Stability Analysis of Impulsive Dynamical Systems
Summary
Impulsive dynamical systems describe processes in which a continuous‐time evolution is punctuated by abrupt, discrete changes at specified instants. Such models arise naturally in biological systems subject to treatment or harvesting, networked control systems with packet drops, financial systems under sudden shocks and mechanical systems with impacts. Stability analysis seeks to determine whether trajectories remain bounded or converge to an equilibrium despite these impulses. Core challenges include handling the interplay between continuous flows and instantaneous jumps, ensuring robustness to external disturbances and verifying stability under varying impulse timing. Lyapunov methods, dwell‐time conditions and input‐to‐state stability criteria have become principal tools. Recent efforts focus on relaxing conservatism in impulse intervals, accommodating delays in impulses and establishing finite‐time convergence, thereby broadening applicability to high‐dimensional, delayed and hybrid systems.
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Research from all publishers
Recent advances have refined stability conditions for infinite‐dimensional impulsive systems. One study developed novel dwell‐time criteria that allow simultaneous instability in continuous and discrete dynamics, using Lyapunov‐like techniques to guarantee input‐to‐state stability even under external perturbations in both finite and infinite dimensions. Another work on delayed systems with bounded‐delay impulses introduced a Halanay‐type inequality tailored to impulsive delays, deriving delay‐independent criteria for input‐to‐state stability via an enriched Lyapunov framework and illustrating results with concrete examples. A further investigation into finite‐time stability of hybrid impulsive systems established sufficient Lyapunov conditions for convergence in a fixed time, combining differential and difference inequalities for continuous flows and discrete resets. This approach yielded universal control laws for rapid stabilisation and demonstrated practical scenarios such as networked consensus requiring swift convergence.
Stability Analysis of Impulsive Dynamical Systems publication trend
The graph below shows the total number of articles in stability analysis of impulsive dynamical systems across all publications each year (not limited to Nature Index journals).
Technical terms
Impulsive dynamical system: A model combining smooth continuous evolution with instantaneous state jumps at specified times.
Lyapunov function: A scalar measure that decreases along system trajectories, used to certify stability.
Input-to-state stability (ISS): A property ensuring that bounded inputs produce bounded system states.
Dwell time: The minimum interval between successive impulses required to preserve stability.
Finite-time stability: The condition under which trajectories reach equilibrium within a predetermined finite time.
References
- Stability conditions for impulsive dynamical systems. Mathematics of Control, Signals, and Systems (2021).
- Input-to-state stability of delayed systems with bounded-delay impulses. Mathematical Modelling and Control (2022).
- On finite-time stability and stabilization of nonlinear hybrid dynamical systems. AIMS Mathematics (2021).
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