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Hierarchical Stability Notions and Lyapunov Functions for PDEs

Matthew M. Peet

Abstract

Unlike linear ordinary differential equations (ODEs), linear partial differential equations (PDEs) admit a multitude of non-equivalent notions of stability. This variety makes interpretation of Lyapunov stability results challenging. To simplify this interpretation, we propose a framework for hierarchical classification of notions of stability and Lyapunov conditions. To do this, for every well-posed PDE and set of boundary conditions, we define a fundamental state on $L_2$ corresponding to the minimal information needed to uniquely forward propagate the solution. Stability notions and Lyapunov functions are then defined in terms of this fundamental state. This gives rise to a hierarchy of stability notions, the weakest being fundamental state to PDE state stability. Other stability notions and Lyapunov conditions may then be interpreted relative to this weakest notion. Hierarchies are established for: Lyapunov, exponential and finite-energy stability. Sufficient Lyapunov conditions are defined in terms of operator inequalities. Illustrative examples and computational tools are provided.

Hierarchical Stability Notions and Lyapunov Functions for PDEs

Abstract

Unlike linear ordinary differential equations (ODEs), linear partial differential equations (PDEs) admit a multitude of non-equivalent notions of stability. This variety makes interpretation of Lyapunov stability results challenging. To simplify this interpretation, we propose a framework for hierarchical classification of notions of stability and Lyapunov conditions. To do this, for every well-posed PDE and set of boundary conditions, we define a fundamental state on corresponding to the minimal information needed to uniquely forward propagate the solution. Stability notions and Lyapunov functions are then defined in terms of this fundamental state. This gives rise to a hierarchy of stability notions, the weakest being fundamental state to PDE state stability. Other stability notions and Lyapunov conditions may then be interpreted relative to this weakest notion. Hierarchies are established for: Lyapunov, exponential and finite-energy stability. Sufficient Lyapunov conditions are defined in terms of operator inequalities. Illustrative examples and computational tools are provided.
Paper Structure (14 sections, 10 theorems, 29 equations, 1 table)

This paper contains 14 sections, 10 theorems, 29 equations, 1 table.

Table of Contents

  1. Introduction
  2. Fundamental (PIE) State and Bijection $\mathcal{ T}$
  3. Positivity, Boundedness, and Negativity
  4. Stability Notions and Lyapunov Functions
  5. Notions of Lyapunov Stability
  6. Lyapunov Conditions for Lyapunov Stability
  7. Notions of Exponential Stability
  8. More Exotic Flavours: Finite Energy Stability
  9. Sufficient Conditions for Notions of Stability
  10. Operator Inequalities for Lyapunov Stability
  11. Operator Inequalities for Exponential Stability
  12. Operator Inequalities for Finite Energy Stability
  13. Summary
  14. Conclusion

Key Result

Lemma 1

For $X,\alpha_{i,j},\beta_{i,j}$ as defined in Eq. eqn:gohberg_domain2, $[N_a]_{i,j}=\alpha_{i,j}$, $[N_b]_{i,j}=\beta_{i,j}$ if $det(N_a+N_b W(b-a))\neq 0$, let $\mathcal{ H}:=D^n$ and where $P=(N_a+N_b W(b-a))^{-1}N_bW(b-a)$ and Then for any $\mathbf{ u} \in X$ and $\mathbf{ x} \in L_2$, we have $\mathcal{ H} \mathcal{ T} \mathbf{ x}=\mathbf{ x}$, $\mathcal{ T} \mathcal{ H} \mathbf{ u}=\mathbf

Theorems & Definitions (24)

  • Definition 1
  • Lemma 1
  • Definition 2: Lyapunov positivity
  • Definition 3: Lyapunov bounds
  • Definition 4: Lyapunov Derivative Conditions
  • Definition 5
  • Lemma 2
  • proof
  • Lemma 3: Conditions for Lyapunov stability
  • proof
  • ...and 14 more