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Input-to-state stability for bilinear feedback systems

René Hosfeld, Birgit Jacob, Felix Schwenninger, Marius Tucsnak

Abstract

Input-to-state stability estimates with respect to small initial conditions and input functions for infinite-dimensional systems with bilinear feedback are shown. We apply the obtained results to controlled versions of a viscous Burger equation with Dirichlet boundary conditions, a Schrödinger equation, a Navier--Stokes system and a semilinear wave equation.

Input-to-state stability for bilinear feedback systems

Abstract

Input-to-state stability estimates with respect to small initial conditions and input functions for infinite-dimensional systems with bilinear feedback are shown. We apply the obtained results to controlled versions of a viscous Burger equation with Dirichlet boundary conditions, a Schrödinger equation, a Navier--Stokes system and a semilinear wave equation.
Paper Structure (9 sections, 10 theorems, 107 equations)

This paper contains 9 sections, 10 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Local input-to-state stability for bilinear feedback systems
  4. Global input-to-state stability for bilinear feedback systems
  5. Examples
  6. The Burgers equation
  7. The Schrödinger equation
  8. The Navier--Stokes equations
  9. A wave equation

Key Result

Theorem 3

Let $A$ be the generator of an exponentially stable semigroup $(T(t))_{t \geq 0}$ with $M,\lambda>0$ such that eq:exp_stable holds, and let eq:LS be well-posed. Then for every $\omega \in (0, \lambda)$ there exists a constant $\varepsilon>0$ such that for all input data $z_0 \in X$ and $u_1 \in \mat it holds that eq:BFS admits a unique solution $z \in C([0,\infty);X)$ and an output $y \in \mathrm{

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Theorem 3
  • proof
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9
  • ...and 16 more