a coercive quadratic ISS Lyapunov function, even if the input operator is bounded. If, however, the semigroup is similar to a contraction semigroup on a Hilbert space, then a quadratic ISS Lyapunov function always exists for any input operator that is bounded, or more generally, $p$-admissible with $p<2$. The constructions are semi-explicit and, in the case of self-adjoint generators, coincide with the canonical Lyapunov function being the norm squared. Finally, we construct a family of non-coercive ISS Lyapunov functions for analytic ISS systems under weaker assumptions on $B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_15093 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Coercive quadratic converse ISS Lyapunov theorems for linear analytic systems Mironchenko, Andrii Schwenninger, Felix Optimization and Control 37C75, 93C25, 93D09 We derive converse Lyapunov theorems for input-to-state stability (ISS) of linear infinite-dimensional analytic systems. We show that input-to-state stability of a linear system does not imply existence of a coercive quadratic ISS Lyapunov function, even if the input operator is bounded. If, however, the semigroup is similar to a contraction semigroup on a Hilbert space, then a quadratic ISS Lyapunov function always exists for any input operator that is bounded, or more generally, $p$-admissible with $p<2$. The constructions are semi-explicit and, in the case of self-adjoint generators, coincide with the canonical Lyapunov function being the norm squared. Finally, we construct a family of non-coercive ISS Lyapunov functions for analytic ISS systems under weaker assumptions on $B$. |
| title | Coercive quadratic converse ISS Lyapunov theorems for linear analytic systems |
| topic | Optimization and Control 37C75, 93C25, 93D09 |
| url | https://arxiv.org/abs/2303.15093 |