Robust Stability and Boundedness of Nonlinear Hybrid Stochastic Differential Delay Equations

Structured Robust Stability and Boundedness of Nonlinear Hybrid Delay Systems

“…Hu, Mao, and Zhang [9] were the first to investigate robust stability and boundedness for nonlinear hybrid SDDEs without the linear growth condition (i.e., the coefficients are not bounded by a linear function, and we will refer to these coefficients as highly nonlinear functions). The significant contribution of [9] lies in that it shows that a given stable hybrid SDDE can tolerate not only the linear-type perturbation but also the highly nonlinear perturbation without loss of the stability, while the papers up to 2013 could only cope with the linear-type perturbation.…”

“…The significant contribution of [9] lies in that it shows that a given stable hybrid SDDE can tolerate not only the linear-type perturbation but also the highly nonlinear perturbation without loss of the stability, while the papers up to 2013 could only cope with the linear-type perturbation. In other words, Hu, Mao, and Zhang [9] opened a new chapter in the study of robust stability for highly nonlinear hybrid SDDEs. However, the progress in this direction is due somewhat to the difficulty of high nonlinearity, and [9] is the only paper so far, to the best of our knowledge.…”

“…In other words, Hu, Mao, and Zhang [9] opened a new chapter in the study of robust stability for highly nonlinear hybrid SDDEs. However, the progress in this direction is due somewhat to the difficulty of high nonlinearity, and [9] is the only paper so far, to the best of our knowledge. The aim of this paper is to make some further progress in this area.…”

“…As we know, hybrid SDDEs have been used to model practical systems that may experience abrupt changes in their structures and parameters (see, e.g., [3,5,13,23]). The theory in [9] is good at dealing with hybrid SDDEs that may experience abrupt changes in their parameters. To explain this, assume that a population system operates in two modes, dry and rain, and it switches from one mode to the other according to a two-state Markov chain with state 1 for dry and state 2 for rain.…”

“…This can be regarded as a stochastically perturbed system of the hybrid delay system dx(t)/dt = x(t)[a r(t) - b r(t) x 2 (t)] with the highly nonlinear stochastic perturbation \sigma r(t) x(t)x(t -\tau )dB(t). Given the asymptotic boundedness of the delay system dx(t)/dt = x(t)[a r(t) -b r(t) x 2 (t)], the theory in [9] shows the upper bounds on \sigma 1 and \sigma 2 for the SDDE to remain asymptotically bounded. We observe that in this example, when the system switches from one mode to the other, only the system parameters change, but the structure of the system remains the same type of Lotka--Volterra.…”