Razumikhin-Type Stability Criteria for Differential Equations with Delayed Impulses
1Department of Computer Sciences, Mathematics, and Engineering Shepherd University, Shepherdstown, WV 25443, USA.
Electronic Journal of Qualitative Theory of Differential Equations
|February 4, 2014
Summary
This study explores stability in impulsive differential equations with time delays. It reveals that impulses can stabilize unstable systems, offering new criteria for complex applications.
Area of Science:
- Dynamical Systems and Control Theory
- Nonlinear Analysis
- Applied Mathematics
Background:
- Impulsive differential equations model systems with sudden changes.
- Time delays are crucial in many real-world dynamic systems.
- Stability analysis is fundamental for predicting system behavior.
Purpose of the Study:
- To investigate stability criteria for impulsive differential equations with time delays.
- To develop methods for analyzing systems where impulses depend on past states.
- To explore conditions under which unstable systems can be stabilized by impulses.
Main Methods:
- Lyapunov functions method
- Razumikhin technique
- Mathematical induction
- Analysis of differential and difference equations with delays
Main Results:
- New stability criteria for impulsive differential equations with delayed impulses.
- Demonstration that impulses can achieve exponential stabilization even for unstable system matrices.
- Less restrictive conditions for maintaining stability under specific impulsive perturbations.
Conclusions:
- The findings provide valuable insights into the stability of complex delayed impulsive systems.
- The developed criteria can be applied to systems where impulses are state-dependent.
- This research contributes to the theoretical understanding and practical application of stability analysis in dynamic systems.
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