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

TL;DR

This paper extends classes of functions used in stability analysis to solve a peak computation problem in discrete-time dynamical systems, introducing new methods and Lyapunov-like functions for better bounds and analysis.

Contribution

It develops two alternative methods using KL-like functions and constructs new Lyapunov-like functions, extending existing theories for peak analysis in discrete-time systems.

Findings

Established existence of a KL-like upper bound using a pair of functions and sequences.

Proved the equivalence between Lyapunov-like functions and specific function-sequence pairs.

Constructed a new Lyapunov-like function tailored for peak computation problems.

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.