Switching diffusion with past dependent switching and countable switching space: Existence and uniqueness of solutions, recurrence, and weak stabilization

This work is an extension of our recent paper published in SIAM Journal on Control and Optimization. Starting with the setup in [11], this paper focuses on positive recurrence and weak stabilization of regime-switching diffusion consisting of continuous and discrete components, in which the discrete component switches in a countably infinite set and its switching rates at current time depend on the past value of the continuous component. After presenting the existence and uniqueness of solutions of switching diffusions, the paper departs from [11] and concentrates on recurrence and related issues. Assuming that the system is linearizable, this work provides more feasible conditions on the coefficients of the systems for positive recurrence. Then we obtain linear feedback controls for weak stabilization. Some illustrative examples are also given.

[1]  Vidyadhar G. Kulkarni,et al.  Fluid models for single buffer systems , 1998 .

[2]  G. Yin,et al.  Stability of regime-switching diffusions , 2007 .

[3]  George Yin,et al.  Modeling and Analysis of Switching Diffusion Systems: Past-Dependent Switching with a Countable State Space , 2016, SIAM J. Control. Optim..

[4]  Mu-Fa Chen,et al.  From Markov Chains to Non-Equilibrium Particle Systems , 1992 .

[5]  G. Yin,et al.  Hybrid Switching Diffusions: Properties and Applications , 2009 .

[6]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[7]  Xuerong Mao,et al.  Stochastic Differential Equations With Markovian Switching , 2006 .

[8]  Gang George Yin,et al.  Regularization and Stabilization of Randomly Switching Dynamic Systems , 2012, SIAM J. Appl. Math..

[9]  George Yin,et al.  Coexistence and exclusion of stochastic competitive Lotka–Volterra models , 2016, 1601.06334.

[10]  Gang George Yin,et al.  Almost Sure and pth-Moment Stability and Stabilization of Regime-Switching Jump Diffusion Systems , 2014, SIAM J. Control. Optim..

[11]  Gang George Yin,et al.  On Strong Feller, Recurrence, and Weak Stabilization of Regime-Switching Diffusions , 2009, SIAM J. Control. Optim..

[12]  Fubao Xi,et al.  Stability and Recurrence of Regime-Switching Diffusion Processes , 2014, SIAM J. Control. Optim..

[13]  W. J. Anderson.  Continuous-Time Markov Chains: An Applications-Oriented Approach , 1991 .

[14]  W. Wonham.  Liapunov criteria for weak stochastic stability , 1966 .

[15]  Jinghai Shao,et al.  Criteria for transience and recurrence of regime-switching diffusion processes * , 2014, 1403.3135.

[16]  Gang George Yin,et al.  Asymptotic Properties of Hybrid Diffusion Systems , 2007, SIAM J. Control. Optim..

[17]  Gang George Yin,et al.  Adaptive Search Algorithms for Discrete Stochastic Optimization: A Smooth Best-Response Approach , 2014, IEEE Transactions on Automatic Control.

[18]  Robert F. Luck,et al.  Evaluation of natural enemies for biological control: A behavioral approach , 1990 .

[19]  Sze-Bi Hsu,et al.  A ratio-dependent food chain model and its applications to biological control. , 2003, Mathematical biosciences.

[20]  Fubao Xi,et al.  Strong ergodicity of the regime-switching diffusion processes , 2013 .