Delay-Dependent Guaranteed Cost Controller Design for Uncertain Neural Networks with Interval Time-Varying Delay

Abstract

This paper studies the problem of guaranteed cost control for a class of uncertain delayed neural networks. 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). Numerical examples are given to illustrate the effectiveness of the obtained result.

1. Introduction

The last few decades have witnessed the use of artificial neural networks (ANNs) in many real-world applications and have offered an attractive paradigm for a broad range of adaptive complex systems. In recent years, ANNs have enjoyed a great deal of success and have proven useful in wide variety pattern recognition feature-extraction tasks. Examples include optical character recognition, speech recognition, and adaptive control, to name a few. To keep the pace with the huge demand in diversified application areas, many different kinds of ANN architecture and learning types have been proposed to meet varying needs as robustness and stability. Stability and control of neural networks with time delay have attracted considerable attention in recent years [1–8]. In many practical systems, it is desirable to design neural networks which are not only asymptotically or exponentially stable but can also guarantee an adequate level of system performance. In the area of control, signal processing, pattern recognition, and image processing, delayed neural networks have many useful applications. Some of these applications require that the equilibrium points of the designed network be stable. In both biological and artificial neural systems, time delays due to integration and communication are ubiquitous and often become a source of instability. The time delays in electronic neural networks are usually time varying, and sometimes vary violently with respect to time due to the finite switching speed of amplifiers and faults in the electrical circuitry. Guaranteed cost control problem [9–12] 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. The Lyapunov-Krasovskii functional technique has been among the popular and effective tool in the design of guaranteed cost controls for neural networks with time delay. Nevertheless, despite such diversity of results available, most existing work either assumed that the time delays are constant or differentiable [13–16]. Although, in some cases, delay-dependent guaranteed cost control for systems with time-varying delays was considered in [12, 13, 15], the approach used there can not be applied to systems with interval, nondifferentiable time-varying delays. To the best of our knowledge, the guaranteed cost control and state feedback stabilization for uncertain neural networks with interval, non-differentiable time-varying delays have not been fully studied yet (see, e.g., [4–26] and the references therein), which are important in both theories and applications. This motivates our research.

In this paper, we investigate the guaranteed cost control for uncertain delayed neural networks problem. The novel features here are that the delayed neural network under consideration is with various globally Lipschitz continuous activation functions, and the time-varying delay 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 functionals combined with Newton-Leibniz formula, new delay-dependent criteria for guaranteed cost control via memoryless feedback control are established in terms of LMIs, which allow simultaneous computation of two bounds that characterize the exponential stability rate of the solution and can be easily determined by utilizing Matlabs LMI control toolbox.

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 guaraneed cost control and a numerical examples showing the effectiveness of the result are presented in Section 3. The paper ends with conclusions and cited references.

2. Preliminaries

The following notation will be used in this paper. ℝ⁺ denotes the set of all real nonnegative numbers; ℝ^𝑛 denotes the 𝑛-dimensional space with the scalar product ⟨𝑥,𝑦⟩ or 𝑥^𝑇𝑦 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)}.𝑥_𝑡∶={𝑥(𝑡+𝑠)∶𝑠∈[−ℎ,0]}, ‖𝑥_𝑡‖=sup_{𝑠∈[−ℎ,0]}‖𝑥(𝑡+𝑠)‖; 𝐶¹([0,𝑡],ℝ^𝑛) denotes the set of all ℝ^𝑛-valued continuously differentiable functions on [0,𝑡]; 𝐿₂([0,𝑡],ℝ^𝑚) denotes the set of all the ℝ^𝑚-valued square integrable functions on [0,𝑡].

Matrix 𝐴 is called semipositive definite (𝐴≥0) if ⟨𝐴𝑥,𝑥⟩≥0, for all 𝑥∈ℝ^𝑛;𝐴 is positive definite (𝐴>0) if ⟨𝐴𝑥,𝑥⟩>0 for all 𝑥≠0;𝐴>𝐵 means 𝐴−𝐵>0. The notation diag{⋯} stands for a block-diagonal matrix. The symmetric term in a matrix is denoted by ∗.

Consider the following uncertain neural networks with interval time-varying delay: 𝑥̇(𝑡)=−(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+𝑊₀+Δ𝑊₀(𝑡)𝑊₀𝑓(𝑥(𝑡))+𝑊₁+Δ𝑊₁(𝑡)𝑔(𝑥((𝑡(𝑡)(𝑡)(𝑡)𝑡−ℎ)))+𝐵𝑢,𝑡≥0,𝑥=𝜙,𝑡∈−ℎ₁,0, (2.1) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇∈ℝ^𝑛 is the state of the neural; 𝑢(⋅)∈𝐿₂([0,𝑡],ℝ^𝑚) is the control; 𝑛 is the number of neurals, and 𝑓(𝑥(𝑡))=𝑓₁𝑥₁(𝑡),𝑓₂𝑥₂(𝑡),…,𝑓_𝑛𝑥_𝑛(𝑡)^𝑇,𝑔(𝑥(𝑡−ℎ(𝑡)))=𝑔₁𝑥₁(𝑡−ℎ(𝑡))(𝑡),𝑔₂𝑥₂(𝑡−ℎ(𝑡))(𝑡),…,𝑔_𝑛𝑥_𝑛(𝑡−ℎ(𝑡))^𝑇, (2.2) are the activation functions; 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛),𝑎_𝑖>0 represents the self-feedback term; 𝐵∈𝑅^𝑛×𝑚 is control input matrix; 𝑊₀,𝑊₁ denote the connection weights, the discretely delayed connection weights and the distributively delayed connection weight, respectively; the time-varying uncertain matrices Δ𝐴(𝑡),Δ𝑊₀(𝑡), and Δ𝑊₁(𝑡) are defined by Δ𝐴(𝑡)=𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎,Δ𝑊₀(𝑡)=𝐸_𝑤0𝐹_𝑤0(𝑡)𝐻_𝑤0,Δ𝑊₁(𝑡)=𝐸_𝑤1𝐹_𝑤1(𝑡)𝐻_𝑤1, (2.3) where 𝐸_𝑎,𝐸_𝑤0,𝐸_𝑤1,𝐻_𝑎,𝐻_𝑤0,and𝐻_𝑤1 are known constant real matrices with appropriate dimensions. 𝐹_𝑎(𝑡),𝐹_𝑤0(𝑡),and𝐹_𝑤1(𝑡) are unknown uncertain matrices satisfying 𝐹^𝑇_𝑎(𝑡)𝐹_𝑎(𝑡)≤𝐼,𝐹^𝑇_𝑤0(𝑡)𝐹_𝑤0(𝑡)≤𝐼,𝐹^𝑇_𝑤1(𝑡)𝐹_𝑤1(𝑡)≤𝐼,𝑡∈𝑅⁺. (2.4) The time-varying delay function ℎ(𝑡) satisfies the condition 0≤ℎ₀≤ℎ(𝑡)≤ℎ₁. (2.5) The initial functions 𝜙(𝑡)∈𝐶¹([−ℎ₁,0],𝑅^𝑛), with the norm 𝜙=sup_{𝑡∈−ℎ1,0}𝜙(𝑡)²+𝜙̇(𝑡)². (2.6) 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: 𝑓_𝑖𝜉₁−𝑓_𝑖𝜉₂≤𝑓_𝑖𝜉₁−𝜉₂,𝑖=1,2,…,𝑛,∀𝜉₁,𝜉₂𝑔∈ℝ,_𝑖𝜉₁−𝑔_𝑖𝜉₂≤𝑒_𝑖𝜉₁−𝜉₂,𝑖=1,2,…,𝑛,∀𝜉₁,𝜉₂∈ℝ. (2.7) The performance index associated with the system (2.1) is the following function: 𝐽=^∞₀𝑓⁰(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡))𝑑𝑡, (2.8) where 𝑓⁰(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡))∶𝑅⁺×𝑅^𝑛×𝑅^𝑛×𝑅^𝑚→𝑅⁺ is a nonlinear cost function that satisfies ∃𝑄₁,𝑄₂,𝑅∶𝑓⁰𝑡,𝑥,𝑦,𝑢≤𝑄₁𝑥,𝑥+𝑄₂𝑦,𝑦+⟨𝑅𝑢,𝑢⟩, (2.9) for all (𝑡,𝑥,𝑢)∈𝑅⁺×𝑅^𝑛×𝑅^𝑚 and 𝑄₁,𝑄₂∈𝑅^𝑛×𝑛,𝑅∈𝑅^𝑚×𝑚 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 (2.1) and the cost function (2.8) such that the resulting closed-loop system 𝑥̇(𝑡)=−𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎−𝐵𝐾𝑥(𝑡)+𝑊₀+𝐸_𝑤0𝐹_𝑤0(𝑡)𝐻_𝑤0𝑓(𝑥(𝑡+𝑊))₁+𝐸_𝑤1𝐹_𝑤1(𝑡)𝐻_𝑤1𝑔(𝑥(𝑡−ℎ(𝑡))) (2.10) is exponentially stable and the closed-loop value of the cost function (2.10) is minimized.

Definition 2.1. Given 𝛼>0. The zero solution of closed-loop system (2.8) is 𝛼-exponentially stabilizable if there exists a positive number 𝑁>0 such that every solution 𝑥(𝑡,𝜙) satisfies the following condition: 𝑥𝑡,𝜙≤𝑁𝑒^−𝛼𝑡𝜙,∀𝑡≥0. (2.11)

Definition 2.2. Consider the control system (2.1). If there exists 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 (2.10) is exponentially stable and the cost function (2.8) satisfies 𝐽≤𝐽^∗, then the value 𝐽^∗ is a guaranteed constant 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 2.3 (Schur complement lemma [27]). Given constant matrices 𝑋,𝑌, and 𝑍 with appropriate dimensions satisfying 𝑋=𝑋^𝑇,𝑌=𝑌^𝑇>0, then 𝑋+𝑍^𝑇𝑌⁻¹𝑍<0 if and only if 𝑋𝑍^𝑇𝑍−𝑌<0. (2.12)

Proposition 2.4 (integral matrix inequality [28]). 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 ^𝛾₀𝜔(𝑠)𝑑𝑠^𝑇𝑀^𝛾₀𝜔(𝑠)𝑑𝑠≤𝛾^𝛾₀𝜔^𝑇(𝑠)𝑀𝜔(𝑠)𝑑𝑠. (2.13)

3. Design of Guaranteed Cost Controller

In this section, we give a design of memoryless guaranteed feedback cost control for uncertain neural networks (2.1). Let us set 𝑊₁₁=−𝑃𝐴^𝑇−𝐴𝑃−2𝛼𝑃+0.25𝐵𝑅𝐵^𝑇+¹_𝑖=0𝐺_𝑖+2𝜖₁𝐸^𝑇_𝑎𝐸_𝑎+6𝜖₁⁻¹𝑃𝐻^𝑇_𝑎𝐻_𝑎𝑃+4𝜖₂⁻¹𝑃𝐹𝐻^𝑇_𝑤0𝐻_𝑤0𝑊𝐹𝑃,₁₂=𝑃+𝐴𝑃−0.5𝐵𝐵^𝑇,𝑊₁₃=𝑒^−2𝛼ℎ0𝐻₀+0.5𝐵𝐵^𝑇𝑊+𝐴𝑃,₁₄=𝑒^−2𝛼ℎ1𝐻₁+0.5𝐵𝐵^𝑇+𝐴𝑃,𝑊₁₅=𝑃+0.5𝐵𝐵^𝑇𝑊+𝐴𝑃,₂₂=¹_𝑖=0𝑊_𝑖𝐷_𝑖𝑊^𝑇_𝑖+¹_𝑖=0ℎ²_𝑖𝐻_𝑖+ℎ₁−ℎ₀𝑈−2𝑃−𝐵𝐵^𝑇+𝜖₁𝐸^𝑇_𝑎𝐸_𝑎+𝜖₂𝐸^𝑇_𝑤0𝐸_𝑤0+𝜖₃𝐸^𝑇_𝑤1𝐸_𝑤1,𝑊₂₃=𝑃,𝑊₂₄=𝑃,𝑊₂₅𝑊=𝑃,₃₃=−𝑒^−2𝛼ℎ0𝐺₀−𝑒^−2𝛼ℎ0𝐻₀−𝑒^−2𝛼ℎ1𝑈+¹_𝑖=0𝑊_𝑖𝐷_𝑖𝑊^𝑇_𝑖+𝜖₁𝐸^𝑇_𝑎𝐸_𝑎+𝜖₂𝐸^𝑇_𝑤0𝐸_𝑤0+𝜖₃𝐸^𝑇_𝑤1𝐸_𝑤1,𝑊₃₄=0,𝑊₃₅=𝑒^−2𝛼ℎ1𝑊𝑈,₄₄=¹_𝑖=0𝑊_𝑖𝐷_𝑖𝑊^𝑇_𝑖−𝑒^−2𝛼ℎ1𝑈−𝑒^−2𝛼ℎ1𝐺₁−𝑒^−2𝛼ℎ1𝐻₁+𝜖₁𝐸^𝑇_𝑎𝐸_𝑎+𝜖₂𝐸^𝑇_𝑤0𝐸_𝑤0+𝜖₃𝐸^𝑇_𝑤1𝐸_𝑤1,𝑊₄₅=𝑒^−2𝛼ℎ1𝑊𝑈,₅₅=−𝑒^−2𝛼ℎ1𝑈+𝑊₀𝐷₀𝑊^𝑇₀+4𝜖₃⁻¹𝑃𝐸𝐻^𝑇_𝑤1𝐻_𝑤1𝐸𝑃+𝜖₁𝐸^𝑇_𝑎𝐸_𝑎+𝜖₂𝐸^𝑇_𝑤0𝐸_𝑤0+𝜖₃𝐸^𝑇_𝑤1𝐸_𝑤1,𝐸=diag𝑒_𝑖,𝑖=1,…,𝑛,𝐹=diag𝑓_𝑖,𝜆,𝑖=1,…,𝑛₁=𝜆_min𝑃⁻¹,𝜆₂=𝜆_max𝑃⁻¹+ℎ₀𝜆_max𝑃⁻¹¹_𝑖=0𝐺_𝑖𝑃⁻¹+ℎ²₁𝜆_max𝑃⁻¹¹_𝑖=0𝐻_𝑖𝑃⁻¹+ℎ₁−ℎ₀𝜆_max𝑃⁻¹𝑈𝑃⁻¹. (3.1)

Theorem 3.1. Consider control system (2.1) and the cost function (2.8). If there exist symmetric positive definite matrices 𝑃,𝑈,𝐺₀,𝐺₁,𝐻₀,and 𝐻₁, and diagonal positive definite matrices 𝐷_𝑖,𝑖=0,1, and 𝜖_𝑖>0,𝑖=1,2,3 satisfying the following LMIs 𝑊ℰ=¹¹𝑊₁₂𝑊₁₃𝑊₁₄𝑊₁₅∗𝑊₂₂𝑊₂₃𝑊₂₄𝑊₂₅∗∗𝑊₃₃𝑊₃₄𝑊₃₅∗∗∗𝑊₄₄𝑊₄₅∗∗∗∗𝑊₅₅𝒮<0,₁=−𝑃𝐴−𝐴^𝑇𝑃−∑¹_𝑖=0𝑒^{−2𝛼ℎ𝑖}𝐻_𝑖2𝑃𝐹𝑃𝑄₁∗−𝐷₀0∗∗−𝑄₁⁻¹𝒮<0,₂=𝑊₁𝐷₁𝑊^𝑇₁−𝑒^−2𝛼ℎ1𝑈2𝑃𝐸𝑃𝑄₂∗−𝐷₁0∗∗−𝑄₂⁻¹<0, (3.2)(3.3)(3.4) then 𝑢(𝑡)1=−2𝐵^𝑇𝑃⁻¹𝑥(𝑡),𝑡≥0 (3.5) is a guaranteed cost control and the guaranteed cost value is given by 𝐽^∗=𝜆₂𝜙². (3.6) Moreover, the solution 𝑥(𝑡,𝜙) of the system satisfies 𝑥𝑡,𝜙≤𝜆₂𝜆₁𝑒^−𝛼𝑡𝜙,∀𝑡≥0. (3.7)

Proof. Let 𝑌=𝑃⁻¹,𝑦(𝑡)=𝑌𝑥(𝑡). Using the feedback control (2.8) we consider the following Lyapunov-Krasovskii functional: 𝑉𝑡,𝑥_𝑡=⁶_𝑖=1𝑉_𝑖𝑡,𝑥_𝑡,𝑉₁=𝑥^𝑇(𝑡)(𝑡),𝑉𝑌𝑥₂=^𝑡_𝑡−ℎ0𝑒^{2𝛼(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑌𝐺₀(𝑠)𝑉𝑌𝑥𝑑𝑠,₃=^𝑡_𝑡−ℎ1𝑒^{2𝛼(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑌𝐺₁(𝑠)𝑉𝑌𝑥𝑑𝑠,₄=ℎ₀⁰_−ℎ0^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑡)}𝑥̇^𝑇(𝜏)𝑌𝐻₀𝑌𝑥̇(𝜏)𝑉𝑑𝜏𝑑𝑠,₅=ℎ₁⁰_−ℎ1^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑡)}𝑥̇^𝑇(𝜏)𝑌𝐻₁𝑌𝑥̇(𝜏)𝑉𝑑𝜏𝑑𝑠,₆=ℎ₁−ℎ₀^𝑡−ℎ0_𝑡−ℎ1^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑡)}𝑥̇^𝑇(𝜏)𝑌𝑈𝑌𝑥̇(𝜏)𝑑𝜏𝑑𝑠. (3.8) It is easy to check that 𝜆₁‖𝑥(𝑡)‖²≤𝑉𝑡,𝑥_𝑡≤𝜆₂𝑥_𝑡²,∀𝑡≥0. (3.9) Taking the derivative of 𝑉₁ we have 𝑉̇₁=2𝑥^𝑇(𝑡)𝑌𝑥̇(𝑡)=𝑦^𝑇(𝑡)−𝑃𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎^𝑇−𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃𝑦(𝑡)−𝑦^𝑇(𝑡)𝐵𝐵^𝑇𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑊₀+𝐸_𝑤0𝐹_𝑤0(𝑡)𝐻_𝑤0𝑓(⋅)𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑊₁+𝐸_𝑤1𝐹_𝑤1(𝑡)𝐻_𝑤1𝑔(⋅)𝑦(𝑡),𝑉̇₂=𝑦^𝑇(𝑡)𝐺₀𝑦(𝑡)−𝑒^−2𝛼ℎ0𝑦^𝑇𝑡−ℎ₀𝐺₀𝑦𝑡−ℎ₀−2𝛼𝑉₂,𝑉̇₃=𝑦^𝑇(𝑡)𝐺₁𝑦(𝑡)−𝑒^−2𝛼ℎ1𝑦^𝑇𝑡−ℎ₁𝐺₁𝑦𝑡−ℎ₁−2𝛼𝑉₃,𝑉̇₄=ℎ²₀𝑦̇^𝑇(𝑡)𝐻₀𝑦̇(𝑡)−ℎ₁𝑒^{−2𝛼ℎ0𝑡}_𝑡−ℎ0𝑥̇^𝑇(𝑠)𝐻₀𝑥̇(𝑠)𝑑𝑠−2𝛼𝑉₄,𝑉̇₅=ℎ²₁𝑦̇^𝑇(𝑡)𝐻₁𝑦̇(𝑡)−ℎ₁𝑒^{−2𝛼ℎ1𝑡}_𝑡−ℎ1𝑦̇^𝑇(𝑠)𝐻₁𝑦̇(𝑠)𝑑𝑠−2𝛼𝑉₄,𝑉̇₆=ℎ₁−ℎ₀²𝑦̇^𝑇(𝑡)𝑈𝑦̇(𝑡)−ℎ₁−ℎ₀𝑒^{−2𝛼ℎ1𝑡−ℎ0}_𝑡−ℎ1𝑦̇^𝑇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠−2𝛼𝑉₆, (3.10) Applying Proposition 2.4 and the Leibniz-Newton formula 𝑦̇^𝑡_𝑠(𝜏)𝑑𝜏=𝑦(𝑡)−𝑦(𝑠). (3.11) We have for 𝑗=1,2,𝑖=0,1−ℎ_𝑖^𝑡_{𝑡−ℎ𝑖}𝑦̇^𝑇(𝑠)𝐻_𝑗𝑦̇(𝑠)𝑑𝑠≤−^𝑡_{𝑡−ℎ𝑖}𝑦̇(𝑠)𝑑𝑠^𝑇𝐻_𝑗^𝑡_{𝑡−ℎ𝑖}𝑦̇(𝑠)𝑦(𝑡)((𝑡𝑑𝑠≤−−𝑦𝑡−ℎ))^𝑇𝐻_𝑗𝑦(𝑡)−𝑦(𝑡−ℎ(𝑡))=−𝑦^𝑇(𝑡)𝐻_𝑖𝑦(𝑡)+2𝑥^𝑇(𝑡)𝐻_𝑗𝑦(𝑡−ℎ(𝑡))−𝑦^𝑇𝑡−ℎ_𝑖𝐻_𝑗𝑦𝑡−ℎ_𝑖. (3.12) Note that 𝑦̇^𝑡−ℎ0_𝑡−ℎ1^𝑇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠=^{𝑡−ℎ(𝑡)}_𝑡−ℎ1𝑦̇^𝑇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠+^𝑡−ℎ0_{(𝑡)𝑡−ℎ}𝑦̇^𝑇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠. (3.13) Applying Proposition 2.4 gives ℎ₁−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_𝑡−ℎ1𝑦̇^𝑇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠≥^{𝑡−ℎ(𝑡)}_𝑡−ℎ1𝑦̇(𝑠)𝑑𝑠^𝑇𝑈^{𝑡−ℎ(𝑡)}_𝑡−ℎ1𝑦̇(𝑠)≥𝑦(𝑡)𝑑𝑠𝑡−ℎ−𝑦𝑡−ℎ₁^𝑇𝑈𝑦𝑡−ℎ(𝑡)−𝑦𝑡−ℎ₁. (3.14) Since ℎ₁−ℎ(𝑡)≤ℎ₁−ℎ₀, we have ℎ₁−ℎ₀^{𝑡−ℎ(𝑡)}_𝑡−ℎ1𝑦̇^𝑇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠≥𝑦𝑡−ℎ(𝑡)−𝑦𝑡−ℎ₁^𝑇𝑈𝑦𝑡−ℎ(𝑡)−𝑦𝑡−ℎ₁, (3.15) then −ℎ₁−ℎ₀^{𝑡−ℎ(𝑡)}_𝑡−ℎ1𝑦̇^𝑇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠≤−𝑦𝑡−ℎ(𝑡)−𝑦𝑡−ℎ₁^𝑇𝑈𝑦𝑡−ℎ(𝑡)−𝑦𝑡−ℎ₁. (3.16) Similarly, we have −ℎ₁−ℎ₀^𝑡−ℎ0_{(𝑡)𝑡−ℎ}𝑦̇^𝑇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠≤−𝑦𝑡−ℎ₀−𝑦(𝑡−ℎ(𝑡))^𝑇𝑈𝑦𝑡−ℎ₀−𝑦(𝑡−ℎ(𝑡)). (3.17) Then, we have 𝑉̇(⋅)+2𝛼𝑉(⋅)≤𝑦^𝑇(𝑡)−𝑃𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎^𝑇−𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃𝑦(𝑡)−𝑦^𝑇(𝑡)𝐵𝐵^𝑇𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑊₀+𝐸_𝑤0𝐹_𝑤0(𝑡)𝐻_𝑤0𝑓(⋅)+2𝑦^𝑇(𝑡)𝑊₁+𝐸_𝑤1𝐹_𝑤1(𝑡)𝐻_𝑤1𝑔(⋅)+𝑦^𝑇(𝑡)¹_𝑖=0𝐺_𝑖𝑦(𝑡)(𝑡)(𝑡+𝑦̇+2𝛼𝑃𝑦,𝑦)_𝑇(𝑡)¹_𝑖=0ℎ²_𝑖𝐻_𝑖𝑦̇(𝑡)+ℎ₁−ℎ₀𝑦̇^𝑇(𝑡)𝑈𝑦̇(𝑡)−¹_𝑖=0𝑒^{−2𝛼ℎ𝑖}𝑦^𝑇𝑡−ℎ_𝑖𝐺_𝑖𝑦𝑡−ℎ_𝑖−𝑒^−2𝛼ℎ0𝑦(𝑡)−𝑦𝑡−ℎ₀^𝑇𝐻₀𝑦(𝑡)−𝑦𝑡−ℎ₀−𝑒^−2𝛼ℎ1𝑦(𝑡)−𝑦𝑡−ℎ₁^𝑇𝐻₁𝑦(𝑡)−𝑦𝑡−ℎ₁−𝑒^−2𝛼ℎ1𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₁^𝑇𝑈𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₁−𝑒^−2𝛼ℎ1𝑦𝑡−ℎ₀−𝑦(𝑡−ℎ(𝑡))^𝑇𝑈𝑦𝑡−ℎ₀−𝑦(𝑡−ℎ(𝑡)). (3.18) Using (2.8) 𝑃𝑦̇(𝑡)+𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃𝑦(𝑡)−𝑊₀+𝐸_𝑤0𝐹_𝑤0(𝑡)𝐻_𝑤0𝑓(⋅)−𝑊₁+𝐸_𝑤1𝐹_𝑤1(𝑡)𝐻_𝑤1𝑔(⋅)+0.5𝐵𝐵^𝑇𝑦(𝑡)=0, (3.19) and multiplying both sides with [2𝑦(𝑡),−2𝑦(̇𝑡),2𝑦(𝑡−ℎ₀),2𝑦(𝑡−ℎ₁),2𝑦(𝑡−ℎ(𝑡))]^𝑇, we have 2𝑦^𝑇(𝑡)𝑃𝑦̇(𝑡)+2𝑦^𝑇(𝑡)𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃𝑦(𝑡)−2𝑦^𝑇(𝑡)𝑊₀+𝐸_𝑤0𝐹_𝑤0(𝑡)𝐻_𝑤0𝑓(⋅)−2𝑦^𝑇(𝑡)𝑊₁+𝐸_𝑤1𝐹_𝑤1(𝑡)𝐻_𝑤1𝑔(⋅)+𝑦^𝑇(𝑡)𝐵𝐵^𝑇𝑦(𝑡)𝑦̇=0,−2^𝑇(𝑡)𝑃𝑦̇(𝑡)−2𝑦̇^𝑇(𝑡)𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃𝑦(𝑡)+2𝑦̇^𝑇(𝑡)𝑊₀+𝐸_𝑤0𝐹_𝑤0(𝑡)𝐻_𝑤0𝑓(⋅)+2𝑦̇^𝑇(𝑡)𝑊₁+𝐸_𝑤1𝐹_𝑤1(𝑡)𝐻_𝑤1𝑔(⋅)−𝑦̇^𝑇(𝑡)𝐵𝐵^𝑇𝑦(𝑡)=0,2𝑦^𝑇𝑡−ℎ₀𝑃𝑦̇(𝑡)+2𝑦^𝑇𝑡−ℎ₀𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃𝑦(𝑡)−2𝑦^𝑇𝑡−ℎ₀𝑊₀+𝐸_𝑤0𝐹_𝑤0(𝑡)𝐻_𝑤0×𝑓(⋅)−2𝑦^𝑇𝑡−ℎ₀𝑊₁+𝐸_𝑤1𝐹_𝑤1(𝑡)𝐻_𝑤1𝑔(⋅)+𝑦^𝑇𝑡−ℎ₀𝐵𝐵^𝑇𝑦(𝑡)=0,2𝑦^𝑇𝑡−ℎ₁𝑃𝑦̇(𝑡)+2𝑦^𝑇𝑡−ℎ₁𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃𝑦(𝑡)−2𝑦^𝑇𝑡−ℎ₁𝑊₀+𝐸_𝑤0𝐹_𝑤0(𝑡)𝐻_𝑤0×𝑓(⋅)−2𝑦^𝑇𝑡−ℎ₁𝑊₁+𝐸_𝑤1𝐹_𝑤1(𝑡)𝐻_𝑤1𝑔(⋅)+𝑦^𝑇𝑡−ℎ₁𝐵𝐵^𝑇𝑦(𝑡)=0,2𝑦^𝑇(𝑡−ℎ(𝑡))𝑃𝑦̇(𝑡)+2𝑦^𝑇(𝑡−ℎ(𝑡))𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃𝑦(𝑡)−2𝑦^𝑇((𝑡×𝑊𝑡−ℎ))₀+𝐸_𝑤0𝐹_𝑤0(𝑡)𝐻_𝑤0𝑓(⋅)−2𝑦^𝑇(𝑡−ℎ(𝑡))𝑊₁+𝐸_𝑤1𝐹_𝑤1(𝑡)𝐻_𝑤1𝑔(⋅)+𝑦^𝑇(𝑡−ℎ(𝑡))𝐵𝐵^𝑇𝑦(𝑡)=0. (3.20) Adding all the zero items of (3.20) and 𝑓⁰(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡))−𝑓⁰(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡))=0, respectively, into (3.18) and using the condition (2.7) for the following estimations: 𝑓⁰(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡))≤𝑄₁𝑥(𝑡),𝑥(𝑡)+𝑄₂𝑥((𝑡((𝑡+⟨(𝑡)(𝑡=𝑡−ℎ)),𝑥𝑡−ℎ))𝑅𝑢,𝑢)⟩𝑃𝑄₁𝑃𝑦(𝑡),𝑦(𝑡)+𝑃𝑄₂𝑃𝑦(𝑡−ℎ(𝑡)),𝑦(𝑡−ℎ(𝑡))+0.25𝐵𝑅𝐵^𝑇𝑦(𝑡)(𝑡,2𝑊,𝑦)₀𝑓(𝑥),𝑦≤𝑊₀𝐷₀𝑊^𝑇₀𝑦,𝑦+𝐷₀⁻¹𝑓(𝑥)(𝑥,2𝑊,𝑓)₁𝑔(𝑧),𝑦≤𝑊₁𝐷₁𝑊^𝑇₁𝑦,𝑦+𝐷₁⁻¹𝑔(𝑧)(𝑧,2𝐷,𝑔)₀₋₁𝑓(𝑥),𝑓(𝑥)≤𝐹𝐷₀⁻¹,2𝐷𝐹𝑥,𝑥₁₋₁𝑔(𝑧),𝑔(𝑧)≤𝐸𝐷₁⁻¹,2𝐸𝐸𝑧,𝑧_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃𝑦,𝑦≤𝜖₁𝐸^𝑇_𝑎𝐸_𝑎𝑦,𝑦+𝜖₁⁻¹𝑃𝐻^𝑇_𝑎𝐻_𝑎𝑃𝑦,𝑦,𝜖₁2𝐸>0,_𝑤0𝐹_𝑤0(𝑡)𝐻_𝑤0𝑃𝑓(𝑥),𝑦≤𝜖₂𝐸^𝑇_𝑤0𝐸_𝑤0𝑦,𝑦+𝜖₂⁻¹𝑃𝐷₀𝐻^𝑇_𝑤0𝐻_𝑤0𝐷₀𝑃𝑦,𝑦,𝜖₂2𝐸>0,_𝑤1𝐹_𝑤1(t)𝐻_𝑤1𝑃𝑔(𝑧),𝑦≤𝜖₃𝐸^𝑇_𝑤1𝐸_𝑤1𝑦,𝑦+𝜖₃⁻¹𝑃𝐷₁𝐻^𝑇_𝑤1𝐻_𝑤1𝐷₁𝑃𝑧,𝑧,𝜖₃>0, (3.21) we obtain 𝑉̇(⋅)+2𝛼𝑉(⋅)≤𝜁^𝑇(𝑡)ℰ𝜁(𝑡)+𝑦^𝑇(𝑡)𝑆₁𝑦(𝑡)+𝑦^𝑇(𝑡−ℎ(𝑡))𝑆₂𝑦(𝑡−ℎ(𝑡))−𝑓⁰(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡)), (3.22) where 𝜁(𝑡)=[𝑦(𝑡),𝑦(̇𝑡),𝑦(𝑡−ℎ₀),𝑦(𝑡−ℎ₁),𝑦(𝑡−ℎ(𝑡))], and 𝑊ℰ=¹¹𝑊₁₂𝑊₁₃𝑊₁₄𝑊₁₅∗𝑊₂₂𝑊₂₃𝑊₂₄𝑊₂₅∗∗𝑊₃₃𝑊₃₄𝑊₃₅∗∗∗𝑊₄₄𝑊₄₅∗∗∗∗𝑊₅₅,𝑆₁=−𝑃𝐴−𝐴^𝑇𝑃−¹_𝑖=0𝑒^{−2𝛼ℎ𝑖}𝐻_𝑖+4𝑃𝐹𝐷₀⁻¹𝐹𝑃+𝑃𝑄₁𝑆𝑃,₂=𝑊₁𝐷₁𝑊^𝑇₁−𝑒^−2𝛼ℎ2𝑈+4𝑃𝐸𝐷₁⁻¹𝐸𝑃+𝑃𝑄₂𝑃. (3.23) Note that by the Schur complement lemma, Proposition 2.3, the conditions 𝑆₁<0 and 𝑆₂<0 are equivalent to the conditions (3.3) and (3.4), respectively. Therefore, by condition (3.2), (3.3), and (3.4), we obtain from (3.22) that 𝑉̇𝑡,𝑥_𝑡≤−2𝛼𝑉𝑡,𝑥_𝑡,∀𝑡≥0. (3.24) Integrating both sides of (3.24) from 0 to 𝑡, we obtain 𝑉𝑡,𝑥_𝑡≤𝑉𝜙𝑒^−2𝛼𝑡,∀𝑡≥0. (3.25) Furthermore, taking condition (3.9) into account, we have 𝜆₁𝑥(𝑡,𝜙)²≤𝑉𝑥_𝑡≤𝑉𝜙𝑒^−2𝛼𝑡≤𝜆₂𝑒^−2𝛼𝑡𝜙², (3.26) then 𝑥𝑡,𝜙≤𝜆₂𝜆₁𝑒^−𝛼𝑡𝜙,𝑡≥0, (3.27) which concludes the exponential stability of the closed-loop system (2.8). To prove the optimal level of the cost function (2.4), we derive from (3.22) and (3.2)–(3.4) that 𝑉̇𝑡,𝑧_𝑡≤−𝑓⁰(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡)),𝑡≥0. (3.28) Integrating both sides of (3.28) from 0 to 𝑡 leads to ^𝑡₀𝑓⁰(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡))𝑑𝑡≤𝑉0,𝑧₀−𝑉𝑡,𝑧_𝑡≤𝑉0,𝑧₀, (3.29) dute to 𝑉(𝑡,𝑧_𝑡)≥0. Hence, letting 𝑡→+∞, we have 𝐽=^∞₀𝑓⁰(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡))𝑑𝑡≤𝑉0,𝑧₀≤𝜆₂𝜙²=𝐽^∗. (3.30) This completes the proof of the theorem.

Remark 3.2. Note that ℎ(𝑡) is non-differentiable and interval time-varying delay; therefore, the stability criteria proposed in [5–8, 12, 15–26] are not applicable to this system.

Example 3.3. Consider the uncertain neural networks with interval time-varying delays (2.1), where 𝐴=0.1000.3,𝑊₀=0.10.10.20.3,𝑊₁=,0.20.20.10.4,𝐵=0.10.2𝐸=0.3000.4,𝐹=0.2000.3,𝑄₁=0.20.10.10.4,𝑄₂=,0.30.20.20.5𝑅=0.10.10.10.3,𝐸_𝑎=0.10.10.10.3,𝐸_𝑤0=0.10.10.10.2,𝐸_𝑤1=,𝐻0.20.10.10.3_𝑎=0.30.20.20.2,𝐻_𝑤0=0.20.10.10.2,𝐻_𝑤1=,ℎ(𝑡)0.30.10.10.3=0.1+1.3sin²𝑡if𝑡∈ℐ=_𝑘≥0[()𝜋]ℎ(𝑡)2𝑘𝜋,2𝑘+1=0if𝑡∈𝑅⁺⧵ℐ. (3.31) Note that ℎ(𝑡) is non-differentiable; therefore, the stability criteria proposed in [4–8, 12, 15–26] are not applicable to this system. Given 𝛼=0.1,ℎ₀=0.1,andℎ₁=1.4, by using the Matlab LMI toolbox, we can solve for 𝑃,𝑈,𝐺₀,𝐺₁,𝐻₀,𝐻₁,𝐷₀, and 𝐷₁ which satisfy the conditions (3.2)–(3.4) in Theorem 3.1. A set of solutions are 𝜖₁=0.0017,𝜖₂=0.0013,𝜖₃=0.0012, ,𝐺𝑃=1.1578−0.1128−0.11281.0597,𝑈=2.3269−0.3820−0.38202.6681₀=1.45960.13970.13971.2369,𝐺₁=,𝐻2.26940.81140.81141.0125₀=0.64550.04520.04520.5104,𝐻₁=,𝐷0.30050.02330.02330.2306₀=0.0011000.0011,𝐷₁=.0.7809000.7809 (3.32) Then 𝑢(𝑡)=−0.2292𝑥₁(𝑡)−0.1816𝑥₂(𝑡),𝑡≥0 (3.33) is a guaranteed cost control law and the cost given by 𝐽^∗=5.4631𝜙². (3.34) Moreover, the solution 𝑥(𝑡,𝜙) of the system satisfies 𝑥𝑡,𝜙≤3.6984𝑒^−0.1𝑡𝜙,∀𝑡≥0. (3.35)

The exponential convergence dynamics of the network (2.1) are shown in Figure 1.

Figure 1 

The simulation of the solutions 𝑥1(𝑡) and 𝑥2(𝑡) with the initial condition 𝜙(𝑡[105]𝑇,𝑡∈[−0.4,0].

Example 3.4. Consider the uncertain neural networks with interval time-varying delays (2.1), where 𝐴=1002,𝑊₀=1123,𝑊₁=12,2214,𝐵=𝐸=3004,𝐹=2003,𝑄₁=2114,𝑄₂=,3225𝑅=1113,𝐸_𝑎=1113,𝐸_𝑤0=1112,𝐸_𝑤1=,𝐻2113_𝑎=3222,𝐻_𝑤0=2112,𝐻_𝑤1=,ℎ(𝑡)3113=0.1+0.7sin²𝑡if𝑡∈ℐ=_𝑘≥0[()𝜋]ℎ(𝑡)2𝑘𝜋,2𝑘+1=0if𝑡∈𝑅⁺⧵ℐ. (3.36) Note that ℎ(𝑡) is non-differentiable; therefore, the stability criteria proposed in [5–8, 12, 15–26] are not applicable to this system. Given 𝛼=0.3,ℎ₀=0.1,ℎ₁=0.8, by using the Matlab LMI toolbox, we can solve for 𝑃,𝑈,𝐺₀,𝐺₁,𝐻₀,𝐻₁,𝐷₀, and 𝐷₁ which satisfy the conditions (3.2)–(3.4) in Theorem 3.1. A set of solutions are 𝜖₁=0.9,𝜖₂=0.8,𝜖₃=0.7, ,𝐺𝑃=0.7832−0.0213−0.02130.0011,𝑈=0.1297−0.0019−0.00190.0197₀=0.17950.01370.01370.2211,𝐺₁=,𝐻1.21970.96480.96480.7391₀=0.89310.11830.11830.7197,𝐻₁=,𝐷0.68510.12970.12970.5726₀=0.1397000.2278,𝐷₁=.0.6812000.6813 (3.37) Then 𝑢(𝑡)=−0.7314𝑥₁(𝑡)−0.0196𝑥₂(𝑡),𝑡≥0, (3.38) is a guaranteed cost control law and the cost given by 𝐽^∗=24.3219𝜙². (3.39) Moreover, the solution 𝑥(𝑡,𝜙) of the system satisfies 𝑥𝑡,𝜙≤12.3690𝑒^−0.3𝑡𝜙,∀𝑡≥0. (3.40)

The exponential convergence dynamics of the network (2.1) are shown in Figure 2.

Figure 2 

The simulation of the solutions 𝑥1(𝑡) and 𝑥2(𝑡) with the initial condition 𝜙(𝑡)=[1−1]𝑇,𝑡∈[−1,0.8].

4. Conclusions

In this paper, the problem of guaranteed cost control for uncertain neural networks with interval nondifferentiable time-varying delay 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. By constructing a set of time-varying Lyapunov-Krasovskii functionals combined with Newton-Leibniz formula, a memoryless state feedback guaranteed cost controller design has been presented, and sufficient conditions for the existence of the guaranteed cost state-feedback for the system have been derived in terms of LMIs.

Acknowledgments

This work was supported by the Thai Research Fund Grant, the Higher Education Commission, and Faculty of Science, Maejo University, Thailand. The second author is supported by the Center of Excellence in Mathematics, Thailand, and Commission for Higher Education, Thailand. The authors thank anonymous reviewers for valuable comments and suggestions, which allowed us to improve the paper.