Stochastic Passivity of Uncertain Neural Networks with Time-Varying Delays
نویسندگان
چکیده
and Applied Analysis 3 dx t [ − C ΔC t x t A ΔA t f x t B ΔB t f x t − τ t u t ] dt σ t, x t , x t − τ t dω t 2.1 for t ≥ 0, where x t x1 t , x2 t , . . . , xn t T ∈ R is the state vector of the network at time t, n corresponds to the number of neurons; C diag c1, c2, . . . , cn is a positive diagonal matrix, A aij n×n, and B bij n×n are known constant matrices; ΔC t , ΔA t and ΔB t are time-varying parametric uncertainties; σ t, x t , x t − τ t ∈ Rn×n is the diffusion coefficient matrix and ω t ω1 t , ω2 t , . . . , ωn t T is an n-dimensional Brownian motion defined on a complete probability space Ω, F, {Ft}t≥0,P with a filtration {Ft}t≥0 satisfying the usual conditions i.e., it is right continuous and F0 contains all P -null sets ; f x t f1 x1 t , f2 x2 t , . . . , fn xn t T denotes the neuron activation at time t; u t u1 t , u2 t , . . . , un t T ∈ R is a varying external input vector; τ t > 0 is the timevarying delay, and is assumed to satisfy 0 ≤ τ t ≤ τ , where τ is constant. The initial condition associated with model 2.1 is given by x s φ s , s ∈ −τ, 0 . 2.2 Let x t, φ denote the state trajectory of model 2.1 from the above initial condition and x t, 0 the corresponding trajectory with zero initial condition. Throughout this paper, we make the following assumptions. H1 33 The time-varying uncertainties ΔC t , ΔA t and ΔB t are of the form ΔC t H1G1 t E1, ΔA t H2G2 t E2, ΔB t H3G3 t E3, 2.3 where H1, H2, H3, E1, E2, and E3 are known constant matrices of appropriate dimensions, G1 t , G2 t , and G3 t are known time-varying matrices with Lebesgue measurable elements bounded by GT1 t G1 t ≤ I, G T 2 t G2 t ≤ I, G T 3 t G3 t ≤ I. 2.4 H2 10 For any j ∈ {1, 2, . . . , n}, fj 0 0 and there exist constants F− j and F j such that F− j ≤ fj α1 − fj α2 α1 − α2 ≤ F j 2.5 for all α1 / α2. H3 15 There exist two scalars ρ1 > 0, ρ2 > 0 such that the following inequality: trace [ σ t, u, v σ t, u, v ] ≤ ρ1uu ρ2vv 2.6 holds for all t, u, v ∈ R × R × R. 4 Abstract and Applied Analysis Definition 2.1 see 33 . System 2.1 is called globally passive in the sense of expectation if there exists a scalar γ > 0 such that
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