Almost sure stability criteria for linear Markovian switching systems

Area of Science:

• Control Theory
• Systems Engineering
• Stochastic Processes

Background:

• Continuous-time Markovian switching systems are crucial in various applications.
• Understanding subsystem interactions is key to achieving overall system stability.
• Existing methods may not fully capture the dynamics of switching systems.

Purpose of the Study:

• To establish novel stability criteria for continuous-time Markovian switching systems.
• To provide insights into the interaction mechanisms driving system stability.
• To offer a more comprehensive framework for analyzing these complex systems.

Main Methods:

• Direct computation of the Lyapunov exponent using the state-transition matrix.
• Utilizing a method adapted from discrete-time settings to analyze subsystem interactions.
• Employing a family of coupled Lyapunov inequalities to understand switching signal properties.
• Investigating stability in the mean-square sense as a specific case.

Main Results:

• The proposed criteria effectively capture subsystem interactions for stability analysis.
• The Lyapunov exponent computation method successfully integrates discrete-time techniques.
• Coupled Lyapunov inequalities provide a robust framework for analyzing switching signal orchestration.
• Theoretical results are validated through three illustrative examples.

Conclusions:

• The developed stability criteria enhance the understanding of continuous-time Markovian switching systems.
• The methods offer practical tools for analyzing and designing stable switching systems.
• This work contributes to the theoretical foundation of stability analysis in dynamic systems.