On construction of Lyapunov functions for scalar linear time-varying systems☆
Section snippets
Introduction and motivation
We consider the following scalar linear time-varying (LTV) system where is a given continuous function. In this paper, we are interested in the construction of Lyapunov functions for system (1), which is an important and a classic problem [1], [2], [3], [4]. Construction of Lyapunov functions and Lyapunov-Krasovskii functionals, especially for time-varying systems, has attracted much attention in the control
The main result
Let denote the set of th differentiable functions defined in the set . We first present the following simple lemma, which is the starting point of our development.
Lemma 3 Let be a constant and be some function. Then satisfy the following differential equation If, moreover, satisfies the following two conditions then
Two illustrative examples
We provide two examples to illustrate the construction of strict Lyapunov functions.
Example 1 We consider the scalar function [14] where is a constant and . Since which corresponds to (4) with and
Conclusion
This paper has established a systematic method for constructing Lyapunov functions for scalar linear time-varying systems, which are assumed to be uniformly exponentially stable and uniformly exponentially bounded. The constructed Lyapunov functions involve an integral of the system parameter with a weighting function over a finite interval. Conditions are imposed on the weighting function and the integral interval such that the Lyapunov function is both positive definite and uniformly bounded,
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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2021, IFAC-PapersOnLineFurther results on the construction of strict Lyapunov–Krasovskii functionals for time-varying time-delay systems
2020, Journal of the Franklin InstituteCitation Excerpt :Initially, strictification problem was studied for systems without time-delay [26,30,31]. A quiet general method for constructing simple Lyapunov functions was studied in [32]. Recently, the strictification problem has been extended to time-delay systems in [33], where the ISS for both continuous systems and discrete systems are analyzed.
Input-to-state stability for discrete-time time-varying impulsive switched delayed systems with asynchronous phenomena
2023, Asian Journal of Control
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This work was supported in part by the Natural Science Foundation of China under the grant number 61773140 and by the GRF HKSAR 17200918.