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Extensions of $\mathcal{KL}$ and Lyapunov Functions for Discrete-time Dynamical System Peaks Analysis

Assalé Adjé

Abstract

In this paper, we extend two classes of functions involved in asymptotic stability analyses. The goal of this extension is to study a maximization problem on the reachable values of a discrete-time dynamical system. This specific maximization problem is called a peak computation problem. The problem consists in finding a pair composed of an initial state and a time that maximizes a given function over states. The paper focuses on the time component of the optimal solution, which is an integer as the time is discrete. We apply a method developed in previous papers to compute an upper bound of the greatest index maximizer of a real sequence. The method uses a formula based on a pair of a strictly increasing and continuous function on [0,1] and a convergent geometric sequence that provides an upper bound of the analyzed sequence. This pair is proven to exist. However, in practice, the computation cannot be done in a general setting. In this paper, we developed two alternative methods. The first uses discontinuous and non-strictly increasing/decreasing KL-like functions. We prove that the existence of a KL-like upper bound is equivalent to the existence of a pair of a strictly increasing and continuous function on [0,1] and a convergent geometric sequence. The construction of a strictly increasing continuous function from a KL-like function requires an extension of the famous Sontag's lemma. Finally, we construct a new type of Lyapunov function that is well designed for our peak computation problem. These Lyapunov-like functions are suitable as we establish an equivalence theorem between the existence of a Lyapunov-like function and that of a pair of a strictly increasing and continuous function on [0,1] and a convergent geometric sequence. The construction of a Lyapunov-like function is inspired by the Yoshizawa construction of Lyapunov functions.

Extensions of $\mathcal{KL}$ and Lyapunov Functions for Discrete-time Dynamical System Peaks Analysis

Abstract

In this paper, we extend two classes of functions involved in asymptotic stability analyses. The goal of this extension is to study a maximization problem on the reachable values of a discrete-time dynamical system. This specific maximization problem is called a peak computation problem. The problem consists in finding a pair composed of an initial state and a time that maximizes a given function over states. The paper focuses on the time component of the optimal solution, which is an integer as the time is discrete. We apply a method developed in previous papers to compute an upper bound of the greatest index maximizer of a real sequence. The method uses a formula based on a pair of a strictly increasing and continuous function on [0,1] and a convergent geometric sequence that provides an upper bound of the analyzed sequence. This pair is proven to exist. However, in practice, the computation cannot be done in a general setting. In this paper, we developed two alternative methods. The first uses discontinuous and non-strictly increasing/decreasing KL-like functions. We prove that the existence of a KL-like upper bound is equivalent to the existence of a pair of a strictly increasing and continuous function on [0,1] and a convergent geometric sequence. The construction of a strictly increasing continuous function from a KL-like function requires an extension of the famous Sontag's lemma. Finally, we construct a new type of Lyapunov function that is well designed for our peak computation problem. These Lyapunov-like functions are suitable as we establish an equivalence theorem between the existence of a Lyapunov-like function and that of a pair of a strictly increasing and continuous function on [0,1] and a convergent geometric sequence. The construction of a Lyapunov-like function is inspired by the Yoshizawa construction of Lyapunov functions.
Paper Structure (32 sections, 35 theorems, 161 equations, 2 figures)

This paper contains 32 sections, 35 theorems, 161 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Statement And Motivated Example
  3. The motivation : a discrete-time peak computation problem
  4. Definition of the problem
  5. A problem on integers
  6. Problem statement
  7. Classical $\mathcal{K}\mathcal{L}$ and Lyapunov approaches
  8. The classical $\mathcal{K}\mathcal{L}$ approach
  9. The Lyapunov approach
  10. Problem statement continues
  11. A running example from Leslie population evolution models
  12. Introduction
  13. Peak computation problem description
  14. A rapid preliminary analysis
  15. Asymptotic divergence
  16. ...and 17 more sections

Key Result

Proposition 1

If $\nu_{\rm opt}<+\infty$. The following assertions are made:

Figures (2)

  • Figure 1: Previous results obtained using $\mathcal{K}\mathcal{L}$ and Lyapunov functions and the results developed in this paper.
  • Figure 2: The scheme showing the mappings between $\mathcal{KL}_{\rm gen}$, $\mathcal{OL}_{\rm comp}$ and $\mathbb{\Gamma}(\nu)$

Theorems & Definitions (80)

  • Proposition 1
  • Definition 1: Useful Strictly Increasing Continuous Functions
  • Remark 1
  • Theorem 1
  • Definition 2: The Class $\mathcal{KL}_{\rm gen}$
  • Example 1: A pertinent $\mathcal{KL}_{\rm gen}$ for Leslie models
  • Definition 3: $\mathcal{KL}_{\rm gen}$-$\operatorname{Fin}\left(X^\mathrm{in}\right)$ upper bounds
  • Example 2: A useful $\mathcal{KL}_{\rm gen}$-$\operatorname{Fin}\left(X_\ell\right)$ upper bound for $(X_\ell,L,\varphi_{\ell})$
  • Theorem 2: $\mathbb{\Gamma}(\nu)$ to $\mathcal{KL}_{\rm gen}$
  • proof
  • ...and 70 more