A NOVEL GUARANTEED COST CONTROL FOR HOPFIELD NEURAL NETWORKS WITH MULTIPLE TIME-VARYING DELAYS

This paper studies the problem of guaranteed cost control for a class of Hopfield delayed neural networks with multiple time-varying delays. The time delay is a continuous function belonging to a given interval, but not necessary to be differentiable. A cost function is considered as a nonlinear performance measure for the closed-loop system. The stabilizing controllers to be designed must satisfy some exponential stability constraints on the closed-loop poles. By constructing a set of augmented Lyapunov-Krasovskii functionals combined with Newton-Leibniz formula, a guaranteed cost controller is designed via memoryless state feedback control and new sufficient conditions for the existence of the guaranteed cost state-feedback for the system are given in terms of linear matrix inequalities (LMIs). AMS Subject Classification: 92B05, 93D20, 37C75


Introduction
There has been great interest recently in dynamical characteristics of neural Received: June 2, 2014 c 2014 Academic Publications, Ltd. url: www.acadpubl.eunetworks or neural nets since Hopfield constructed a simplified neural network model, in which each neuron [1][2][3][4][5][6][7][8] is represented by a linear circuit consisting of a resistor and a capacitor, and is connected to other neurons via nonlinear sigmoidal activation functions, called transfer functions.Based on the Hopfield neural network model, Marcus and Westervelt argued that the nonlinear sigmoidal activation functions which connected to the other neurons would include discrete delays.
Guaranteed cost control problem [9][10][11][12] has the advantage of providing an upper bound on a given system performance index and thus the system performance degradation incurred by the uncertainties or time delays is guaranteed to be less than this bound.Nevertheless, despite such diversity of results available, most existing work either assumed that the time delays are constant or differentiable [13,14].Although, in some cases, delay-dependent guaranteed cost control for systems with time-varying delays were considered in [12][13][14], the approach used there can not be applied to systems with interval, non-differentiable time-varying delays.
In this paper, we investigate the guaranteed cost control for Hopfield delayed neural networks problem with multiple time-varying delays.The novel features here are that the delayed neural network under consideration is with various globally Lipschitz continuous activation functions, and the multiple time-varying delays function is interval, non-differentiable.A nonlinear cost function is considered as a performance measure for the closed-loop system.The stabilizing controllers to be designed must satisfy some exponential stability constraints on the closed-loop poles.Based on constructing a set of augmented Lyapunov-Krasovskii functional combined with Newton-Leibniz formula, new delay-dependent criteria for guaranteed cost control via memoryless feedback control is established in terms of LMIs.
The outline of the paper is as follows.Section 2 presents definitions and some well-known technical propositions needed for the proof of the main result.LMI delay-dependent criteria for guaranteed cost control is presented in Section 3. The paper ends with conclusions and cited references.

Preliminaries
The following notation will be used in this paper.R + denotes the set of all real non-negative numbers; R n denotes the n−dimensional space with the scalar product x, y or x T y of two vectors x, y, and the vector norm .; M n×r denotes the space of all matrices of (n × r)−dimensions.A T denotes the transpose of matrix A; A is symmetric if A = A T ; I denotes the identity matrix; λ(A) denotes the set of all eigenvalues of A; λ max (A) = max{Reλ; λ ∈ λ(A)}.
The notation diag{. ..} stands for a block-diagonal matrix.The symmetric term in a matrix is denoted by * .
Consider the following Hopfield neural networks with interval multiple timevarying delays: where is the control; n is the number of neurals, and are the activation functions; A = diag(a 1 , a 2 , . . ., a n ), a i > 0 represents the self-feedback term; B ∈ R n×m is control input matrix; W 0 , W 1 denote the connection weights, the discretely delayed connection weights and the distributively delayed connection weight, respectively; The multiple time-varying delays function h i (t), i = 1, 2, . . ., m satisfies the condition In this paper we consider various activation functions and assume that the activation functions f (.), g(.) are Lipschitzian with the Lipschitz constants f i , e i > 0 : The performance index associate with the system (1) is the following function where for all (t, x, u , are given symmetric positive definite matrices.The objective of this paper is to design a memoryless state feedback controller u(t) = Kx(t) for system (1) and the cost function ( 3) such that the resulting closed-loop system is exponentially stable and the closed-loop value of the cost function ( 3) is minimized.
Definition 1.Given α > 0. The zero solution of closed-loop system (5) is α−exponentially stabilizable if there exist a positive number N > 0 such that every solution x(t, φ) satisfies the following condition: x(t, φ) ≤ N e −αt φ , ∀t ≥ 0. Definition 2. Consider the control system (1).If there exist a memoryless state feedback control law u * (t) = Kx(t) and a positive number J * such that the zero solution of the closed-loop system (5) is exponentially stable and the cost function (3) satisfies J ≤ J * , then the value J * is a guaranteed costant and u * (t) is a guaranteed cost control law of the system and its corresponding cost function.
We introduce the following technical well-known propositions, which will be used in the proof of our results.Proposition 1.(Schur complement lemma [15]).Given constant matrices X, Y, Z with appropriate dimensions satisfying Proposition 2.(Integral matrix inequality [15]).For any symmetric positive definite matrix M > 0, scalar γ > 0 and vector function ω : [0, γ] → R n such that the integrations concerned are well defined, the following inequality holds

Main Results
In this section, we give a design of memoryless guaranteed feedback cost control for neural networks (1).Let us set Theorem 3.1.Consider control system (1) and the cost function (3).If there exist symmetric positive definite matrices P, U, G 0 , G 1 , H 0 , H 1 , and diagonal positive definite matrices D 0 , D 1i , i = 1, 2, . . ., m satisfying the following LMIs is a guaranteed cost control and the guaranteed cost value is given by Moreover, the solution x(t, φ) of the system satisfies Using the feddback control (2.5) we consider the following Lyapunov-Krasovskii functional It easy to check that Taking the derivative of V i , i = 1, 2, . . ., 6, we have =y T (t)[−P A T − AP ]y(t) − y T (t)BB T y(t) we obtain where , and . ., m.Note that by the Schur complement lemma, Proposition 2.1, the conditions S 1i < 0 and S 2i < 0 are equivalent to the conditions ( 7) and ( 8), respectively.Therefore, by condition ( 6), ( 7), ( 8), we obtain from ( 14) that Integrating both sides of (15 ) from 0 to t, we obtain Furthermore, taking condition (3.5) into account, we have
This completes the proof of the theorem.

Conclusion
In this paper, the problem of guaranteed cost control for Hopfield neural networks with interval multiple nondifferentiable time-varying delays has been studied.A nonlinear quadratic cost function is considered as a performance measure for the closed-loop system.The stabilizing controllers to be designed must satisfy some exponential stability constraints on the closed-loop poles.