Stochastic Control of Semi-Markov Jump Systems

Semi-Markov jump systems represent dynamical processes in which the system parameters switch according to a stochastic rule that allows the dwelling time in each mode to follow an arbitrary distribution. This framework generalises classical Markov jump systems by introducing sojourn-time dependence, thereby capturing more realistic behaviours in engineering, economics and networked applications. Stochastic control of such systems addresses the design of feedback or event-triggered laws to ensure stability, performance and robustness under probabilistic switching. Central to this endeavour are concepts of mean-square stability, stochastic admissibility and finite-time reachability of designated surfaces or states. Control strategies often combine Lyapunov-Krasovskii functionals with techniques such as linear matrix inequalities, sliding-mode control and neural network adaptation to handle uncertainties, delays and partial mode information. Practical applications span uncertain electrical circuits, airborne vehicles with mode-dependent dynamics and networked sensors operating under communication constraints. Recent advances have reduced conservatism in stability criteria, improved resilience to unknown transition rates and shown how event-triggered mechanisms can alleviate communication burdens without compromising stochastic performance.

Research from Nature Portfolio

Recent studies have investigated adaptive neural-network controllers for hybrid systems combining deterministic switching and stochastic jumps. By modelling the stochastic subsystems as Markov or semi-Markov processes, researchers have constructed radial-basis-function networks to approximate unknown nonlinearities and to generate adaptive control laws. Stability analyses based on Lyapunov methods and energy attenuation theory establish global asymptotic convergence and almost-sure exponential stability. Numerical examples illustrate that the adaptive scheme effectively regulates tracking errors in uncertain dual-switching scenarios, even when mode-transition information is partially unknown.

Stochastic Control of Semi-Markov Jump Systems publication trend

The graph below shows the total number of articles in stochastic control of semi-markov jump systems across all publications each year (not limited to Nature Index journals).

Technical terms

Semi-Markov process: A stochastic process in which transitions between modes follow a Markov chain but the sojourn time in each mode can have a general probability distribution.

Sojourn time: The random duration for which a system remains in a given mode before jumping to the next.

Mean-square stability: A form of stochastic stability requiring that the second moment of the state remains bounded or converges to zero over time.

Sliding-mode control: A robust control technique that drives system trajectories onto a predefined manifold and maintains them there despite uncertainties.

Lyapunov-Krasovskii functional: A generalisation of Lyapunov functions for systems with delays, used to assess stability by incorporating state history.

Linear matrix inequality (LMI): A convex constraint formulation used to derive tractable conditions for system stability and controller synthesis.

Event-triggered control: A control strategy in which updates are applied only when a certain condition is met, reducing unnecessary communication or computation.

References

1. Adaptive neural network control for uncertain dual switching nonlinear systems. Scientific Reports (2022).
2. Stabilization of Semi-Markovian Jumping Uncertain Complex-Valued Networks with Time-Varying Delay: A Sliding-Mode Control Approach. Neural Processing Letters (2024).
3. The Event-Triggered Resilient Control of Discrete-Time Nonlinear Semi-Markov Jump Systems Based on Incremental Quadratic Constraints. Mathematics (2024).
4. New stability criteria for semi-Markov jump linear systems with time-varying delays. AIMS Mathematics (2021).