Neutral Stochastic Differential Delay Equations with Markovian Switching

Abstract Neutral stochastic differential delay equations (NSDDEs) have recently been studied intensively (see Kolmanovskii, V.B. and Nosov, V.R., Stability and Periodic Modes of Control Systems with Aftereffect; Nauka: Moscow, 1981 and Mao X., Stochastic Differential Equations and Their Applications; Horwood Pub.: Chichester, 1997). Given that many systems are often subject to component failures or repairs, changing subsystem interconnections and abrupt environmental disturbances etc., the structure and parameters of underlying NSDDEs may change abruptly. One way to model such abrupt changes is to use the continuous‐time Markov chains. As a result, the underlying NSDDEs become NSDDEs with Markovian switching which are hybrid systems. So far little is known about the NSDDEs with Markovian switching and the aim of this paper is to close this gap. In this paper we will not only establish a fundamental theory for such systems but also discuss some important properties of the solutions e.g. boundedness and stability.

[1]  R. Brayton.  Nonlinear oscillations in a distributed network , 1967 .

[2]  X. Mao,et al.  Stability of Stochastic Differential Equations With Respect to Semimartingales , 1991 .

[3]  D. Elworthy.  ASYMPTOTIC METHODS IN THE THEORY OF STOCHASTIC DIFFERENTIAL EQUATIONS , 1992 .

[4]  Jan C. Willems,et al.  Feedback stabilizability for stochastic systems with state and control dependent noise , 1976, Autom..

[5]  Michael Athans,et al.  Command and control (C2) theory: A challenge to control science , 1986 .

[6]  Decision Systems.,et al.  Stochastic stability research for complex power systems , 1980 .

[7]  John R. Broussard,et al.  Application of precomputed control laws in a reconfigurable aircraft flight control system , 1989 .

[8]  Gopal K. Basak,et al.  Stability of a Random diffusion with linear drift , 1996 .

[9]  V. Lakshmikantham,et al.  Random differential inequalities , 1980 .

[10]  X. Mao.  Stability of stochastic differential equations with Markovian switching , 1999 .

[11]  L. Arnold.  Stochastic Differential Equations: Theory and Applications , 1992 .

[12]  Kai Liu.  Stochastic Stability of Differential Equations in Abstract Spaces , 2019 .

[13]  Xuerong Mao,et al.  Stability in distribution of stochastic differential delay equations with Markovian switching , 2003, Syst. Control. Lett..

[14]  Pavel Pakshin,et al.  Robust stability and stabilization of the family of jumping stochastic systems , 1997 .

[15]  R. Rogers,et al.  An LQ-solution to a control problem associated with a solar thermal central receiver , 1983 .

[16]  X. Mao,et al.  Asymptotic properties of neutral stochastic differential delay equations , 2000 .

[17]  Jong Hae Kim,et al.  H∞-output feedback controller design for linear systems with time-varying delayed state , 1998, IEEE Trans. Autom. Control..

[18]  X. Mao,et al.  Exponential Stability of Stochastic Di erential Equations , 1994 .

[19]  H. Chizeck,et al.  Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control , 1990 .

[20]  Z. Gao,et al.  Feedback stabilizability of non-linear stochastic systems with state-dependent noise , 1987 .

[21]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[22]  M. Mariton,et al.  Jump Linear Systems in Automatic Control , 1992 .

[23]  Xuerong Mao,et al.  Stochastic differential equations and their applications , 1997 .

[24]  V. Dragan,et al.  Stability and robust stabilization to linear stochastic systems described by differential equations with markovian jumping and multiplicative white noise , 2002 .

[25]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[26]  Y. Bar-Shalom,et al.  Stabilization of jump linear gaussian systems without mode observations , 1996 .

[27]  X. Mao,et al.  Stochastic differential delay equations with Markovian switching , 2000 .

[28]  M. K. Ghosh,et al.  Optimal control of switching diffusions with application to flexible manufacturing systems , 1993 .

[29]  X. Mao,et al.  A note on the LaSalle-type theorems for stochastic differential delay equations , 2002 .