A Mironchenko. Lyapunov method for stability of infinite dimensional systems
Talk at the Colloquium of the Department of Mathematics, University of Bayreuth, Germany.
Title:
Lyapunov criteria for stability of infinite-dimensional systems
Speaker: Andrii Mironchenko, http://www.mironchenko.com/
Slides:
http://mironchenko.com/__My_php_sourc...
00:28 Introduction
02:17 Outline
03:15 Lyapunov functions for ODEs
07:53 LFs for nonlinear infinite-dimensional systems
11:52 LFs for linear systems: Lyapunov view on Datko, Pazy and Littman theorems
17:19 Differences between nonlinear and linear Lyapunov theories
18:43 Non-coercive Lyapunov functions
23:11 Quadratic coercive Lyapunov functions for linear systems
27:42 Take-Home Message I
============== Part 2 ===============
29:47 Systems with inputs
31:03 ISS and why it is needed
34:29 ISS Lyapunov functions and Lyapunov theorems
42:49 Linear systems
46:56 Non-coercive ISS Lyapunov theorem for boundary control systems
48:31 Lyapunov functions for a heat equation with Neumann and Dirichlet input
52:55 Coercive quadratic L^2-ISS Lyapunov functions for analytic systems
56:54 Conclusion for 2nd part & Literature
The talk is based on the following papers:
1. Mironchenko, Wirth. Non-coercive Lyapunov functions for infinite-dimensional systems. JDE, 2019.
2. Mironchenko, Wirth. Existence of non-coercive Lyapunov functions is equivalent to integral uniform global asymptotic stability, MCSS, 2019.
3. Jacob, Mironchenko, Partington, Wirth. Non-coercive Lyapunov functions for
input-to-state stability of infinite-dimensional systems. SICON, 2020.
4. Mironchenko, Schwenninger. On coercivity of ISS Lyapunov functions for linear
infinite-dimensional systems. Submitted to MCSS, 2023.
Abstract:
We start by a short recap of classical Lyapunov characterizations of asymptotic stability of
nonlinear infinite-dimensional systems.
Next we present the Lyapunov view on classical criteria for exponential stability of linear systems, namely
theorems of Datko, Pazy and Littman, and argue that there is a
deep difference between Lyapunov results used in linear and nonlinear systems theory.
We build a bridge between these results by introducing non-coercive Lyapunov functions,
and showing that for many system classes existence of a
non-coercive Lyapunov function is sufficient to ensure the asymptotic stability of a dynamical system.
In the second part of the talk, we consider the infinite-dimensional systems with inputs (boundary control systems)
and show how coercive and non-coercive Lyapunov functions can be used to analyze the input-to-state stability of such systems.
Finally, we present non-coercive and coercive quadratic ISS Lyapunov theorems for ISS of linear boundary control systems.