Abstract
This paper focuses on a three-species food chain system which is formulated as stochastic differential equations with regime switching represented by a hidden Markov chain. Firstly, using the Wonham filter, we estimate the hidden Markov chain through the observable solution of the Markov chain in Gaussian white noise. Then two kinds of special dissipative control strategy are proposed to study the given model. That is, under H_{infty} control and passive control, the sufficient conditions for global asymptotic stability are established, respectively. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results.
Highlights
1 Introduction The dynamic relationship between predator and prey has been considerably studied in ecology and mathematical ecology
In order to illustrate such sudden shift in different regimes, we introduce the Markov chain into the underlying three-species food chain stochastic model ( . )
Motivated by the above discussions, in this paper we investigate the global asymptotic stability of equation ( . ) under H∞ control and passive control
Summary
The dynamic relationship between predator and prey has been considerably studied in ecology and mathematical ecology. ), the Markov chain can only be observed in Gaussian white noise It has been proved in [ ] that the posterior probability α(·) satisfies the following stochastic differential equations:. ) at the positive equilibrium point x∗ is equivalent to the global asymptotic stability in probability of equation ) is the globally asymptotically stable in probability Under this control, the L gain of equation Assume that equilibrium point x∗ = of the equation dx = f (x) dt + l (x) dw is asymptotic stable in probability and there is a function V (x) ≥ , for any ε > , it is positive semidefinite and satisfies. ) is zero-state detectable, the equilibrium point x∗ = of the equation dx = f (x) dt + l (x) dw is asymptotically stable in probability.
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