Thai Journal of Mathematics Vol. 18, No. 1 (2020), Pages 435 - 452 ISSN 1686-0209 NEW RESULTS ON DELAY-INTERVAL-DEPENDENT ROBUST EXPONENTIAL STABILITY FOR UNCERTAIN NEUTRAL-TYPE SYSTEMS WITH MIXED TIME-VARYING DELAYS AND NONLINEAR PERTURBATIONS Nayika Samorn1 , Kanit Mukdasai1 , Prem Junsawang2 , Sirada Pinjai3,∗ of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand, e-mail : nayika@kkumail.com (N. Samorn), kanit@kku.ac.th (K. Mukdasai) 2 Department of Statistics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand, e-mail : prem@kku.ac.th 3 Department of Mathematics, Faculty of Science and Agricultural, Technology, Ragamangala University of Technology Lanna, Chiang Mai 50300, Thailand, e-mail : siradapinjai@gmail.com 1 Department Abstract The design problem of delay-interval-dependent robust exponential stability for uncertain neutral-type system with distributed and discrete time-varying delays, and nonlinear perturbations was studied. We concentrated on norm-bounded uncertainties and nonlinear time-varying parameter perturbations. By using mixed model transformation, Peng-Park’s integral inequality, Wirtinger-based integral inequality, and proper Lyapunov-Krasovskii functional, new delay-inteval-dependent robust exponential stability criterion was received and formulated in the form of linear matrix inequalities (LMIs). Moreover, exponential stability criterion was also suggested for a neutral-type system with distributed and discrete time-varying delays, and nonlinear perturbations. Finally, numerical examples showed that the recommended approach achieves the expected results and the predominance of our results to those in the literature. MSC: 93D05; 93D09; 37C75 Keywords: Lyapunov-Krasovskii functional; linear matrix inequality; robust exponential stability; interval time-varying delay Submission date: 01.11.2019 / Acceptance date: 10.01.2020 1. Introduction Nowadays, neutral-type system is popularly in discussed because it can be applied in many fields which composed delays both in its derivatives and state variables [1, 2]. In practical applications, this delays can be noticed in various fields such as mechanics, *Corresponding author. Published by The Mathematical Association of Thailand. c 2020 by TJM. All rights reserved. Copyright ⃝ 436 Thai J. Math. Vol. 18, No. 1 (2020) / Samorn et al. vibrating masses attached to population ecology, distributed networks, heat exchangers, robots in contact with rigid environments, automatic control, [3, 4] and so on. Besides, in terms of applications is able to modelled by systems with distributed delay which appeared in [5, 6]. Recently, the issue of delay-dependent stability on uncertain neutral-type systems with time-varying delays has been studied in [7–9], and the considered system has interval timevarying delays and uncertainties in [10, 11]. In addition, several authors have designed the topic of stability for systems with time-varying delays, and nonlinear perturbations such as [12–14], and have presented some stability conditions for uncertain neutral-type systems with interval time-varying delays, and nonlinear perturbations which appeared in [15– 17]. In [14], Cheng et al. have expanded the novel criteria on uncertain delay-differential systems for neutral-type and nonlinear uncertainties, which the variation interval of time delay was divided into two subintervals by introducing the central point. Mohajerpoor, et al. [16] have taken advantage of descriptor transformation and utilizing triple integral terms, Which improved above system. Furthermore, the stability condition for neutraltype systems with mixed time delays and distributed delay have been studied in [18–21]. Pinjai and Mukdasai [18] have considered the issue of a class of delayed neutral-type systems with mixed time-varying delays, and nonlinear uncertain by using decomposition technique of coefficient matrix and the combination of descriptor model transfomation. The relationships between discrete delay, neutral delay and distributed delay for uncertain nutral-type systems has been studied in [20]. On the other hand, the exponential stability of various systems has also been received a lot of attention from researcher as well (for examples, see [18, 22–26]). Ali.[23] has used generalized eigenvalue problem approach for presented a novel exponential stability criterion for the neutral-type differential system with nonlinear uncertainties. Maharajan et al. have discussed the problem of exponential stability for BAM-type neural networks with non-fragile state estimator by fabricating a suitable LyapunovKrasovskii functional and enrolling some analysis techniques in [24]. As far as we can tell, there have proposed few results in the literature interesting the problem of delay-interval-dependent robust exponential stability of the uncertain neutraltype systems with time-varying delays, and nonlinear uncertainties. The exponential stability is important toward an analyzing stability because it can identify the rates convergence of system states to equilibrium point, so we have established the robustness of the exponential stability in Euclidean spaces. Besides, the characteristic of interval time-varying delay shows the ability of time delay on varying in an interval in which the lower bound of delay is not limited to zero. In this paper, the delay-interval-dependent robust exponential stability criterion was designed for uncertain neutral-type system with distributed and discrete time-varying delays, and nonlinear perturbations. we concentrated on norm-bounded uncertainties and nonlinear time-varying parameter perturbations. First, the problem of delay-intervaldependent exponential stability criterion was desined for neutral-type system with distributed and discrete time-varying delays, and nonlinear perturbations. By using the Leibniz-Newton formula, utilization of zero equation, mixed model transformation, PengPark’s integral inequality, Wirtinger-based integral inequality, and proper LyapunovKrasovskii functional, new delay-interval-dependent robust exponential stability criterion NEW RESULTS ON DELAY-INTERVAL-DEPENDENT ROBUST EXPONENTIAL . . . 437 was received and formulated in the form of LMIs. Then, the problem of delay-intevaldependent robust exponential stability criterion was suggested for uncertain neutral system with mixed time-varying delays, and nonlinear perturbations. Finally, we represented the numerical examples to indicate the advantage of the new results, which are superior to the containing results. 2. Preliminaries Notations: R+ denotes the set of all real non-negative numbers; Rn and Rn×r denotes the n-dimensional Euclidean space and the set of all n × r real matrices, respectively ; AT denotes the transpose of the matrix A; A is symmetric if A = AT ; λ(A) denotes the set of all eigenvalues of A; λmax (A) = max{Re λ : λ ∈ λ(A)}; λmin (A) = min{Re λ : λ ∈ λ(A)}; C([−δ̄, 0], Rn ) denotes the space of all continuous vector functions mapping [−δ̄, 0] into Rn where δ̄ = max{ηU , δU , gU }, ηU , δU , gU ∈ R+ ; xt = x(t + s), s ∈ [−δ̄, 0]; δU L = δU − δU ; ηU L = ηU − ηU ; gU L = gU − gU ; ∗ represents the elements below the main diagonal of a symmetric matrix. We consider uncertain neutral-type delayed systems and nonlinear perturbations ∫t ż(t) = A(t)z(t) + B(t)z(t − δ(t)) + C(t)ż(t − η(t)) + D(t) t−g(t) z(s)ds (2.1) +ga (t, z(t)) + gb (t, z(t − δ(t))) + gc (t, ż(t − η(t))), t ≥ 0, z(t) = Φ(t), ż(t) = Ψ(t), ∀t ∈ [−δ̄, 0], where z(t) ∈ Rn is the state variable. δ(t), η(t) and g(t) are discrete, neutral and distributed interval time-varying delays, respectively, satisfying 0 ≤ δL ≤ δ(t) ≤ δU , δ̇(t) ≤ δd , (2.2) 0 ≤ ηL ≤ η(t) ≤ ηU , η̇(t) ≤ ηd , (2.3) 0 ≤ gL ≤ g(t) ≤ gL , (2.4) where δL , δU , δd , ηL , ηU , ηd , gL and gU are given nonnegative real constants. Φ(t) and Ψ(t) are the initial functions that are continuously differentiable on C([−δ̄, 0], Rn ) with the norm ∥Φ∥ = sups∈[−δ̄,0] ∥Φ(s)∥, ∥Ψ∥ = sups∈[−δ̄,0] ∥Ψ(s)∥. The uncertainties gi (·), i = a, b, c, satisfying gi (0, ·) = 0, and gaT (t, z(t))ga (t, z(t)) gbT (t, z(t − δ(t)))gb (t, z(t − δ(t))) gcT (t, ż(t − η(t)))gc (t, ż(t − η(t))) ≤ αa2 z T (t)z(t), (2.5) ≤ αb2 z T (t αc2 ż T (t − δ(t))z(t − δ(t)), (2.6) − η(t))ż(t − η(t)), (2.7) ≤ where αa , αb and αc are nonnegative real constants. A(t) = A + ∆A(t), B(t) = B + ∆B(t), C(t) = C + ∆C(t), D(t) = D + ∆D(t), where A, B, C, D ∈ Rn×n are real constant matrices, and ∆A(t), ∆B(t), ∆C(t), ∆D(t) are uncertainties matrices, which the form is according to [ ] [ ] ∆A(t) ∆B(t) ∆C(t) ∆D(t) = L∆(t) Ga Gb Gc Gd , where L, Ga , Gb , Gc and Gd are real constant matrices with appropriate dimensions. The uncertainty matrix ∆(t) is satisfying ∆(t) = F (t)[I − JF (t)]−1 , (2.8) is said to be admissible where J is an unknown matrix satisfying I − JJ T > 0. (2.9) 438 Thai J. Math. Vol. 18, No. 1 (2020) / Samorn et al. The uncertainty matrix F (t) is satisfying F (t)T F (t) ≥ 0. (2.10) Definition 2.1. The system (2.1) is robustly exponentially stable if there exist real constants α > 0, k > 0 such that, the solution z(t, Φ, Ψ) of the system (2.1) satisfies ∥z(t, Φ, Ψ)∥ ≤ k max{∥Φ∥, ∥Ψ∥}e−αt , ′ t ≥ 0. n×n Lemma 2.2. (Jensen s inequality) Let Q ∈ R , Q = QT > 0 be any constant matrix, n δU be positive real constant and ż : [−δU , 0] → R be vector-valued function. Then, ∫ t ) )T ( ∫ t (∫ t ż(s)ds . ż(s)ds Q ż T (s)Qż(s)ds ≤ − −δU t−δU t−δU t−δU n×n T Lemma 2.3. [27](Sun et al.) Let Q ∈ R , Q = Q > 0 be any constant matrix, δL and δU be positive real constants. Then, ∫ t (∫ t )T ( ∫ t ) T z (s)Qz(s)ds ≤ − z(s)ds Q z(s)ds , −δU t−δU − 2 (δU − 2 2 δL ) t−δU ∫ −δL −δU ≤− (∫ ∫ z T (u)Qz(u)duds t+s −δL ∫ t −δU t−δU t t+s )T ( ∫ z(u)duds Q n×n −δL −δU ∫ t t+s T ) z(u)duds . Lemma 2.4. [17] Let Q ∈ R , Q = Q > 0 be any positive constant matrix, δ(t) be discrete time-varying delays with (2.2), z : [−δU , 0] → Rn be a vector function. Then, ∫ t−δL ∫ t−δL ∫ t−δL z(s)ds z T (s)dsQ z T (s)Qz(s)ds ≤ − −[δU L ] t−δ(t) t−δ(t) t−δU − ∫ t−δ(t) z T (s)dsQ t−δU ∫ t−δ(t) z(s)ds. t−δU Lemma ]2.5. [17] Let Q1 , Q2 , Q3 ∈ Rn×n be any costant matrices which Q1 ≥ 0, Q3 > 0, [ Q1 Q2 ≥ 0, δ(t) be time-varying delays with (2.2), ż : [−δU , 0] → Rn be vector ∗ Q3 function. Then, ][ ]T [ ] ∫ t−δL [ z(s) Q1 Q2 z(s) −[δU L ] ds ∗ Q3 ż(s) ż(s) t−δU   −Q3 Q3 0 −QT2 0  ∗ −Q3 − QT3 Q3 QT2 −QT2    T  ∗ −Q3 0 QT2  ≤ ω1  ∗  ω1 ,  ∗ ∗ ∗ −Q1 0  ∗ ∗ ∗ ∗ −Q1 [ ] ∫ ∫ t−δL t−δ(t) where ω1T = z(t − δL ), z(t − δ(t)), z(t − δU ), t−δ(t) z(s)ds, t−δU z(s)ds . and δ(t) Lemma 2.6. [17] Let χ, Mi ∈ Rn×n , i = 1, 2, . . . , 5 be any constant matrices   be χ M1 M2 time-varying delays with (2.2), z(t) ∈ Rn be a vector-valued function, if  ∗ M3 M4  ≥ ∗ ∗ M5 NEW RESULTS ON DELAY-INTERVAL-DEPENDENT ROBUST EXPONENTIAL . . . [ ] 0 and ω2T = z(t − δL ), z(t − δ(t)), z(t − δU ) , then ∫ t−δL − ż T (s)χż(s)ds t−δU  M1 + M1T −M1T + M2 T  ∗ M1 + M1T − M2 − M2T ≤ ω2 ∗ ∗   M3 M4 0 +(δU L )ω2T  ∗ M3 + M5 M4  ω2 . ∗ ∗ M5 439  0 −M1T + M2  ω2 −M2 − M2T Lemma 2.7. [28] (W irtinger − based integral inequality) Let Q ∈ Rn×n , Q = QT > 0 be any constant matrix, δL , δU be nonnegative real constants and ż : [−δU , −δL ] → Rn be a vector-valued function. Then,   ∫ t−δL −4Q −2Q 6Q −4Q 6Q  ω3 , ż T (s)Qż(s)ds ≤ ω3T  ∗ −(δU L ) t−δU ∗ ∗ −12Q [ ] ∫ t−δ where ω3T = z(t − δL ), z(t − δU ), δU1L t−δUL z(s)ds . Lemma 2.8. [29, 30] (P eng −[ P ark ′]s integral inequality) Let Q, S ∈ Rn×n be any Q S costant matrices which Q ≥ 0, ≥ 0, δ(t) be time-varying delay with 0 ≤ δ(t) ≤ ∗ Q n δU , ż : [−δU , 0] → R be a vector-valued function. Then,   ∫ t −Q Q−S S −2Q + S + S T Q − S  ω4 , ż T (s)Qż(s)ds ≤ ω4T  ∗ −δU t−δU ∗ ∗ −Q [ ] where ω4T = z(t), z(t − δ(t)), z(t − δU ) . Lemma 2.9. [31] For any real constant matrices of appropriate dimensions M , S and N with M = M T , and ∆(t) is given constant by (2.8)-(2.10), then M + S∆(t)N + N T ∆(t)S T < 0, holds if and only if  M ∗ ∗ where any positive real constant β. S −βI ∗  βN T βJ T  < 0, −βI 3. Main Results First, the exponential stability criterion will be offered for the following system ∫ t z(s)ds + ga (t, z(t)) ż(t) = Az(t) + Bz(t − η(t)) + C ż(t − η(t)) + D t−g(t) +gb (t, z(t − η(t))) + gc (t, ż(t − η(t))), t ≥ 0, z(t) = φ(t), ż(t) = ϕ(t), ∀t ∈ [−δ̄, 0]. (3.1) 440 Thai J. Math. Vol. 18, No. 1 (2020) / Samorn et al. Then, the new criterion of system (3.1) will be introduced via LMIs approach, which we present the following notations for later use. [ ] Λ = Λi,j 21×21 , (3.2) for Λi,j = ΛTj,i , where Λ1,1 = P1 A1 + AT1 P1 + 2QT1 + 2QT5 + QT9 A1 + AT1 Q9 + 2α(P1 + P2 ) + QT13 A1 2 (P5 + P6 ) + (δU L )2 (W2 + W3 ) + e−2αδU (M1 + M1 T ) +AT1 Q13 + P3 + P4 + δU 2 −2αδU 2 +δU e M3 − 4e−2αδU P8 − e−2αδU P9 + δU R1 − e−2αδU R3 + (δU L )2 R4 2 −4αδU 2 2 −δU e P10 − (δU L )2 e−4αδU W7 + gU P12 + (gU L )2 W10 + νa αA I + N1T A1 +AT1 N1 , Λ1,2 = P1 (B + A2 ) − QT1 + Q2 − QT5 + Q6 + QT9 (B + A2 ) + AT1 Q10 2 −2αδU +QT13 (B + A2 ) + δU e−2αδU (−M1T + M2 ) + δU e M4 + e−2αδU (P9 − S) +e−2αδU R3 + N1T (B + A2 ) + AT1 N2 , Λ1,3 = −2e−2αδU P8 + e−2αδU S, 2 R2 + (δU L )R5 − N1T + AT1 N3 , Λ1,4 = Q4 + Q8 − QT9 + AT1 Q12 + δU Λ1,5 = (P1 + QT9 + QT13 )C + N1T C + AT1 N4 , Λ1,6 = −e−2αδU R2T + δU e−4αδU P10 , Λ1,7 = δU e−4αδU P10 + (δU L )e−4αδU W7 , Λ1,8 = 6e−2αδU P8 , Λ1,9 = −QT1 + Q3 −QT5 + Q7 + AT1 Q11 + (P1 + QT9 + QT13 )A2 , Λ1,10 = P1 + QT9 + QT13 + N1T + AT1 N5 Λ1,11 = P1 + QT9 + QT13 + N1T + AT1 N6 , Λ1,12 = P1 + QT9 + QT13 + N1T + AT1 N7 , Λ1,14 = (δU L )e−4αδU W7 , Λ1,17 = P2 + AT1 Q14 − QT13 , Λ1,20 = (P1 + QT9 + QT13 )D +N1T D + AT1 N8 , Λ2,2 = −2QT2 − 2QT6 + QT10 (B + A2 ) + (B T + AT2 )Q10 2 −2αδU e (M3 + M5 ) −e−2αδU P4 + δd P4 + δU e−2αδU (M1 + M1T − M2 − M2T ) + δU −2e−2αδU (P9 − S − S T ) + (δU L )e−2αδU (M6 + M6T − M7 − M7T ) + (δU L )2 e−2αδU ×(M8 + M10 ) − e−2αδU (R3 + R3T ) − e−2αδU (R6 + R6T ) + νb αb2 I + N2T (B + A2 ) 2 −2αδU +(B T + AT2 )N2 , Λ2,3 = −δU e−2αδU (M1T − M2 ) + δU e M4 + e−2αδU (P9 − S) −(δU L )e−2αδU (M6T − M7 ) + (δU L )2 e−2αδU M9 + e−2αδU R3 + e−2αδU R6 , Λ2,4 = −Q4 − Q8 − QT10 + (B T + AT2 )Q12 − N2T + (B T + AT2 )N3 , Λ2,5 = QT10 C + N2T C + B T N4 + AT2 N4 , Λ2,6 = e−2αδU R2T , Λ2,7 = −e−2αδU R2T − e−2αδU R5T , Λ2,9 = −QT2 − Q3 − Q6 − Q7 + B T Q11 +AT2 Q11 , Λ2,10 = QT10 + N2T + (B T + AT2 )N5 , Λ2,11 = Q10 + N2T + (B T +AT2 )N6 , Λ2,12 = Q10 + N2T + (B T + AT2 )N7 , Λ2,13 = −(δU L )e−2αδU (M6 −M7T ) + (δU L )2 e−2αδU M9T + e−2αδU R6 , Λ2,14 = e−2αδU R5T , Λ2,17 = QT14 ×(B + A2 ), Λ2,20 = QT10 D + N2T D + (B T + AT2 )N8 , Λ3,3 = −e−2αδU W1 2 −2αδU −e−2αδU P3 − δU e−2αδU (M2T + M2T ) + δU e M5 − 4e−2αδU P8 − e−2αδU P9 −(δU L )e−2αδU (M7 + M7T ) + (δU L )2 e−2αδU M10 − 4e−2αδU W5 − e−2αδU R3 −e−2αδU R6 , Λ3,7 = e−2αδU R2T + e−2αδU R5T , Λ3,8 = 6e−2αδU P8 , 441 NEW RESULTS ON DELAY-INTERVAL-DEPENDENT ROBUST EXPONENTIAL . . . 2 Λ3,13 = −2e−2αδU W5 , Λ3,15 = 6e−2αδU W5 , Λ4,4 = −2QT12 + δU (P7 + P8 2 +P9 ) + (δU L )2 e−2αδU (W4 + W5 + W6 ) + δU R3 + (δU L )2 R6 + 4 δU P10 4 2 2 2 (δU − δL ) T W7 + P11 + (δU L )2 W8 + W9 − 2N3T + 2N10 , Λ4,5 = QT12 C 4 +N3T C − N4 , Λ4,9 = −QT4 − QT8 − Q11 + QT12 A2 + N3T A2 − N9 , + Λ4,10 = QT12 + N3T − N5 , Λ4,11 = QT12 + N3T − N6 , Λ4,12 = QT12 + N3T − N7 , T Λ4,17 = −N10 + N11 , Λ4,20 = QT12 D + N3T D − N8 , Λ5,5 = −e−2αηU P11 2 +ηd P11 + νc αC I + N4T C + C T N4 , Λ5,9 = C T Q11 + N4T A2 + C T N9 , Λ5,10 = N4T + C T N5 , Λ5,11 = N4T + C T N6 , Λ5,12 = N4T + C T N7 , Λ5,17 = C T QT14 , Λ5,20 = N4T D + C T N8 , Λ6,6 = −e−2αδU P6 − e−2αδU R1 −e−4αδU P10 , Λ6,7 = −e−4αδU P10 , Λ7,7 = −e−2αδU P6 − e−2αδU W3 − e−2αδU R1 2 −2αδU e −e−2αδU R4 − e−4αδU P10 − e−4αδU W7 , Λ7,14 = e−4αδU W7 , Λ8,8 = −δU ×P5 − 12e−2αδU P8 , Λ9,9 = −2QT3 − 2QT7 + N1T A2 + AT2 N9 , Λ9,10 = QT11 +AT2 N5T + N9T , Λ9,11 = QT11 + AT2 N6T + N9T , Λ9,12 = QT11 + AT2 N7T + N9T , Λ9,20 = N9T D + QT11 D + AT2 N8 , Λ10,10 = −νa I + 2N5T , Λ10,11 = N5T + N6 , Λ10,12 = N5T + N7 , Λ10,17 = Q14 , Λ10,20 = N5T D + N8 , Λ11,11 = −νb +2N6T , Λ11,12 = N6T + N7 , Λ11,17 = Q14 , Λ11,20 = N6T D + N8 , Λ12,12 = −νc I + 2N7T , Λ12,17 = Q14 , Λ12,20 = N7T D + N8 , Λ13,13 = e−2αδL ×W1 + (δU L )e−2αδU (M6 + M6T ) + (δU L )2 e−2αδU M8 − 4e−2αδU W5 − e−2αδU ×R6 , Λ13,14 = −e−2αδU R5T , Λ13,15 = 6e−2αδU W5 , Λ14,14 = −e−2αδL W3 −e−2αδU R4 − e−4αδU W7 , Λ15,15 = −(δU L )2 e−2αδU W2 − 12e−2αδU W5 , T Λ16,16 = −e−2αδU W6 , Λ17,17 = −2QT14 − 2N11 , Λ17,20 = QT14 D, Λ18,18 = (ηU − ηL )e−2αηU W8 − e−2αηL W9 , Λ19,19 = −(ηU L )e−2αηU W8 , Λ20,20 = −e−2αgU P12 + N8T D + DT N8 , Λ21,21 = −e−2αgU W10 , and others are equal to zero. Theorem 3.1. For ∥C∥ + αc < 1, α > 0, if there exist positive definite symmetric matrices Pi , Wj , i = 1, 2, . . . , 12, j = 1, 2, . . . , 10, any appropriate dimensional matrices S, Qk , Ml , Nm , Rn , k = 1, 2, . . . , 14, l = 1, 2, . . . , 10, m = 1, 2, . . . , 11, n = 1, 2, ..., 6, and positive real constants αs , νs , s = a, b, c, satisfying the following LMIs   P7 ∗ ∗ W4  ∗ ∗ M1 M3 ∗ M6 M8 ∗  M2 M4  M5  M7 M9  M10 ≥ 0, (3.3) ≥ 0, (3.4) 442 Thai J. Math. Vol. 18, No. 1 (2020) / Samorn et al. [ R1 ∗ R2 R3 [ ] ] R 4 R5 ∗ R6 [ ] P9 S ∗ P9 Λ ≥ 0, (3.5) ≥ 0, (3.6) ≥ 0, (3.7) < 0, (3.8) then the system (3.1) is exponentially stable. Proof. Firstly, we improve the bound of interval time-varying delays by using the decomposition technique. Let constant matrix A as A = A1 + A2 , (3.9) where A1 , A2 ∈ Rn×n are constant matrices. Ensure the exponential stability of the system (3.1) by choosing to take advantage of the zero equation as follows ∫ t 0 = z(t) − z(t − δ(t)) − ż(s)ds. (3.10) t−δ(t) By (3.9) and (3.10), the system (3.1) can be represented in the form of the descriptor system ∫ t ż(s)ds + C ż(t − η(t)) ż(t) = A1 z(t) + (A2 + B)z(t − δ(t)) + A2 t−δ(t) +D ∫ t z(s)ds + ga (t, z(t)) + gb (t, z(t − δ(t))) t−g(t) +gc (t, ż(t − η(t))). (3.11) Modify the system (3.11) in term of descriptor systems, which is the form as follows ż(t) = 0 = (3.12) w(t), −w(t) + A1 z(t) + (A2 + B)z(t − δ(t)) + A2 +C ż(t − η(t)) + D ∫ t ∫ t ż(s)ds t−δ(t) z(s)ds + ga (t, z(t)) t−g(t) +gb (t, z(t − δ(t))) + gc (t, ż(t − η(t))). (3.13) Next, we consider Lyapunov-Krasovskii functional for a class of neutral-type delayed systems (3.11), (3.12) and (3.13) as V (t) = 9 ∑ Vi (t), (3.14) i=1 where V1 (t) = T z(t)  z(t − δ(t))   ∫ z T (t)P1 z(t) =   t ż(s)ds t−δ(t) ż(t)   I 0  0 0 0 0 0 0 0 0 0 0  0 0  0 0 NEW RESULTS ON DELAY-INTERVAL-DEPENDENT ROBUST EXPONENTIAL . . .   z(t) 0 0  z(t − δ(t))  Q3 Q4  ,  ∫ t Q7 Q8   t−δ(t) ż(s)ds Q11 Q12 ż(t) [ ]T [ ][ ][ ] z(t) I 0 P2 0 z(t) T z (t)P2 z(t) = , w(t) 0 0 Q13 Q14 w(t) ∫ t ∫ t e2α(s−t) z T (s)P3 z(s)ds + e2α(s−t) z T (s)P4 z(s)ds  0 Q2 Q6 Q10 P1 Q1 × Q5 Q9 V2 (t) = V3 (t) = t−δU + V4 (t) = ∫ δU e2α(s−t) z T (s)W1 z(s)ds, t−δU ∫ 0 −δU +(δU L ) V5 (t) = δU ∫ 0 −δU +(δU L ) V6 (t) = V7 (t) = V8 (t) = 0 + = ∫ ∫ [ ] e2α(θ−t) z T (θ) P5 + P6 z(θ)dθds t+s −δL −δU t ∫ ∫ t ∫ t t+s [ ] e2α(θ−t) z T (θ) W2 + W3 z(θ)dθds, ] [ e2α(θ−t) ż T (θ) P7 + P8 + P9 ż(θ)dθds t+s −δL −δU t ∫ t t+s [ ] e2α(θ−t) ż T (θ) W4 + W5 + W6 ż(θ)dθds, gU t−ηL ∫ 0 −gU +(gU L ) ∫ e2α(θ−t) [ ]T [ ][ ] R1 R2 z(θ) dθds ∗ R3 ż(θ) −δU t+s ][ ] [ ]T [ ∫ −δL ∫ t R4 R5 z(θ) 2α(θ−t) z(θ) e dθds, +(δU L ) ∗ R6 ż(θ) ż(θ) t+s −δU 2 ∫ 0 ∫ 0∫ t δU e2α(u+θ−t) ż T (u)P10 ż(u)dudθds 2 −δU s t+θ 2 2 ∫ −δL ∫ 0 ∫ t (δU − δL ) + e2α(u+θ−t) ż T (u)W7 ż(u)dudθds, 2 t+θ s −δU ∫ t ∫ t−ηL e2α(s−t) ż T (s)W8 ż(s)ds e2α(s−t) ż T (s)P11 ż(s)ds + (ηU L ) δU ∫ t−η(t) ∫ t V9 (t) t−δ(t) t−δL z(θ) ż(θ) t−ηU e2α(s−t) ż T (s)W9 ż(s)ds, ∫ ∫ t e2α(θ−t) z T (θ)P12 z(θ)dθds t+s −gL −gU ∫ t e2α(θ−t) z T (θ)W10 z(θ)dθds. t+s The differential of V1 (t) along the trajectory of system (3.11), we obtian V̇1 (t) = 2z T (t)P1 ż(t) 443 444 Thai J. Math. Vol. 18, No. 1 (2020) / Samorn et al. = [ T 2z (t)P1 A1 z(t) + (A2 + B)z(t − δ(t)) + A2 +C ż(t − η(t)) + D ] ∫ t ×QT5 T + z (t − −z(t − δ(t)) − + ∫ t ż T t−δ(t) ][ [ t−δ(t) (s)QT11 ds ∫ T (t)QT1 T + z (t − z(t) − z(t − δ(t)) − t ×z(t − δ(t)) + A2 ż(s)ds t−δ(t) z(s)ds + ga (t, z(t)) + gb (t, z(t − δ(t))) δ(t))QT6 ∫ t t−g(t) +gc (t, ż(t − η(t))) + 2 z ×QT3 ds + ż(t)QT4 ∫ + ∫ t ż T t−δ(t) ∫ δ(t))QT2 t t ż T (s) t−δ(t) ] [ ż(s)ds + 2 z T (t) t−δ(t) (s)QT7 ds + ∫ + ż(t)QT8 ][ z(t) ] [ ż(s)ds + 2 z T (t)QT9 + z T (t − δ(t))QT10 + ż(t)QT12 t ][ − ż(t) + A1 z(t) + (A2 + B) ż(s)ds + C ż(t − η(t)) + D t−δ(t) ] ∫ t z(s)ds t−g(t) +ga (t, z(t)) + gb (t, z(t − δ(t))) + gc (t, ż(t − η(t))) + 2αz T (t)P1 z(t) −2αV1 (t). Calculating V̇2 (t) in accordance with the solutions of the systems (3.12) and (3.13), we get ]T [ ][ ] P2 QT13 ż(t) V̇2 (t) = 0 0 QT14 [ = 2z T (t)P2 w(t) + 2z T (t)QT13 − w(t) + A1 z(t) + (A2 + B)z(t − δ(t)) [ z(t) 2z (t)P2 ż(t) = 2 w(t) T +A2 ∫ t ż(s)ds + C ż(t − η(t)) + D t−δ(t) ∫ t z(s)ds + ga (t, z(t)) t−g(t) [ ] +gb (t, z(t − δ(t))) + gc (t, ż(t − η(t))) + 2wT (t)QT14 − w(t) + A1 z(t) +(A2 + B)z(t − δ(t)) + A2 +D ∫ t ∫ t ż(s)ds + C ż(t − η(t)) t−δ(t) z(s)ds + ga (t, z(t)) + gb (t, z(t − δ(t))) + gc (t, ż(t − η(t))) t−g(t) +2αz T (t)P2 z(t) − 2αV2 (t). The time derivative of V3 (t) is calculated as V̇3 (t) ≤ z T (t)[P3 + P4 ]z(t) − e−2αδU z T (t − δU )P3 z(t − δU ) ] NEW RESULTS ON DELAY-INTERVAL-DEPENDENT ROBUST EXPONENTIAL . . . 445 −e−2αδU z T (t − δ(t))P4 z(t − δ(t)) + δd z T (t − δ(t))P4 z(t − δ(t)) +e−2αδL z T (t − δL )W1 z(t − δL ) − e−2αδU z T (t − δU )W1 z(t − δU ) −2αV3 (t). For any scalar s ∈ [t − δU , t], we obviously e−2αδU ≤ e−2αδL ≤ e2α(s−t) ≤ 1. Combine with Lemma 2.2 and Lemma 2.4, we obtain V˙4 (t) as the follow V˙4 (t) ≤ 2 T (δU ) z (t)[P5 + P6 ]z(t) − e × ( (∫ 1 δU t ∫ t t−δU ) z(s)ds − e−2αδU ) z(s)ds + t−δ(t) 2 T (∫ t−δ(t) ( 1 δU [( ∫ t −2αδU ∫ t z (s)ds (δU )2 P5 t−δU t−δ(t) ) z T (s)ds P6 ) T z (s)ds P6 × t−δU −2αδU ) T ( 1 (∫ ∫ t−δ(t) z(s)ds t−δU t−δL T )] ) z (s)ds δU L t−δU ( ) [( ∫ t−δL ) ∫ t−δL 1 ×(δU L )2 W2 z T (s)ds z(s)ds − e−2αδU δU L t−δU t−δ(t) ) ( ∫ t−δ(t) ( ∫ t−δL ) ( ∫ t−δ(t) )] T z(s)ds z (s)ds W3 ×W3 z(s)ds + +(δU L ) z (t)[W2 + W3 ]z(t) − e t−δ(t) t−δU t−δU −2αV4 (t). By taking advantage of the Lemma 2.6 - Lemma 2.8, the differential of V5 (t) is calculated as V˙5 (t) ≤ T z(t) 2 T δU ż (t)[P7 + P8 + P9 ]ż(t) + δU e−2αδU z(t − δ(t)) z(t − δU )    T T M1 + M1 −M1 + M2 0 z(t) ∗ M1 + M1T − M2 − M2T −M1T + M2  z(t − δ(t)) × z(t − δU ) ∗ ∗ −M2 − M2T )  T   z(t) M3 M4 0 z(t) +δU z(t − δ(t))  ∗ M3 + M5 M4  z(t − δ(t)) z(t − δU ) ∗ ∗ M5 z(t − δU )   T   z(t) z(t) 4P8 2P8 −6P8 − δU )   ∗ − δU )  4P8 −6P8   z(t −e−2αδU  z(t ∫t ∫t 1 1 ∗ ∗ 12P8 δU t−δU z(s)ds δU t−δU z(s)ds  T    P9 −P9 + S −S z(t) z(t) −e−2αδU z(t − δ(t))  ∗ 2P9 − S − S T −P9 + S  z(t − δ(t)) z(t − δU ) z(t − δU ) ∗ ∗ P9 (  446 Thai J. Math. Vol. 18, No. 1 (2020) / Samorn et al. T ( z(t − δL ) +(δU L )2 ż T (t)[W4 + W5 + W6 ]ż(t) + (δU L )e−2αδU z(t − δ(t)) z(t − δU )    T T 0 −M6 + M7 M6 + M6 z(t − δL ) ∗ M6 + M6T − M7 − M7T −M6T + M7  z(t − δ(t)) × z(t − δU ) ∗ ∗ −M7 − M7T    T  z(t − δL ) ) M8 M9 0 z(t − δL ) +(δU L ) z(t − δ(t))  ∗ M8 + M10 M9  z(t − δ(t)) ∗ ∗ M10 z(t − δU ) z(t − δU )   T  z(t − δL ) 4W5 2W5 −6W5   ∗ z(t − δU ) 4W5 −6W5  −e−2αδU  ∫ t−δL 1 ∗ ∗ 12W5 δU L t−δU z(s)ds   ( ∫ t−δL ) ( ∫ t−δL ) z(t − δL ) −2αδU T   z(t − δ ) × −e ż (s)ds W6 ż(s)ds ∫ t−δL U 1 t−δU t−δU z(s)ds δU L t−δU −2αV5 (t). According to Lemma 2.5 and calculating V˙6 (t), we obtain V˙6 (t) ≤ T z(t)  z(t − δ(t))  [ ][ ]T [ ]   R1 R2 z(t) 2 z(t) −2αδU  z(t − δU )  δU +e  ∫ t ∗ R3 ż(t) ż(t)    t−δ(t) z(s)ds  ∫ t−δ(t) z(s)ds t−δU     z(t) −R3 R3 0 −R2T 0  z(t − δ(t))   ∗ −R3 − R3 R3 R2T −R2T      T   z(t − δU )  ∗ ∗ −R 0 R ×   3 ∫ 2   t   ∗ ∗ ∗ −R1 0   t−δ(t) z(s)ds  ∫ t−h( t) ∗ ∗ ∗ ∗ −R1 z(s)ds t−δU  T z(t − δL )  z(t − δ(t))  [ ]T [ ][ ]   R4 R5 z(t) 2 z(t) −2αδU  z(t − δU )  +(δU L ) +e  ∫ t−δ  L ż(t) ∗ R6 ż(t)    t−δ(t) z(s)ds  ∫ t−h( t) z(s)ds t−δU     z(t − δL ) −R6 R6 0 −R5T 0  z(t − δ(t))   ∗ −R6 − R6 R6 R5T −R5T      T   z(t − δU )  ∗ ∗ −R 0 R ×   − 2αV6 (t). 6 ∫ 5    t−δL   ∗ ∗ ∗ −R4 0   t−δ(t) z(s)ds  ∫ t−h( t) ∗ ∗ ∗ ∗ −R4 z(s)ds  t−δU NEW RESULTS ON DELAY-INTERVAL-DEPENDENT ROBUST EXPONENTIAL . . . 447 Now, applying Lamma 2.3, differenting lead to V˙7 (t) ≤ ( ∫ t 4 δU ż T (t)P10 ż(t) − e−4αδU δU z(t) − z(s)ds 4 t−δ(t) ( )T ∫ t−δ(t) ∫ t ∫ − z(s)ds P10 δU z(t) − z(s)ds − t−δU t−δ(t) t−δ(t) z(s)ds t−δU ( ∫ t−δL 2 2 2 (δU − δL ) T −4αδU z(s)ds (δU L )z(t) − ż (t)W7 ż(t) − e + 4 t−δ(t) ( )T ∫ t−δ(t) ∫ t−δL − z(s)ds) z(s)ds W7 (δU L )z(t) − t−δ(t) t−δU − ∫ ) t−δ(t) ) z(s)ds − 2αV7 (t). t−δU Besides, for any scalar s ∈ [t − ηU , t], we obtain e−2αηU ≤ e−2αηL ≤ e2α(s−t) ≤ 1. The time derivative of V8 (t) is calculated as V˙8 (t) ≤ ż T (t)(P11 + W9 )ż(t) − e−2αηU ż T (t − η(t))P11 ż(t − η(t)) +ηd ż T (t − η(t))P11 ż(t − η(t)) + (ηU − ηL )e−2αηU ׿ T (t − ηL )W8 ż(t − ηL ) − (ηU − ηL )e−2αηU ż T (t − ηU ) ×W8 ż(t − ηU ) − e−2αηL ż T (t − ηL )W9 ż(t − ηL − 2αV8 (t). Further, for any scalar s ∈ [t − gU , t], we have e−2αgU ≤ eα(s−t) ≤ 1. Combine with Lamma 2.2, we obtain V̇9 (t) as follows V̇9 (t) ≤ 2 T gU z (t)P12 z(t) − e−2αgU ∫ t z T (s)dsP12 t−g(t) +(gU L )2 z T (t)W10 z(t) − e−2αgU × ∫ t−gL ∫ t−gL ∫ t z(s)ds t−g(t) z T (s)dsW10 t−gU z(s)ds − 2αV9 (t). t−gU Consider (2.5)-(2.7), we inspected that the following inequalities hold: νa (αa2 z T (t)z(t) − gaT (t, z(t))ga (t, z(t)))) ≥ 0, (3.15) νb (αb2 z T (t − δ(t))z(t − δ(t))gbT (t, z(t − δ(t))gb (t, z(t − δ(t)))) ≥ 0, νc (αc2 ż T (t − η(t))ż(t − η(t)) − gcT (t, ż(t − η(t))gc (t, ż(t − η(t)))) ≥ 0, (3.16) where νa , νb , νc , are positive real constants. (3.17) 448 Thai J. Math. Vol. 18, No. 1 (2020) / Samorn et al. By the use zero equations, we obtain the following equations [ 2 z T (t)N1T + z T (t − δ(t))N2T + ż T (t)N3T + ż T (t − η(t)))N4T + ga (t, z(t))N5T +gb (t, z(t − δ(t)))N6T + gc (t, z(t − η(t)))N7T + + ∫ t t−δ(t) +A2 ∫ ż T (s)dsN9T ][ ∫ t t−g(t) z T (s)dsN8T − ż(t) + A1 z(t)) + (A2 + B)z(t − δ(t)) t ż T (s)ds + C ż(t − η(t)) + D t−δ(t) ] ∫ t z(s)ds + ga (t, z(t)) t−g(t) +gb (t, z(t − δ(t))) + gc (t, z(t − η(t))) = 0, (3.18) ][ ] [ T T ż(t) − w(t) = 0, 2 ż T (t)N10 + wT (t)N11 (3.19) where any real matrices Nm , m = 1, 2, ..., 11 with appropriate dimensions. Due to the use (3.15)-(3.19), it is apparently that V̇ (t) + 2αV (t) ≤ ξ T (t)Λξ(t), (3.20) ∫t where ξ T (t) = z(t), z(t − δ(t)), z(t − δU ), ż(t), ż(t − η(t)), t−δ(t) z(s)ds, ∫ t−δ(t) ∫t ∫t z(s)ds, δ1U t−δU z(s)ds, t−δ(t) ż(s)ds, ga (t, z(t)), gb (t, z(t − δ(t))), t−δU ∫ t−δ ∫ t−δ ∫ t−δL gc (t, ż(t − η(t))), z(t − δL ), t−δ(t) z(s)ds, (δU1L ) t−δUL z(s)ds, t−δUL ż(s)ds, ] ∫t ∫ t−g w(t), ż(t − ηL ), ż(t − ηU ), t−g(t) z(s)ds, t−gUL z(s)ds and Λ is defined in (3.2). By condition (3.8), we obtain [ V̇ (t) + 2αV (t) ≤ 0, ∀t ∈ R+ , (3.21) V̇ (0) ≤ V (0)e−2αt , ∀t ∈ R+ . (3.22) which gives It is readily visible that λmin (P1 )∥z(t)∥2 ≤ V (t) ≤ V (0)e−2αt ≤ N max{∥Φ}, ∥Ψ∥}2 e−2αt , ∀t ∈ R+ , and ∥z(t, Φ, Ψ)∥ ≤ √ N max{∥Φ}, ∥Ψ∥}e−αt , λmin (P1 ) ∀t ∈ R+ , where N = 3 λmax (P1 + P2 ) + δU λmax (P3 + P4 ) + (δU L )λmax (W1 ) + δU λmax (P5 +P6 + P7 + P8 + P9 ) + (δU L )3 λmax (W2 + W3 + W4 + W5 + W6 ) ] [ ] [ δ5 R1 R2 R 4 R5 3 3 +δU λmax + (δU L ) λmax + U λmax (P10 ) ∗ R3 ∗ R6 2 2 2 (δU − δL ) (δU L )3 λmax (W7 ) + ηU λmax (P11 ) + (ηU L )2 λmax (W8 ) 2 3 +ηL λmax (W9 ) + gU λmax (P12 ) + (gU L )3 λmax (W10 ). + (3.23) NEW RESULTS ON DELAY-INTERVAL-DEPENDENT ROBUST EXPONENTIAL . . . 449 Therefore, if the LMIs conditions (3.3)-(3.8) hold, we conclude that the system (3.1) is exponentially stable, This proof is complete. Based on Theorem 3.1, we consider the new delay-interval-dependent robust exponential stability for uncertain neutral-type system with distributed and discrete time-varying delays, and nonlinear perturbations for (2.1). Then, the corresponding result is summarized in Theorem 3.2. Theorem 3.2. For ∥C(t)∥ + αc < 1, α > 0, if there exist positive definite symmetric matrices Pi , Wj , i = 1, 2, . . . , 12, j = 1, 2, . . . , 10, any appropriate dimensional matrices S, Qk , Ml , Nm , Rn , k = 1, 2, . . . , 14, l = 1, 2, . . . , 10, m = 1, 2, . . . , 11, n = 1, 2, ..., 6, and positive real constants β αs , νs , s = a, b, c, satisfying the following LMIs (3.3)-(3.7) and   Λ Γ1 βΓT2  ∗ −βI βJ T  < 0, (3.24) ∗ ∗ −I then the system (2.1) is robustly exponentially stable. Proof. Result of the use the similar method in the proof of Theorem 3.1, and substitution A1 , B, C and D in LMI (3.8) with A1 + L∆(t)Ga , B + L∆(t)Gb , C + L∆(t)Gc and D + L∆(t)Gd , respectively, we conclude that condition (3.8) for system (2.1) is equivalent to following are required Λ + Γ1 ∆(t)Γ2 + ΓT2 ∆T (t)ΓT1 < 0, (3.25) [ where ΓT1 = (P1 + QT9 + QT13 + N1T )L, (QT10 + N2T )L, 0, (QT12 + N3T )L, N4T L, 0, 0, 0, ] [ (QT11 + N9T )L, N5T L, N6T L, N7T L, 0, 0, 0, 0, QT13 L, 0, 0, N8T L, 0 , Γ2 = Ga , Gb , 0, 0, ] Gc , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Gd , 0 , and Λ is defined in (3.2). By using Lemma 2.9, condition (3.25) is equivalent to the condition (3.24). Thus, if the LMIs conditions (3.3) - (3.7) and (3.24) hold, we conclude that the system (2.1) is robustly exponentially stable. This proof is complete. 4. Numerical examples In this section, we allow the numerical examples to show the performance of the systems (2.1) and (3.1). Example 4.1. Consider the uncertain neutral-type system with distributed and discrete interval[ time-varying] delays, and (2.1), where ] ] [ [ [ nonlinear]perturbations −0.2 0 −1.1 −0.2 −0.1 0.2 −0.8 0 , , C = , B = , A2 = A1 = 0.2 −0.1 −0.1 −1.1 0 −0.1 [ 0.1 −0.8 ] −0.12 −0.12 D= , L = I, Ga = Gb = Gc = Gd = 0.1I, −0.12 0.12 αa = 0.1, αb = αc = 0.05. We have found that the LMI (3.24) is feasible, which consider for ηL = gL = 0.1, ηU = gU = 0.2, and ηd = δd . In Table 1, we show the maximum allowable bound δU for ensuring Theorem 3.2 of the system(2.1), which for the exponential convergence rate α = 0.5, ηd = δd = 0.1 and δL = 0.5, we obtain δU = 0.7487. 450 Thai J. Math. Vol. 18, No. 1 (2020) / Samorn et al. Table 1. Upper bounds of time delay δU for various values of α and δL . δd = η d 0.0 0.1 δL α = 0.0 0.5 1.2526 1.0 1.4990 0.5 1.2108 1.0 1.4625 α = 0.1 1.0711 1.3550 1.4010 1.3352 α = 0.3 0.8731 1.2069 0.8510 1.1951 α = 0.5 0.7487 1.0560 0.7340 1.0507 α = 0.7 0.6250 0.6121 - α = 0.9 0.5442 0.5346 - Example 4.2. Consider the uncertain neutral-type system with distributed and discrete where ] interval[ time-varying] delays (2.1), [ [ ] [ ] −0.8 0 −0.1 0.2 −1.1 −0.2 −0.2 0 A1 = , A2 = , B = , C = , −0.1 −1.1 −0.1 0.2 −0.1 0 [ 0.1 −0.8 ] −0.12 −0.12 D= , L = I, Ga = Gb = Gc = Gd = 0.1I, −0.12 0.12 ga (t, z(t)) = gb (t, z(t − δ(t))) = gc (t, ż(t − η(t))) = 0. By appying Theorem 3.2, we show the upper bounds on distributed time delay gU for different α, which apply the conditions in [19], [20] and [21], where ηL = δL = gL = 0, ηU = δU = 0.1 and ηd = δd = 0. It is clear that our result (Theorem 3.2) are better than those results, which appeared in [19], [20] and [21]. Moreover, we show the ensuring exponential stability of system (2.1). Table 2. Upper bounds of time delay gU for various values of α. Method Chen et al. [19] Chen et al. [20] Zhu et al. [21] Theorem 3.2 α = 0.0 6.67 6.67 6.8925 7.2682 α = 0.1 4.4142 α = 0.2 3.3197 α = 0.3 2.6887 α = 0.5 1.9472 5. Conclusions The problem of delay-interval-dependent robust exponential stability criterion for uncertain neutral-type system with distributed and discrete time-varying delays, and nonlinear perturbations was studied. We concentrated on norm-bounded uncertainties and nonlinear time-varying parameter perturbations. New delay-interval-dependent robust exponential stability criterion for uncertain neutral-type system with distributed and discrete interval time-varying delays, and nonlinear perturbations was received and formulated in terms of LMIs by using mixed model transformation, Peng-Park’s integral inequality, Wirting-based integral inequality and proper Lyapunov-Krasovskii function. 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