Stability Analysis of Time-Varying Dynamical Systems
Summary
Stability analysis of time-varying dynamical systems concerns the behaviour of solutions when the governing laws or parameters evolve explicitly over time. Unlike autonomous systems, non-autonomous models capture external influences, seasonal variations or adaptive control laws. Central questions include whether trajectories remain bounded, converge to desired states and resist perturbations. Classical approaches extend Lyapunov’s direct method by constructing scalar functions that decrease along trajectories, while modern techniques employ Lyapunov–Krasovskiĭ functionals for systems with memory or delays. The theory embraces continuous, discrete and hybrid formulations, often unified through the framework of time scales. Applications span power-grid synchronisation under fluctuating loads, robotic manipulators subject to variable payloads, and networked systems with time-dependent link weights. Recent advances have focused on relaxing conservatism in sufficient conditions, guaranteeing practical stability in the presence of uncertainties, and enabling constructive design of feedback laws. By linking stability notions to controllability and observability in non-stationary contexts, researchers have developed separation principles and robust filtering schemes that ensure safe operation of complex infrastructures under dynamic environments.
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Stability Analysis of Time-Varying Dynamical Systems publication trend
The graph below shows the total number of articles in stability analysis of time-varying dynamical systems across all publications each year (not limited to Nature Index journals).
Technical terms
Non-autonomous system: A dynamical model in which system equations or parameters depend explicitly on time rather than solely on the state variables.
Lyapunov–Krasovskiĭ functional: A generalised energy-like measure for delayed or distributed-parameter systems, constructed to assess stability when memory effects are present.
Uniform asymptotic stability: A property whereby solutions not only converge to an equilibrium as time tends to infinity but do so at a rate uniform over a family of initial conditions.
Time scale: A mathematical framework that unifies continuous and discrete analysis by treating time as a closed subset of the real numbers.
Delay differential equation: A differential equation in which the derivative of the state at a given time depends on past values of the state, often with time-varying delay.
References
- A separation principle of time‐varying dynamical systems: A practical stability approach. Mathematical Modelling and Analysis (2007).
- Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach. Mathematics (2021).
- Exponential stability of dynamic equations on time scales. Advances in Continuous and Discrete Models (2005).
- Delay-Dependent Stability, Integrability and Boundedeness Criteria for Delay Differential Systems. Axioms (2021).
- Controllability of Impulsive Non–Linear Delay Dynamic Systems on Time Scale. IEEE Access (2020).
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