Mean-Square Exponential Synchronization of Markovian Switching Stochastic Complex Networks with Time-Varying Delays by Pinning Control

and Applied Analysis 3 2. Preliminaries 2.1. Notations Throughout this paper, R shall denote the n-dimensional Euclidean space and Rn×n the set of all n × n real matrices. The superscript T shall denote the transpose of a matrix or a vector, Tr · the trace of the corresponding matrix A A A /2 and 1n 1, 1, . . . , 1 T ∈ R and In the n-dimensional identity matrix. For square matrices M, the notation M > 0 resp., < 0 shall mean that M is a positive-definite resp., negative-definite matrix and λmax A , and λmin A shall denote the greatest and least eigenvalues of a symmetric matrix, respectively. Let Ω,F, {Ft}t≥0,P be a complete probability space with a filtration {Ft}t≥0 that is right continuous with F0 containing all the P-null sets. C −τ, 0 ;R shall denote the family of continuous functions φ from −τ, 0 to R with the uniform norm ‖φ‖ sup−τ≤s≤0|φ s | andC2 F0 −τ, 0 ;R the family of allF0 measurable,C −τ, 0 ;R -valued stochastic variables ξ {ξ θ : −τ ≤ θ ≤ 0} such that ∫0 −τ E|ξ s |ds ≤ ∞, where E stands for the correspondent expectation operator with respect to the given probability measure P. Consider a complex network consisting of N identical nodes with nondelayed and time-varying delayed linear coupling and Markovian switching dxi t ⎧ ⎨ ⎩ f t, xi t , xi t − τ t N ∑ j 1,i / j aij ( raij t ) Σ ( xj t − xi t ) N ∑ j 1,i / j bij ( rbij t ) Σ ( xj t − τc t − xi t − τc t ) ⎫ ⎬ ⎭ dt σi t, x t , x t − τ t , x t − τc t , rσi t dwi t , i 1, 2, . . . ,N, 2.1 where xi t xi1 t , xi2 t , . . . , xin t T ∈ R is the state vector of the ith node of the network, f t, xi t , xi t − τ t f1 t, xi t , xi t − τ t , f2 t, xi t , xi t − τ t , . . . , fn t, xi t , xi t − τ t T is a continuous vector-valued function, Σ diag 1, 2, . . . , n is an inner coupling of the networks that satisfies j > 0, j 1, 2, . . . , n, and raij t , rbij t and rσi t are the continuous-time Markov processes that describe the evolution of the modes at time t. Here, A ra t aij raij t ∈ Rn×n and B rb t bij rbij t ∈ Rn×n are the outer coupling matrices of the network at time t at nodes raij t , t − τc t and rbij t , respectively, such that aij raij t ≥ 0 for i / j, aii raii t − ∑N j i,j / i aij raij t , bij rbij t ≥ 0 for i / j, and bii rbii t − ∑N j i,j / i bij rbij t . τ t is the inner time-varying delay satisfying τ ≥ τ t ≥ 0 and τc t is the coupling time-varying delay satisfying τc ≥ τc t ≥ 0. Finally, σi t, x t , x t − τ t , x t−τc t , rσi t σi t, x1 t , . . . , xn t , x1 t−τ t , . . . , xn t−τ t , x1 t−τc t , . . . , xn t− τc t , rσi t ∈ Rn×n and wi t wi1 t , wi2 t , . . . , win t T ∈ R is a bounded vector-form Weiner process, satisfying Ewij t 0, Ew2 ij t 1, Ewij t wij s 0 s / t . 2.2 In this paper, A ra t is assumed to be irreducible in the sense that there are no isolated nodes. 4 Abstract and Applied Analysis The initial conditions associated with 2.1 are xi s ξi s , −τ̌ ≤ s ≤ 0, i 1, 2, . . . ,N, 2.3 where τ̌ max{τ t , τc t }, ξi ∈ C F0 −τ̌ , 0 ,R with the norm ‖ξi‖ 2 sup−τ̌≤s≤0ξi s T ξi s , and our objective is to control system 2.1 so that it stays in the trajectory s t ∈ R of the system ds t f t, s t , s t − τ t dt 2.4 by adding pinning controllers to some of the nodes. Without loss of generality, let the first l nodes be controlled. Then the network is described by dxi t ⎧ ⎨ ⎩ f t, xi t , xi t − τ t N ∑ j 1,i / j aij ( raij t ) Σ ( xj t − xi t ) N ∑ j 1,i / j bij ( rbij t ) Σ ( xj t − τc t − xi t − τc t ) ui t ⎫ ⎬ ⎭ dt σi t, x t , x t − τ t , x t − τc t , rσi t dwi t , i 1, 2, . . . ,N, 2.5 where ui t i 1, 2, . . . ,N are the linear state feedback controllers that are defined by ui t { −εi xi t − s t , i 1, 2, . . . , l, 0, i l 1, l 2, . . . ,N, 2.6 where εi > 0 i 1, 2, . . . , l are the control gains, denoted by Ξ diag{ε1, ε2, . . . , εl, 0, . . . , 0} ∈ R n×n. Define ei t xi t − s t i 1, 2, . . . ,N as the synchronization error. Then, according to the controller 2.6 , the error system is dei t ⎧ ⎨ ⎩ f t, xi t , xi t − τ t − f t, si t , si t − τ t N ∑ j 1,i / j aij ( raij t ) Σ ( ej t − ei t ) N ∑ j 1,i / j bij ( rbij t ) Σ ( ej t − τc t − ei t − τc t ) ui t ⎫ ⎬ ⎭ dt σi t, e t , e t − τ t , e t − τc t , rσi t dwi t , i 1, 2, . . . ,N. 2.7 Remark 2.1. Since the Markov chains raij t , rbij t , and rσi t are independent, we have an equivalent system as follows. Abstract and Applied Analysis 5 Let r t , t > 0 be a right-continuous Markov chain on a probability space that takes values in a finite state space S 1, 2, . . . ,M with a generator Γ γij ∈ RM×M given byand Applied Analysis 5 Let r t , t > 0 be a right-continuous Markov chain on a probability space that takes values in a finite state space S 1, 2, . . . ,M with a generator Γ γij ∈ RM×M given by P { r t Δ j | r t i { γijΔ o Δ if i / j, 1 γiiΔ o Δ if i j, 2.8 for some Δ > 0. Here γij 0 is the transition rate from i to j if i / j and γii − ∑ i / j γij , dei t ⎧ ⎨ ⎩ f t, xi t , xi t − τ t − f t, si t , si t − τ t N ∑ j 1 aij r t Σej t N ∑ j 1 bij r t Σej t − τc t ui t ⎫ ⎬ ⎭ dt σi t, e t , e t − τ t , e t − τc t , r t dwi t , i 1, 2, . . . ,N. 2.9 Definition 2.2. The complex network 2.5 is said to be exponentially synchronized in mean square if the trivial solution of system 2.9 is such that