Robust Stability of a Class of Uncertain Lur’e Systems of Neutral Type

and Applied Analysis 3 The rest of this paper is organized as follows. In Section 2, we give notations, definition, propositions, and lemma to be used in the proof of the main results. Delay-dependent sufficient conditions for uncertain neutral and Lur’e dynamical systems with sector-bounded nonlinearity are presented in Section 3. Numerical examples illustrated the obtained results and are given in Section 4. The paper ends with conclusions in Section 5 and cited references. 2. Problem Formulation and Preliminaries The following notation will be used in this paper: R denotes the set of all real nonnegative numbers; R denotes the n-dimensional space and the vector norm ‖ · ‖; Mn×r denotes the space of all matrices of n × r -dimensions. A denotes the transpose of matrix A; A is symmetric if A A ; I denotes the identity matrix; λ A denotes the set of all eigenvalues of A; λmax A max{Reλ; λ ∈ λ A }. xt : {x t s : s ∈ −h, 0 }, ‖xt‖ sups∈ −h,0 ‖x t s ‖; C 0, t ,R denotes the set of all R-valued continuous functions on 0, t ; Matrix A is called semipositive definite A ≥ 0 if xAx ≥ 0, for all x ∈ R; A is positive definite A > 0 if xAx > 0 for all x / 0;A > B means A − B > 0; diag c1, c2, . . . , cm denotes block diagonal matrix with diagonal elements ci, i 1, 2, . . . , m. The symmetric term in a matrix is denoted by ∗. Consider the following uncertain Lur’e system of neutral type with interval timevarying delays and sector-bounded nonlinearity: ẋ t − Cẋt − η t ) A ΔA t x t A1 ΔA1 t x t − h t B ΔB t f σ t , σ t Hx t [ h1 h2 · · · hm ]T x t , ∀t ≥ 0, x t s φ t s , ẋ t s φ t s , s ∈ −m, 0 , m maxh2, η } , 2.1 where x t ∈ R is the state vector; σ t ∈ R is the output vector; A ∈ Rn×n, B ∈ Rn×m, C ∈ Rn×n, A1 ∈ Rn×n, H ∈ Rn×m are constant known matrices; f σ t ∈ R is the nonlinear function in the feedback path, which is denoted as f for simplicity in the sequel. Its form is formulated as f σ t [ f1 σ1 t f2 σ2 t · · · fm σm t ]T , σ t [ σ1 t σ2 t · · · σm t ]T [ hT1x t h T 2x t · · · hmx t ] , 2.2 wherein, each term fi σi t , i 1, 2, . . . , m satisfies any one of the following sector conditions: fi σi t ∈ K 0,ki { fi σi t | fi 0 0, 0 < σi t fi σi t ≤ kiσi t , σi t / 0 } 2.3