Automatica. Exponential stability of linear distributed parameter systems with time-varying delays. Emilia Fridman a,, Yury Orlov b.

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1 Automatica 45 (9) Contents lists available at ScienceDirect Automatica journal homepage: Brief paper Exponential stability of linear distributed parameter systems with time-varying delays Emilia Fridman a,, Yury Orlov b a School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel b CICESE Research Center, Km.17, Carretera Tijuana-Ensenada, Ensenada, B.C. 86, Mexico a r t i c l e i n f o a b s t r a c t Article history: Received 31 May 7 Received in revised form 17 March 8 Accepted 1 June 8 Available online 3 November 8 Keywords: Distributed parameter systems Time-varying delay Stability Lyapunov method LMI Exponential stability analysis via the Lyapunov Krasovskii method is extended to linear time-delay systems in a Hilbert space. The operator acting on the delayed state is supposed to be bounded. The system delay is admitted to be unknown and time-varying with an a priori given upper bound on the delay. Sufficient delay-dependent conditions for exponential stability are derived in the form of Linear Operator Inequalities (LOIs), where the decision variables are operators in the Hilbert space. Being applied to a heat equation and to a wave equation, these conditions are reduced to standard Linear Matrix Inequalities (LMIs). The proposed method is expected to provide effective tools for stability analysis and control synthesis of distributed parameter systems. 8 Elsevier Ltd. All rights reserved. 1. Introduction Time-delay naturally appears in many control systems, and it is frequently a source of instability (Kolmanovskii & Myshkis, 1999). In the case of distributed parameter systems, even arbitrarily small delays in the feedback may destabilize the system (see e.g. Datko (1988), Logemann, Rebarber, and Weiss (1996) and Nicaise and Pignotti (6)). The stability issue of systems with delay is, therefore, of theoretical and practical value. During the last decade, a considerable amount of attention has been paid to stability of Ordinary Differential Equations (ODEs) with uncertain constant or time-varying delays (see e.g. Gu, Kharitonov, and Chen (3), Kolmanovskii and Myshkis (1999), Niculescu (1) and Richard (3)). Special forms of Lyapunov Krasovskii functionals have been used for derivation of simple finite dimensional conditions in terms of LMIs (Boyd, El Ghaoui, Feron, & Balakrishnan, 1994). These conditions are either delay-independent or delay-dependent. The stability analysis of Partial Differential Equations (PDEs) with delay is essentially more complicated. There are only a few This paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Faryar Jabbari under the direction of Editor Roberto Tempo. Corresponding author. Tel.: ; fax: addresses: emilia@eng.tau.ac.il (E. Fridman), yorlov@cicese.mx (Y. Orlov). works on Lyapunov-based technique for PDEs with delay. The second Lyapunov method was extended to abstract nonlinear time-delay systems in the Banach spaces in Wang (1994a), and was applied to stability analysis of some scalar heat/wave equations with constant delays and with the Dirichlet boundary conditions in Wang (1994b). Stability and instability conditions for delay wave equations were found in Nicaise and Pignotti (6). In the present paper, we study the exponential stability of general distributed parameter systems. A class of linear systems in a Hilbert space is considered, where a bounded operator acts on the delayed state. The system delay is admitted to be unknown and time-varying. Sufficient delay-dependent exponential stability conditions are derived in the form of LOIs, where the decision variables are operators in the Hilbert space. General methods for solving LOI have not been developed yet. Some finite dimensional approximations were considered in Ikeda, Azuma, and Uchida (1). Being applied to a heat equation and to a wave equation, the derived conditions are reduced to standard finite-dimensional LMIs that appear to guarantee the exponential stability of the first order and, respectively, the second order delay-differential equations. The surprising fact is that this reduction of infinitedimensional LOIs to finite-dimensional LMIs is tight in the sense that the stability of the latter delay-differential equations is necessary for the stability of the PDEs in question. Notation and Preliminaries The notation used throughout is fairly standard. The superscript T stands for matrix transposition, R n denotes the n-dimensional 5-198/$ see front matter 8 Elsevier Ltd. All rights reserved. doi:1.116/j.automatica.8.6.6

2 E. Fridman, Y. Orlov / Automatica 45 (9) Euclidean space with the norm, R n m is the set of all n m real matrices. The notation P >, for P R n n means that P is symmetric and positive definite, whereas λ min (P) (λ max (P)) denotes its minimum (maximum) eigenvalue. Let H be a Hilbert space equipped with the inner product, and the corresponding norm. Denote by L(H) bounded linear operators from H to H. Given a linear operator P : H H with a dense domain D(P) H, the notation P stands for the adjoint operator. Such an operator P is strictly positive definite, i.e., P >, iff it is self-adjoint in the sense that P = P and there exists a constant β > such that x, Px β x, x and for all x D(P), whereas P means that P is self-adjoint and nonnegative definite, i.e., x, Px for all x D(P). If an infinitesimal operator A generates a strongly continuous semigroup T(t) on the Hilbert space H (see, e.g., Curtain and Zwart (1995) for details), the domain of the operator A forms another Hilbert space D(A) with the graph inner product (, ) D(A) defined as follows: (x, y) D(A) = x, y + Ax, Ay, x, y D(A). Moreover, the induced norm T(t) of the semigroup T(t) satisfies the inequality T(t) κe σ t everywhere with some constant κ > and growth bound σ. The space of the continuous H-valued functions x : a, b] H with the induced norm x C(a,b],H) = max s a,b] x(s) is denoted by C(a, b], H). The space of the continuously differentiable H- valued functions x : a, b] H with the induced norm x C 1 (a,b],h) = max( x C(a,b],H), ẋ C(a,b],H) ) is denoted by C 1 (a, b], H). L (a, b; H) is the Hilbert space of square integrable H-valued functions on (a, b) with the corresponding norm. W l, (a, b], R) is the Sobolev space of absolutely continuous scalar functions on a, b] with square integrable derivatives of the order l 1. Given x( ) L (a, b; H), we denote x t = x(t + θ) L ( h, ; H) for t a + h, b]. Lemma 1 (Wang (1994b) (Wirtinger s Inequality and its Generalization)). Let z W 1, (a, b], R) be a scalar function with z(a) = z(b) =. Then b a z (ξ)dξ (b a) π b a dz(ξ) dξ. (1) dξ If additionally z W, (a, b], R), then b a dz(ξ) dξ dξ (b a) π b a d z(ξ) dξ. () dξ Lemma (Jensen s Inequality). Let H be a Hilbert space with the inner product,. For any linear bounded operator R : H H, R >, scalar l > and x L (a, b], H) the following holds: l x(s), Rx(s) ds x(s)ds, R x(s)ds. (3) Note that (3) follows from the Cauchy Schwartz inequality l R 1 x(s), R 1 x(s) ds R 1 x(s)ds, R 1 x(s)ds.. Lyapunov method for exponential stability Consider a linear infinite-dimensional system ẋ(t) = Ax(t) + A 1 x(t τ(t)), t t (4) evolving in a Hilbert space H where x(t) H is the instantaneous state of the system. Let the following assumptions be satisfied: A1 the operator A generates a strongly continuous semigroup T(t) and the domain D(A) of the operator A is dense in H; A the linear operator A 1 is bounded in H; A3 the function τ(t) is piecewise-continuous of class C 1 on the closure of each continuity subinterval and it satisfies inf τ(t) >, t sup τ(t) h (5) t with some constant h > for all t t. Let the initial conditions x t = ϕ(θ), θ h, ], φ W (6) be given in the space W = C( h, ], D(A)) C 1 ( h, ], H). (7) Definition 1. A function x(t) C(t h, t + η], D(A)) is said to be a solution of the initial-value problem (4), (6) on t h, t + η] if x(t) is initialized with (6), it is absolutely continuous for t t, t + η], and it satisfies (4) for almost all t t, t + η]. The initial-value problem (4), (6) turns out to be well-posed on the semi-infinite time interval t, ) and its solutions can be found as mild solutions, i.e., as those of the integral equation x(t) = T(t t )x(t ) + t T(t s)a 1 x(s τ(s))ds, t t. (8) Lemma 3. Under A1 A3 there exists a unique solution of the initial value problem (4), (6) on t, ). This solution is also a unique solution of the integral initial value problem (6), (8). Proof. To begin with, let us choose a positive η small enough to ensure that η < inf t τ(t) and the first discontinuity point t 1 > t of τ(t) is such that the difference t 1 t is multiple to η, i.e., t 1 = t + k η for some integer k >. While being viewed over the time segment t, t + η ], the initial-value problem (4) is equivalent to ẋ(t) = Ax(t) + A 1 φ(t t τ(t)), x(t ) = φ() (9) where the inhomogeneous term A 1 φ(t t τ(t)) is of class C 1 on, η ]. By Theorem of Curtain and Zwart (1995), there exists a unique local solution of (9) and this solution satisfies the integral equation (8) on t, t + η ]. The same line of reasoning is step-by-step applied to the time segments t i 1, t i 1 + η ], i = 1,..., k with t i = t i 1 + η and t k = t 1. Following this line, the initial-value problem is demonstrated to possess a unique solution x(t, t, φ) for t t, t 1 ], which satisfies the integral equation (8) on t, t 1 ]. The assertion of Lemma 3 is then concluded by iteration on the time segments t j, tj+1 ], j = 1,,... where t 1 < t < are the successive discontinuity points of the function τ(t). Our aim is to derive exponential stability criteria for linear timedelay system (4). The stability concept under study is based on the initial data norm φ W = Aφ() + φ (1) C 1 ( h,],h) in space (7). Suppose x(t, t, φ) denotes a solution of (4), (6) at a time instant t t. Definition. System (4) is said to be exponentially stable with a decay rate δ > if there exists a constant K 1 such that the following exponential estimate holds: x(t, t, φ) K e δ(t t ) φ W t t. (11)

3 196 E. Fridman, Y. Orlov / Automatica 45 (9) Consider Lyapunov Krasovskii Functionals (LKFs), which depend on x and ẋ (Kolmanovskii & Myshkis, 1999). Given a continuous functional V : R W C( h, ], H) R, its upper right-hand derivative along solutions x t (t, φ), t t of (4), (6) is defined as follows: 1 V(t, φ, φ) = lim sup s + s V(t + s, xt+s (t, φ), ẋ t+s (t, φ)) V(t, φ, φ)]. Lemma 4. Let A1 A3 be in force and let there exist positive numbers δ, β, γ and a continuous functional V : R W C( h, ], H) R such that the function V(t) = V(t, xt, ẋ t ) is absolutely continuous for x t, satisfying (4) and β φ() V(t, φ, φ) γ φ W, (1) V(t, φ, φ) + δv(t, φ, φ). (13) Then (4) is exponentially stable with the decay rate δ and (11) holds with K = γ β. Proof. As in the case of ODE, from (13) with φ = x t we obtain d dt V(t, xt, ẋ t ) + δv(t, x t, ẋ t ), where V(t, x t, ẋ t ) = V(t, φ, φ). Hence, by the comparison principle argument (Khalil, 199), it follows that β x(t) V(t, x t, ẋ t ) V(t, φ, φ)e δ(t t ) γ e δ(t t ) φ W. 3. Exponential stability in a Hilbert space In this section, the delay is assumed to be either slowly-varying with τ d < 1, or fast-varying (with no restrictions on the delayderivative). Let A1 A3 be in force. We derive delay-dependent conditions by using a simple (as defined in Gu et al. (3)) LKF: e δ(s t) x(s), Sx(s) ds + h e δ(s t) ẋ(s), Rẋ(s) dsdθ h t+θ V(t, x t, ẋ t ) = x(t), Px(t) + + e δ(s t) x(s), Qx(s) ds (14) where P : D(A) H is a linear positive definite operator and R, Q, S L(H) are non negative definite operators, satisfying the following inequalities: β x, x x, Px γ P x, x + Ax, Ax ], x, Qx γ Q x, x, x, Rx γ R x, x, x, Sx γ S x, x, x D(A) (15) for some positive constants β, γ P, γ Q, γ S, γ R. Thus condition (1) of Lemma 4 is satisfied. We note that the first inequality (15) allows one to use unbounded operators P which are upper estimated by the unbounded operator A according to (15). In the case of ODE, where A is a matrix, the above upper bound is equivalent to the standard one with A =. For ODE with delay the Lyapunov functional of the form (14) was recently introduced in He, Wang, Lin, and Wu (7) (for δ = ), whereas this functional with S = was introduced earlier in Fridman and Shaked (3) (for δ = ) and in Sun, Zhao, and Hill (6) (for δ > ). Being viewed on solutions of (4), the LKF (14) is absolutely continuous as a function of t because the solutions are absolutely continuous in t. Differentiating V, we find V(t, x t, ẋ t ) + δv(t, x t, ẋ t ) x(t), Pẋ(t) + δ x(t), Px(t) + h ẋ(t), Rẋ(t) he δh ẋ(s), Rẋ(s) ds + x(t), (Q + S)x(t) (1 τ(t)) x(t τ(t)), Qx(t τ(t)) e δh x(t h), Sx(t h) e δh. Following He et al. (7), we employ the representation τ(t) h ẋ(s), Rẋ(s) ds = h ẋ(s), Rẋ(s) ds h ẋ(s), Rẋ(s) ds (16) (17) and apply the Jensen s inequality (3) Rẋ(s) ds ẋ(s), 1 ẋ(s)ds, R ẋ(s)ds, h τ(t) ẋ(s), Rẋ(s) ds 1 τ(t) τ(t) (18) ẋ(s)ds, R ẋ(s)ds. h Then, taking into account that τ d < 1 and following Gouaisbaut and Peaucelle (6), we obtain V(t, x t, ẋ t ) + δv(t, x t, ẋ t ) x(t), Pẋ(t) + δ x(t), Px(t) + h ẋ(t), Rẋ(t) x(t) x(t τ(t)), R(x(t) x(t τ(t))) + x(t τ(t)) x(t h), R(x(t τ(t)) x(t h)) + (1 d) x(t τ(t)), Qx(t τ(t)) ] e δh + x(t), (Q + S)x(t) x(t h), Sx(t h) e δh. (19) We will derive stability conditions in two forms. The first form will subsequently be applied to the wave equation and the second one to the heat equation. The first form is derived by substituting the right-hand side of (4) for ẋ(t). Setting η(t) = col{x(t), x(t h), x(t τ(t))}, we find that the condition (13) of Lemma 4 V(t, x t, ẋ t ) + δv(t, x t, ẋ t ) η(t), Φ h η(t) () is satisfied if the following LOI Φ11 PA 1 A RA A ] RA1 Φ h = + h A 1 P A 1 RA A 1 RA 1 R R e δh (S + R) R, R R R + (1 d)q holds provided that (1) Φ 11 = A P + PA + δp + Q + S. () The resulting inequality (1) is convex with respect to h: given h >, it becomes feasible for all h, h ] whenever it is feasible for h. The convexity follows from the fact that Φ h Φ h since h and e δh multiply the non negative definite operators. Summarizing, the following result is obtained: Theorem 1. Let A1 A3 be in force. Given δ >, let there exist linear operators P > and R, S, Q subject to (15) such that the LOI (1) with notation () holds in the Hilbert space D(A) D(A) D(A). Then system (4) is exponentially stable with

4 E. Fridman, Y. Orlov / Automatica 45 (9) the decay rate δ for all differentiable delays with τ(t) d < 1. The inequality (11) is satisfied with K = max{γ P, h(γ Q + γ S + h γ R /)}/β. Moreover, (4) is exponentially stable for all fast-varying delays τ h if the LOI (1) is feasible with Q =. The conditions of Theorem 1 are delay-dependent, namely h- dependent, even for δ. Taking in the above derivations S = R = we obtain the following quasi delay-independent conditions, which become delay-independent for δ and coincide in the case of ODE with the result of Mondie and Kharitonov (5): Corollary 1. Let A1 A3 be in force. Given δ >, system (4) is exponentially stable with the decay rate δ for all differentiable delays with τ(t) d < 1 if there exist linear operators P > and Q subject to (15) such that the LOI (A + δ) P + P(A + δ) + Q PA1 A 1 P (1 d)q (3) e δh holds in the Hilbert space D(A) D(A). The inequality (11) is satisfied with K = max{γ P, hγ Q }/β. Remark 1. Differently from the finite dimensional case, the feasibility of the strict LOI (1) and (3) for h = (δ = ) does not necessarily imply the feasibility of (1) and (3) for small enough h (δ) because h (δ) is multiplied by the operator, which may be unbounded. It may be difficult to verify the feasibility of (1), if the operator that multiplies h (and depends on A) in Φ h is unbounded. To avoid this, we will derive the second form of LOI by the descriptor method (Fridman, 1), where the right-hand sides of the expressions = x(t), P Ax(t) + A 1x(t τ(t)) ẋ(t)], = ẋ(t), P 3 Ax(t) + A 1x(t τ(t)) ẋ(t)] (4) with some P, P 3 L(H) are added into the right-hand side of (19). Setting η d (t) = col{x(t), ẋ(t), x(t h), x(t τ(t))}, we obtain that V(t, x t, ẋ t ) + δv(t, x t, ẋ t ) η d (t), Φ d η d (t), if the LOI Φ d11 Φ d1 P A 1 + Re δh Φ d = Φ d P 3 A 1 (S + R)e δh Re δh R + (1 d)q ]e δh (5) holds, where Φ d11 = A P + P A + δp + Q + S Re δh, Φ d1 = P P + A P 3, Φ d = P 3 P 3 + h R (6) and denotes the symmetric terms of the operator matrix. Thus, the following result is obtained. Theorem. Let A1 A3 be in force. Given δ >, let there exist linear operators P > and R, S, Q subject to (15) and indefinite operators P, P 3 L(H) such that the LOI (5) with notations given in (6) holds in the Hilbert space D(A) D(A) D(A) D(A). Then system (4) is exponentially stable with the decay rate δ for all differentiable delays (5) with τ d < 1. The inequality (11) is satisfied with K = max{γ P, h(γ Q + γ S + h γ R /)}/β. Moreover, (4) is exponentially stable for all fast-varying delays τ h if the LOI (5) is feasible with Q =. Differently from the LOI (1), the feasibility of the strict LOI (5) for h = implies the feasibility of (5) for small enough h (h is multiplied by the bounded operator R). Remark. Consider now the system (4) with A and A 1 from the uncertain time-invariant polytope Ω = M M f j Ω j for some f j 1, f j = 1, (7) j=1 where Ω j = A (j) A (j) ] (j) 1, A 1 L(H) and the operators A (j) have a common domain, which is dense in H and A = M j=1 f ja (j) generates a strongly continuous semigroup for all f j, satisfying (7). Applying conditions of Theorem to the uncertain system, we conclude that (4) is exponentially stable if LOI (5) is feasible. Since LOI (5) is affine in A and A 1, by the same arguments as for LMIs (see Boyd et al. (1994)) we conclude that (5) is feasible if the LOIs (5) in M vertices are feasible for the same P, P 3 and for different Q (j), S (j) >, R (j) >, P (j) >. j=1 4. Stability of the delay heat equation Consider the heat equation z t (ξ, t) = az ξξ (ξ, t) a z(ξ, t) a 1 z(ξ, t τ(t)), (8) where t t, ξ l, with the constant parameters a >, a and a 1, with the time-varying delay τ(t), satisfying (5), and with the Dirichlet boundary condition z(, t) = z(l, t) =, t t. (9) The boundary-value problem (8) and (9) describes the propagation of heat in a homogeneous one-dimensional rod with a fixed temperature at the ends in the case of the delayed (possibly, due to actuation) heat exchange with the surroundings. Here a and a i, i =, 1 stand for the heat conduction coefficient and for the coefficients of the heat exchange with the surroundings, respectively, z(ξ, t) is the value of the temperature field of the plant at time moment t and location ξ along the rod. In the sequel, the state dependence on time t and spatial variable ξ is suppressed whenever possible. The boundary-value problem (8) and (9) can be rewritten as the differential equation (4) in the Hilbert space H = L (, l) with the infinitesimal operator A = a a ξ with the dense domain ( ) D = {z W, (, l], R) : z() = z(l) = }, (3) ξ and with the bounded operator A 1 = a 1 of the multiplication by the constant a 1. The infinitesimal operator A generates an exponentially stable semigroup (see, e.g., Curtain and Zwart (1995) for details). We will first derive simple delay-independent conditions, based on LOI (3). Consider the LKF of the form V = p z (ξ, t)dξ + q e δ(s t) z (ξ, s)dξ ds (31) with some positive constants p and q. Then the operators P and Q in (3) take the form P = p, Q = q of the bounded operators of the multiplication by positive constants p and q, respectively. Integrating by parts and taking into account (9), we find that for x D(A) x, (A P + PA)x = a = pzz ξξ dξ a pz dξ ( ) π (apzξ + a pz )dξ l a + a pz dξ (3)

5 198 E. Fridman, Y. Orlov / Automatica 45 (9) where the last inequality follows from the Wirtinger s inequality (1). We thus obtain that (3) is satisfied if ( ) π Ψ δ = q l a + a δ p a 1 p <. (33) a 1 p (1 d)qe δh From Ψ < it follows that Ψ δ < for small enough δ, since Ψ δ = Ψ + diag{δp, (1 d)q(1 e δh )}. We proved Theorem 3. Given δ >, let the LMI (33) holds for some scalars p > and q >. Then the Dirichlet boundary-value problem (8), (9) is exponentially stable for all differentiable delays (5) with τ(t) d < 1, and the inequality z (ξ, t)dξ K e δ(t t ) max s t h,t ] z (ξ, s)dξ (34) is satisfied for all t t with K = 1 + hq/p. If (33) holds for δ =, then the inequality (34) is satisfied with K = 1 + hq/p and small enough δ. Remark 3. By Schur complements formula, LMI (33) with δ = is feasible iff for some p > and q > the following holds: q ( π a + a l )pq + a 1 p /(1 d) <. The left part of the latter inequality achieves its minimum at q = ( π a + a l )p and, thus, the inequality holds iff ( ) π π l a + a >, a < 1 l a + a (1 d). (35) To derive delay-dependent conditions, we apply Theorem (it is difficult to find operators that satisfy Theorem 1 since h is multiplied by the unbounded operator in (1)). For simplicity we consider l = π. We choose V of the form V(t, z t, z t s ) = (p 1 p 3 a) + + s r h t+θ z (ξ, t)dξ + p 3 a e δ(s t) z s (ξ, s)dsdθ e δ(s t) z (ξ, s)ds + q zξ (ξ, t)dξ ] e δ(s t) z (ξ, s)ds dξ with some constants p 1 >, p 3 >, s >, r > and q. Then the operators in (14) take the form P = p 3 (a + a) + p ξ 1, R = r, Q = q, S = s. We choose P = p and P 3 = p 3, where p > and p δp 3. (36) Here P is unbounded operator and all the others are bounded operators in L (, π). We note that the above choice of P, depending on the slack variable P 3, is different to that of the ODEs (where these matrices are independent). Thus, for the first time, the slack variable allows one to construct an appropriate LKF. Integrating by parts and utilizing the Wirtinger s inequality (1), we find that for x D(A) x, Px = p3 az ξξ z p 3 az + p 1 z ] dξ = ap3 zξ z ] + p 1 z ] dξ p 1 z dξ >. Moreover, (15) is satisfied, since by the generalized Wirtinger s inequality () the following holds x, Px ap 3 (z ξξ ) + (ap 3 + p 1 )z ]dξ γ P Ax + x ] for some γ P >. We obtain that ẋ, (P P + A P 3 )x = ẋ, (p 1 p (a + a )p 3 )x ; x, A P x + x, P Ax + δ x, Px = a(p δp 3 ) = a(p δp 3 ) (a + a )p + δp 1 ] z ξξ zdξ + δ(p 1 ap 3 ) a p ] z ξ dξ + δ(p 1 ap 3 ) a p ] z dξ, z dξ z dξ where the latter inequality follows from (36) and the Wirtinger s inequality (1). Therefore, (5) holds if φ 11 φ 1 φ 14 p 3 + h r p 3 a 1 (s + r)e δh re δh <, φ 44 (37) φ 11 = (a + a )p + δp 1 + q + s re δh, φ 1 = p 1 p (a + a )p 3, φ 14 = p a 1 + re δh, φ 44 = r + (1 d)q]e δh. Summarizing the following result is obtained Theorem 4. Given δ >, let there exist scalars p 1 >, p >, p 3 >, s >, r > and q such that LMIs (36) and (37) hold. Then the boundary-value problem (8) and (9), where l = π, is exponentially stable with the decay rate δ for all differentiable delays (5) with τ d < 1 and the inequality p 1 z (ξ, t)dξ e δ(t t ) {ap 3 zξ (ξ, t )dξ + max{p 1 p 3 a + hq + hs, h 3 r/} } max z (ξ, s) + z (ξ, s)]dξ t s t h,t ] (38) is satisfied for all t t. Moreover, (8), (9) is exponentially stable with the decay rate δ for all fast varying delays (5) with no restrictions on τ if (36), (37) are feasible with q =. If (37) holds for δ =, then (8), (9) is exponentially stable with a sufficiently small decay rate. Remark 4. The same LMIs (33) (with l = π) and (37) guarantee the exponentially stability of the scalar ODE ẏ(t) + (a + a )y(t) + a 1 y(t τ(t)) =. (39) System (39) corresponds to the first modal dynamics (with k = 1) in the modal representation of the Dirichlet boundary-value problem (8), (9) with l = π ẏ k (t) + (ak + a )y k (t) + a 1 y k (t τ(t)) =, k = 1,,... (4) projected on the eigenfunctions of the operator (this operator ξ has eigenvalues k, see e.g. Wu (1996)). The stability of (8), (9) implies the stability of (4). Thus the reduction of infinitedimensional LOI of Corollary 1 (Theorem ) to finite-dimensional LMI of Theorem 3 (Theorem 4) is tight, since the stability of (39) is necessary for the stability of (8), (9). Remark 5. Consider now the Dirichlet boundary-value problem (8), (9) with the uncertain coefficients from the uncertain timeinvariant polytope Ω given by (7) with Ω j = a (j) a (j) a (j) ] 1. Here M = k and k is the number of uncertain parameters and it may take values from the finite set {1,, 3}. The uncertain infinitesimal operator A = M j=1 f (j) ja a (j) ξ with the

6 E. Fridman, Y. Orlov / Automatica 45 (9) dense domain (3) generates strongly continuous semigroup, whereas the uncertain operator A 1 = M j=1 f ja (j) is bounded. By applying Remark and Theorem 4, we conclude that (8), (9) is exponentially stable if (36) holds and LMIs (37) in the vertices are feasible for the same p, p 3 and for different q (j), s (j) >, r (j) >, p (j) >, j = 1,..., M. By Theorem 3, (8), (9) is exponentially stable if LMIs (33) in the vertices are feasible for the same p > and for different q (j), j = 1,..., M. Example 1. Consider the controlled heat equation z t (ξ, t) = z ξξ (ξ, t) + rz(ξ, t) + u, z(, t) = z(l, t) =, (41) where ξ (, l), t > and where r is uncertain parameter satisfying r β with given β. It was shown in Rebiai and Zinober (1993) that for l = 1 the state-feedback u = γ z(ξ, t) with γ > ( β π ) exponentially stabilizes (41). By verifying LMI (33), we conclude that the closed-loop system is exponentially stable if there exists p > such that (π r + γ )p < for all r β, i.e. if γ > β π. Since β π ( β π ), our method guarantees exponential stabilization of (41) via a lower gain, which becomes essentially lower for big β. Noting that a time-delay often appears in the feedback, we consider next the case of l = π, β = 1.9 and the delayed feedback u = z(ξ, t τ(t)) with uncertain delay satisfying A3. This is a polytopic system reached by choosing r = ±1.9. Applying Theorem 4 (with δ = ) and Remark 5 to the resulting closedloop system, we verify the feasibility of LMI (37) in the two vertices corresponding to r = ±1.9. We use LMI toolbox of Matlab and let d to be and unknown, respectively. We find the maximum values of h for which the system remains exponentially stable: d =, h = 1.5; unknown d, h = 1.1. As shown before, the latter results correspond also to the stability of ẏ = ( 1 + r)y(t) y(t τ(t)), with r Stability of the delay wave equation Consider the wave equation z tt (ξ, t) = az ξξ µ z t (ξ, t) µ 1 z t (ξ, t τ(t)) a z(ξ, t) a 1 z(ξ, t τ(t)), t t, ξ π (4) with the Dirichlet boundary condition (9), where l = π and with the constant parameters a >, µ >, µ 1, a, and a 1, with the time-varying delay τ(t), satisfying (5). The boundary-value problem (9), (4) describes the oscillations of a homogeneous string with fixed ends in the case of the delayed (possibly, due to actuation) stiffness restoration and dissipation. Here a stands for the elasticity coefficient, µ and µ 1 stand for the dissipation coefficients, and a, a 1 stand for the restoring stiffness coefficients, the state vector x = col{z, z t } consists of the deflection z(ξ, t) of the string and its velocity z t (ξ, t) at time moment t and location ξ along the string. Let us introduce the operators 1 A = a ξ a, A µ 1 = (43) a 1 µ 1 where A 1 is a bounded operator of multiplication by the constant matrix and where the domain D( ξ ) of the double differentiation operator is determined by (3). Then the boundary-value problem (9), (4) can be represented as the differential equation (4) in the Hilbert space H = L (, π) L (, π) with the infinitesimal operator A, possessing the domain D(A) = D( ξ ) L (, π) and generating an exponentially stable semigroup (see, e.g., Curtain and Zwart (1995) for details). We first derive quasi delay-independent conditions by choosing V in the form V = ap 3 zξ (ξ, t)dξ + v T (ξ, t)p v(ξ, t)dξ P = + v T (ξ, s)e δ(s t) Q v(ξ, s)dξ ds, (44) p1 p, P p p w = P + diag{ap 3, } >, Q, 3 where v T (ξ, t) = z(ξ, t) z t (ξ, t)]. Then the operators P (unbounded) and Q (bounded) in (3) are given by P = diag{ ap 3, } + P ξ, Q. Now, integrating by parts and taking into account the inequality p 3 > (extracted from P w > ) and Wirtinger s inequality (1) we obtain that x, Px = = a ap3 z ξξ z + z T P z ] dξ p 3 z ξ dξ + x, P x x, P w x λ min (P w ) x > (45) for all x D(A) L (, π). Moreover, by the generalized Wirtinger s inequality () the following holds x, Px ap 3 (z ξξ ) dξ + x, P x γ P ( Ax + x ) (46) with some constant γ P > and thus (15) is satisfied. Finally, integration by parts and application of Wirtinger s inequality (1) under p δp 3 yield x, P(A + δ)x + x, (A + δ)px = z z t ] p δ 1 1 ap 3 p ξ a p p 3 ξ a µ + δ + δ a ξ a p 1 ap 3 p ξ z dξ z 1 µ + δ t p p 3 = a(p p 3 δ) (z ξ ) dξ z z t ] ] p a + p 1 δ p 1 (µ δ)p p 3 a p 1 (µ δ)p p 3 a p (µ δ)p 3 z z dξ z z z t ](P w C δ + Cδ T P w) dξ, (47) t z t where ] δ 1 C δ =. (48) a a µ + δ Therefore (3) is feasible if the following LMI Ω C T δ = δ P w + P w C δ + Q P w A 1 A T 1 P w (1 d)e δh (49) Q is feasible. Since the LMI (49) ensures that the (1,1)-term of its lefthand side satisfies p (a + a) + (p 1 + ap 3 )δ + q 1, whereas q 1 + δ(p 1 + ap 3 ), it follows that p. From Ω < it follows that p > and, thus, p δp 3 for small enough δ. Moreover, Ω < yields Ω δ < for small enough δ, since Ω δ = Ω + diag{δp w, (1 d)q (1 e δh )}. Summarizing, we arrive at the following.

7 E. Fridman, Y. Orlov / Automatica 45 (9) Theorem 5. Given δ >, let the LMIs p δp 3 and (49) hold for some symmetric -matrices P w > and Q, where p and p 3 are respectively (1, ) and (, ) terms of P w. Then the Dirichlet boundary-value problem (4), (9), where l = π, is exponentially stable with the decay rate δ for all differentiable delays (5) with τ(t) d < 1, and the inequality λ min (P w ) z (ξ, t) + z t (ξ, t)]dξ { e δ(t t ) K w max z (ξ, s) + z t (ξ, s)]dξ s t h,t ] } + ap 3 zξ (ξ, t )dξ is satisfied with K w = λ max (P w diag{ap 3, }) + hλ max (Q ) (5) for all t t. If Ω < holds for δ =, then (4), (9) is exponentially stable with a sufficiently small decay rate. Corollary. In a particular case where a = 1 and a = a 1 =, the Dirichlet boundary-value problem (9), (4), where l = π, is exponentially stable for all differentiable delays with τ d < 1 if µ 1 < (1 d)µ. Proof. Since the delay appears only in z t, we choose the ] decision variables of LMI (49) in the form P w = 1 δ δ, Q = 1 µ ], and thus P w A 1 = µ1 δ µ ]. Deleting from 1 Ω δ the column and the row, consisting of zero elements, we obtain the matrix δ 4δ µ δ µ 1 δ µ + 6δ µ 1. (1 d)e δh µ Applying the Schur complements formula to the last column and the last row of this matrix, we conclude that (49) is feasible if the following LMI holds δ + O(δ ) O(δ) µ + O(δ) µ 1 e δh + 6δ (1 d)µ <, (51) where O(δ k ) cδ k (k = 1, ) for some constant c > and for all sufficiently small δ. If µ < 1 (1 d)µ, then the (,)-term of the left-hand side of (51) is negative for small enough δ. Finally, applying the Schur complements formula to the second column and the second row of the matrix in (51), we obtain the expression of the form δ + O(δ ), which is negative for small δ. Therefore, (51) and, thus (49) are feasible for small δ. Remark 6. The condition µ 1 < µ for the stability of the wave equation with constant delay and a = 1, a = a 1 = and with mixed Dirichlet Neumann boundary condition was obtained by Nicaise and Pignotti (6), where it was shown that if µ 1 µ, there exists a sequence of arbitrary small delays that destabilize the system. We will further derive delay-dependent stability conditions for (4), (9) with µ 1 =. We apply the conditions of Theorem 1. We choose V as follows: V = ap 3 zξ (ξ, t)dξ + z(ξ, t) z t (ξ, t)]p z(ξ, t) dξ + hr z z t (ξ, t) (ξ, t s)eδ(s t) dsdθ h t+θ ] + s z (ξ, s)e δ(s t) ds + q t τ z (ξ, s)e δ(s t) ds dξ, where P and p 3 satisfy (44) and where r >, s >, q. Then the operators P, Q, R in (1) are given by } P = diag { ap 3 ξ, + P >, Q = diag{q, }, R = diag{r, }, S = diag{s, }. We have h A RA = diag{, h r}, h A RA 1 =, h A 1 RA 1 =. From (47) it follows that (1) is feasible if p δp 3 and the following LMI are satisfied: ] re φ w P w + δh a 1 (s + r)e δh re δh (r + (1 d)q)e δh <, (5) where φ w = C T δ P w +P w C δ +diag{q+s re δh, h r}. Summarizing the following result is obtained Theorem 6. Given δ >, let there exist a -matrix P w > and scalars q, r >, s > such that satisfy LMIs p δp 3 and (5), where p and p 3 are respectively (1, ) and (, ) terms of P w. Then the wave time-delay equation (4) with µ 1 = and with the Dirichlet boundary condition (9), where l = π, is exponentially stable with the decay rate δ for all differentiable delays (5) with τ d < 1, and the inequality (5) is satisfied with K w = λ max (P w diag{ap 3, }) + max{hq + hs, h 3 r/} for all t t. Moreover, if LMIs (36) and (5) are feasible with q =, then (9), (4) with µ 1 = is exponentially stable with the decay rate δ for all fast varying delays τ h. If the LMI (5) holds for δ =, then (9), (4) with µ 1 = is exponentially stable with a sufficiently small decay rate. Remark 7. The same LMIs (49) and (5) appear to guarantee the exponential stability of ODE with delay z(t) = C z(t) + A 1 z(t τ(t)), z(t) R or, equivalently, of the first modal dynamics (with k = 1) of the modal representation of the Dirichlet boundary-value problem (9), (4) with l = π ÿ k (t) + µ ẏ k (t) + µ 1 ẏ k (t τ(t)) + (ak + a )y k (t) + a 1 y k (t τ(t)) =, k = 1,,... on the eigenfunctions of the operator Theorem 5 and of Theorem 6 are tight. ξ. Hence, the results of Remark 8. One can derive mixed stability conditions for the wave equation (4) with µ 1 : delay-dependent (with respect to delay in z)/ delay-independent (with respect to delay in z t ). This is similar to neutral systems, where the delay in the state derivative is treated in the delay-independent manner (Niculescu, 1). Example. Consider the controlled wave equation z tt (ξ, t) =.1z ξξ (ξ, t) z t (ξ, t) + u, (53) with boundary condition (9) and l = π, t t, ξ π, τ h, τ d < 1. Applying Theorem 6 to the openloop system we find that (53) with u = is exponentially stable with the decay rate δ =.5. Considering next a delayed feedback u = z(ξ, t τ(t)) and verifying conditions of Theorem 6, we find that the closed-loop system is exponentially stable with a greater decay rate δ =.8 for all τ(t).31.

8 E. Fridman, Y. Orlov / Automatica 45 (9) Conclusions A general framework is given for the stability analysis of linear time-delay systems in a Hilbert space with a bounded operator acting on the delayed state. The exponential stability conditions are derived in terms of linear operator inequalities in the Hilbert space. In the case of a heat/wave scalar equation with the Dirichlet boundary conditions, these LOIs are reduced to finite-dimensional LMIs by applying new Lyapunov Krasovskii functionals. The reduced-order LMIs coincide with the stability conditions for appropriate ODEs with delay, whereas the stability of the latter ODEs are necessary for the stability of the original boundary value problems. LOIs are expected to provide effective tools for robust control of distributed parameter systems. References Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequality in systems and control theory. SIAM Frontier Series. Curtain, R., & Zwart, H. (1995). An introduction to infinite-dimensional linear systems. Springer-Verlag. Datko, R. (1988). Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM Journal on Control and Optimization, 6, Fridman, E. (1). New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems. Systems & Control Letters, 43, Fridman, E., & Shaked, U. (3). Delay-dependent stability and H control: Constant and time-varying delays. International Journal on Control, 76(1), Gouaisbaut, F., & Peaucelle, D. (6). Delay-dependent stability of time delay systems. In: Proc. of ROCOND 6. Gu, K., Kharitonov, V., & Chen, J. (3). Stability of time-delay systems. Boston: Birkhauser. He, Y., Wang, Q.-G., Lin, C., & Wu, M. (7). Delay-range-dependent stability for systems with time-varying delay. Automatica, 43(), Ikeda, K., Azuma, T., & Uchida, K. (1). 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In Lecture notes in control and information sciences: Vol. 69. London: Springer- Verlag. Rebiai, S. E., & Zinober, A. S. (1993). Stabilization of uncertain distributed parameter systems. International Journal on Control, 57(5), Richard, J.-P. (3). Time-delay systems: An overview of some recent advances and open problems. Automatica, 39, Sun, X.-M., Zhao, J., & Hill, D. (6). Stability and L -gain analysis for switched delay systems. Automatica, 4, Wang, T. (1994a). Stability in abstract functional-differential equations. I. General theorems. Journal of Mathematical Analysis and Applications, 186(), Wang, T. (1994b). Stability in abstract functional-differential equations. II. Applications. Journal of Mathematical Analysis and Applications, 186, Wu, J. (1996). Theory and applications of partial functional differential equations. New York: Springer-Verlag. Emilia Fridman received the M.Sc. degree from Kuibyshev State University, USSR, in 1981 and the Ph.D. degree from Voroneg State University, USSR, in 1986, all in mathematics. From 1986 to 199 she was an Assistant and Associate Professor in the Department of Mathematics at Kuibyshev Institute of Railway Engineers, USSR. Since 1993 she has been at Tel Aviv University, where she is currently Professor of Electrical Engineering-Systems. Her research interests include time-delay systems, H control, singular perturbations, nonlinear control and asymptotic methods. She has published over 7 articles in international scientific journals. Currently she serves as Associate Editor in Automatica and IMA Journal of Mathematical Control & Information. Yury Orlov received his M.S. degree from Mechanical- Mathematical Faculty of Moscow State University, in 1979, the Ph.D. degree in physics and mathematics from the Institute of Control Science, Moscow, in 1984, and the Dr.Sc. degree also in physics and mathematics from Moscow Aviation Institute, in 199. He is a Professor at the Mexican Scientific Research Center CICESE in Ensenada, since He has been with the Institute of Control Sciences of Russian Academy of Sciences since He was also a part-time Professor at Moscow Aviation Institute from 1989 to 199. His research interests include mathematical methods in control, analysis and synthesis of nonlinear, nonsmooth, discontinuous, time-delay, distributed parameter systems, and applications to electromechanical systems.