INTERIOR FEEDBACK STABILIZATION OF WAVE EQUATIONS WITH TIME DEPENDENT DELAY

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1 Electronic Journal of Differential Equations, Vol. 11 (11), No. 41, pp. 1. ISSN: URL: or ftp ejde.math.txstate.edu INTERIOR FEEDBACK STABILIZATION OF WAVE EQUATIONS WITH TIME DEPENDENT DELAY SERGE NICAISE, CRISTINA PIGNOTTI Abstract. We study the stabilization problem by interior damping of the wave equation with boundary or internal time-varying delay feedback in a bounded and smooth domain. By introducing suitable Lyapunov functionals exponential stability estimates are obtained if the delay effect is appropriately compensated by the internal damping. 1. Introduction Let R n be an open bounded set with boundary Γ of class C. We assume that Γ is divided into two parts Γ and ; i.e., Γ = Γ, with Γ = and meas Γ. We consider the problem u tt (x, t) u(x, t) a u t (x, t) = in (, + ), (1.1) u(x, t) = on Γ (, + ), (1.) µu tt (x, t) = (u + au t) (x, t) ku t (x, t ) on (, + ), (1.3) u(x, ) = u (x), u t (x, ) = u 1 (x) in, (1.4) u t (x, t) = f (x, t) in ( τ(), ), (1.5) where ν(x) denotes the outer unit normal vector to the point x Γ and u is the normal derivative. Moreover, τ = is the time delay, µ, a, k are real numbers, with µ, a >, and the initial datum (u, u 1, f ) belongs to a suitable space. It is well-known that the above model is exponentially stable in absence of delay; that is, if. We refer to [, 18, 17, 19, 14, 16, 15, 3] for the more studied case a =, µ = and to [11,, 7, 8, 9] in the case a, µ >. In presence of a constant delay, when µ = and the condition (1.3) is substituted by u (x, t) = ku t(x, t τ), (, + ), the system becomes unstable for arbitrarily small delays (see [5]). Mathematics Subject Classification. 35L5, 93D15. Key words and phrases. Wave equation; delay feedback; stabilization. c 11 Texas State University - San Marcos. Submitted July 6, 1. Published March 17, 11. 1

2 S. NICAISE, C. PIGNOTTI EJDE-11/41 Then, the 1-d version of the above model with µ = in the boundary condition (1.3) has been considered by Morgül [] who proposed a class of dynamic boundary controllers to solve the stability robustness problem. In the case µ >, (1.3) is a so-called dynamic boundary condition. Dynamic boundary conditions arise in many physical applications, in particular they occur in elastic models. For instance, these conditions appear in modelling dynamic vibrations of linear viscoelastic rods and beams which have attached tip masses at their free ends. See [1, 3, 1, 9] and the references therein for more details. The above model without delay (e.g. τ = ) has been proposed in one dimension by Pellicer and Sòla-Morales [8] as an alternative model for the classical springmass damper system, the case k = being treated in [11]. In both cases, no rates of convergence are proved. In dimension higher than 1, we refer to Gerbi and Said-Houari [9] where a nonlinear boundary feedback is even considered and the exponential growth of the energy is proved if the initial data are large enough. A different problem with a dynamic boundary condition (without delay), motivated by the study of flows of gas in a channel with porous walls, is analyzed in [7] where exponential decay is proved. On the function τ we assume that there exist positive constants τ, τ such that Moreover, we assume < τ τ, t >. (1.6) τ W, ([, T ]), T >, (1.7) τ (t) d < 1 t >. (1.8) Under the above assumptions on the time-delay function we will prove that an exponential stability result holds under a suitable assumption between the coefficients a and k (namely condition (.56) below). We consider also the problem with interior delay u tt (x, t) u(x, t) + a u t (x, t) + a 1 u t (x, t ) = in (, + ) (1.9) u(x, t) = on Γ (, + ) (1.1) u(x, ) = u (x), u t (x, ) = u 1 (x) in (1.11) u t (x, t) = g (x, t) in ( τ(), ), (1.1) where > is the time-varying delay, a and a 1 are real numbers with a >, and the initial datum (u, u 1, g ) belongs to a suitable space. The above model, with a >, a 1 > and a constant delay τ has been studied by the authors [3] in the case of mixed homogeneous Dirichlet-Neumann boundary conditions. Assuming that a 1 < a (1.13) a stabilization result is given, by using a suitable observability estimate. This is done by applying inequalities obtained from Carleman estimates for the wave equation by Lasiecka, Triggiani and Yao in [] and by using compactness-uniqueness arguments. Instability phenomena when (1.13) is not satisfied are also illustrated. We refer to [6, 4] for instability examples of related problems in one dimension. The analogous problem with boundary feedback has been introduced and studied by Xu, Yung, Li [9] in one-space dimension using a fine spectral analysis and in higher space dimension by the authors [3].

3 EJDE-11/41 INTERIOR FEEDBACK STABILIZATION 3 The case of time-varying delay has been already studied in [6] in one space dimension and in general dimension, with a possibly degenerate delay, in [5]. Both these papers deal with boundary feedback. See also [8] for abstract problems also under the assumption of non-degeneracy of. Here, we will give an exponential stability result for problem (1.9)-(1.1) under the condition a 1 < 1 d a, (1.14) where d is the constant in (1.8). The outline of the paper is the following. In section we study well-posedness and exponential stability of the problem (1.1) (1.5) with structural damping and boundary delay in both cases µ > and µ =. In section 3 we analyze the problem with internal delay feedback (1.9) (1.1).. Boundary delay feedback In this section we concentrate on the problem with boundary delay (1.1) (1.5). Let C P be a Poincaré s type constant defined as the smallest positive constant such that v dγ C P v dx, v H (), (.1) where, as usual, H () = {u H 1 () : u = on Γ }. First of all we will give a well-posedness result under the assumption k a 1 d, (.) C P where d is the positive constant of assumption (1.8). We have to distinguish the two cases µ > and µ =..1. Well-posedness in the case of dynamic boundary condition. First we study the well-posedness of (1.1) (1.5) for µ >. We introduce the auxiliary unknown z(x, ρ, t) = u t (x, t ρ), x, ρ (, 1), t >. (.3) Then, problem (1.1) (1.5) is equivalent to u tt (x, t) u(x, t) a u t (x, t) = in (, + ), (.4) z t (x, ρ, t) + (1 τ (t)ρ)z ρ (x, ρ, t) = in (, 1) (, + ), (.5) u(x, t) = on Γ (, + ), (.6) µu tt (x, t) = (u + au t) (x, t) kz(x, 1, t) on (, + ), (.7) z(x,, t) = u t (x, t) on (, ), (.8) u(x, ) = u (x) and u t (x, ) = u 1 (x) in, (.9) z(x, ρ, ) = f (x, ρτ()) in (, 1). (.1) Let us denote U := (u, u t, γ 1 u t, z) T, where γ 1 is the trace operator on. Then the previous problem is formally equivalent to U : = (u t, u tt, γ 1 u tt, z t ) T

4 4 S. NICAISE, C. PIGNOTTI EJDE-11/41 ( = u t, u + a u t, µ 1( (u + au t ) Therefore, problem (1.1) (1.5) can be rewritten as U = A(t)U, ) (x, t) + kz(, 1, ), τ (t)ρ 1 z ρ ) T. U() = ( u, u 1, γ 1 u 1, f (, τ) ) T, (.11) where the time varying operator A(t) is defined by u v A(t) v v 1 := ( (u + av) ) µ 1 (u+av) + kz(, 1), z with domain where D(A(t)) := τ (t)ρ 1 z ρ { (u, v, v 1, z) T H 1 Γ () L ( ) L ( ; H 1 (, 1)) : u + av E(, L ()), v = v 1 = z(, )on }, (u + av) L ( ), E(, L ()) = {u H 1 () : u L ()}. (.1) Recall that for a function u E(, L ()), u belongs to H 1/ ( ) and the next Green formula is valid (see section 1.5 of [1]) u wdx = uwdx + u ; w w H 1 Γ (), (.13) where ; Γ1 means the duality pairing between H 1/ ( ) and H 1/ ( ). Observe that the domain of A(t) is independent of the time t; i.e., D(A(t)) = D(A()), t >. (.14) A(t) is an unbounded operator in H, the Hilbert space defined by H := H 1 Γ () L () L ( ) L ( (, 1)), (.15) equipped with the standard inner product u ũ v v 1, ṽ ṽ := { u(x) ũ(x) + v(x)ṽ(x)}dx H z 1 z 1 + v 1 (x)ṽ 1 (x)dγ + z(x, ρ) z(x, ρ) dρ dγ. (.16) We can obtain a well-posedness result using semigroup arguments by Kato [1, 13, 7]. The following result is proved in [1, Theorem 1.9]. Theorem.1. Assume that (i) D(A()) is a dense subset of H, (ii) D(A(t)) = D(A()) for all t >,

5 EJDE-11/41 INTERIOR FEEDBACK STABILIZATION 5 (iii) for all t [, T ], A(t) generates a strongly continuous semigroup on H and the family A = {A(t) : t [, T ]} is stable with stability constants C and m independent of t (i.e. the semigroup (S t (s)) s generated by A(t) satisfies S t (s)u H Ce ms u H, for all u H and s ), (iv) t A belongs to L ([, T ], B(D(A()), H)), the space of equivalent classes of essentially bounded, strongly measurable functions from [, T ] into the set B(D(A()), H) of bounded operators from D(A()) into H. Then, problem (.11) has a unique solution U C([, T ], D(A())) C 1 ([, T ], H) for any initial datum in D(A()). Therefore, we will check the above assumptions for problem (.11). Lemma.. D(A()) is dense in H. Proof. Let (f, g, g 1, h) T H be orthogonal to all elements of D(A()); that is, u f = v v 1, g g 1 H z h 1 = { u(x) f(x) + v(x)g(x)}dx + v 1 g 1 dγ + z(x, ρ)h(x, ρ)dρdγ, for all (u, v, v 1, z) T D(A()). We first take u = and v = (then v 1 = ) and z D( (, 1)). As (,,, z) T D(A()), we obtain 1 z(x, ρ)h(x, ρ)dρdγ =. Since D( (, 1)) is dense in L ( (, 1), we deduce that h =. In the same way, by taking u =, z = and v D() (then v 1 = ) we see that g =. Therefore, for u =, z = we deduce also g 1 v 1 dγ =, v 1 D( ) and so g 1 =. The above orthogonality condition is then reduced to = u fdx, (u, v, v 1, z) T D(A()). By restricting ourselves to v = and z =, we obtain u(x) f(x)dx =, (u,,, ) T D(A()). But we easily see that (u,,, ) T D(A()) if and only if u E(, L ()) H (). This set is dense in H () (equipped with the inner product.,. H 1 Γ ()), thus we conclude that f =. Assuming (.) we will show that A(t) generates a C semigroup on H and using the variable norm technique of Kato from [13] and Theorem.1, that problem (.11) has a unique solution.

6 6 S. NICAISE, C. PIGNOTTI EJDE-11/41 Let ξ be a positive constant that satisfies k ξ a k. (.17) 1 d C P 1 d Note that this choice of ξ is possible from assumption (.). We define on the Hilbert space H the time dependent inner product u ũ v v 1, ṽ ṽ := { u(x) ũ(x) + v(x)ṽ(x)}dx t z 1 z 1 + µ v 1 (x)ṽ 1 (x)dγ + ξ z(x, ρ) z(x, ρ) dρ dγ. (.18) Using this time dependent inner product and Theorem.1, we can deduce a wellposedness result. Theorem.3. For any initial datum U D(A()) there exists a unique solution of system (.11). Proof. We first observe that U C([, + ), D(A())) C 1 ([, + ), H) φ t φ s e c τ t s, t, s [, T ], (.19) where φ = (u, v, v 1, z) T and c is a positive constant. Indeed, for all s, t [, T ], we have φ t φ se c τ t s = (1 ) { } e c τ t s ( u(x) + v )dx + µ v1dγ + ξ ( ) 1 τ(s)e c τ t s z (x, ρ) dρ dγ. Γ N We notice that 1 e c τ t s. Moreover τ(s)e c τ t s for some c >. Indeed, = τ(s) + τ (a)(t s), where a (s, t), and thus, τ(s) 1 + τ (a) t s. τ(s) By (1.7), τ is bounded on [, T ] and therefore, recalling also (1.6), τ(s) 1 + c t s e c τ t s, τ which proves (.19). Now we calculate A(t)U, U t for a fixed t. Take U = (u, v, v 1, z) T D(A(t)). Then, v u ( (u + av) ) A(t)U, U t = µ 1 (u+av) + kz(, 1), v v 1 t z τ (t)ρ 1 z ρ

7 EJDE-11/41 INTERIOR FEEDBACK STABILIZATION 7 = { v(x) u(x) + v(x) (u(x) + av(x))}dx ξ 1 (1 τ (t)ρ)z ρ (x, ρ)z(x, ρ)dρdγ ( (u + av) ) (x) + kz(x, 1) v(x) dγ. So, by Green s formula, A(t)U, U t = k z(x, 1)v(x)dΓ a v(x) dx 1 ξ (1 τ (t)ρ)z ρ (x, ρ)z(x, ρ)dρdγ. Integrating by parts in ρ, we obtain 1 z ρ (x, ρ)z(x, ρ)(1 τ (t)ρ) dρ dγ = 1 = τ (t) 1 ρ z (x, ρ)(1 τ (t)ρ)dρ dγ 1 z (x, ρ)dρdγ + 1 {z (x, 1)(1 τ (t)) z (x, )}dγ. Therefore, from (.) and (.1), A(t)U, U t = k z(x, 1)v(x)dΓ a v(x) dx ξ {z (x, 1)(1 τ (t)) z (x, )}dγ ξτ (t) 1 z (x, ρ)dρdγ = k z(x, 1)v(x)dΓ a v(x) dx ξ z (x, 1)(1 τ (t))dγ + ξ v (x)dγ ξτ (t) 1 z (x, ρ)dρdγ, (.) (.1) from which, using Cauchy-Schwarz s inequality, a trace estimate and Poincaré s Theorem, it follows that ( A(t)U, U t a k C P 1 d ξ ) C P v(x) dx ( ξ k ) (.) (1 d) 1 d z (x, 1)dΓ + κ(t) U, U t, Γ1 where κ(t) = (τ (t) + 1) 1 Now, observe that from (.17),. (.3) A(t)U, U t κ(t) U, U t, (.4) which means that the operator Ã(t) = A(t) κ(t)i is dissipative.

8 8 S. NICAISE, C. PIGNOTTI EJDE-11/41 Moreover, κ τ (t)τ (t) (t) = τ (t)(τ (t) + 1) 1 (τ (t) + 1) 1 is bounded on [, T ] for all T > (by (1.6) and (1.7)) and we have d dt A(t)U = with τ (t)ρ τ (t)(τ (t)ρ 1) τ (t)ρ τ (t)(τ (t)ρ 1) z ρ bounded on [, T ]. Thus d dtã(t) L ([, T ], B(D(A()), H)), (.5) the space of equivalence classes of essentially bounded, strongly measurable functions from [, T ] into B(D(A()), H). Now, we show that λi A(t) is surjective for fixed t > and λ >. Given (f, g, g 1, h) T H, we seek U = (u, v, v 1, z) T D(A(t)) solution of u f (λi A(t)) v v 1 = g g 1, z h that is verifying λu v = f λv (u + av) = g ( ) (u + av) λv 1 + µ 1 (x) + kz(x, 1) λz + 1 τ (t)ρ z ρ = h. = g 1 Suppose that we have found u with the appropriate regularity. Then and we can determine z. Indeed, by (.1), and, from (.6), (.6) v := λu f (.7) z(x, ) = v(x), for x, (.8) λz(x, ρ) + 1 τ (t)ρ z ρ (x, ρ) = h(x, ρ), for x, ρ (, 1). (.9) Then, by (.8) and (.9), we obtain if τ (t) =, and ρ z(x, ρ) = v(x)e λρ + e λρ h(x, σ)e λσ dσ, z(x, ρ) = v(x)e λ τ (t) ln(1 τ (t)ρ) + e λ τ (t) ln(1 τ (t)ρ) ρ h(x, σ) 1 τ (t)σ e λ τ ln(1 τ (t)σ) (t) dσ,

9 EJDE-11/41 INTERIOR FEEDBACK STABILIZATION 9 otherwise. So, from (.7), if τ (t) =, and z(x, ρ) = λu(x)e λρ f(x)e λρ ρ + e λρ h(x, σ)e λσ dσ, on (, 1), z(x, ρ) = λu(x)e λ τ (t) ln(1 τ (t)ρ) f(x)e λ τ (t) ln(1 τ (t)ρ) + e λ τ (t) ln(1 τ (t)ρ) on (, 1) otherwise. In particular, if τ (t) =, with z L ( ) defined by ρ h(x, σ) 1 τ (t)σ e λ τ ln(1 τ (t)σ) (t) dσ, (.3) (.31) z(x, 1) = λu(x)e λ + z (x), x, (.3) 1 z (x) = f(x)e λ + e λ h(x, σ)e λσ dσ, x, (.33) and, if τ (t), with z L ( ) defined by z(x, 1) = λu(x)e λ τ (t) ln(1 τ (t)) + z (x), x, (.34) z (x) = f(x)e λ τ (t) ln(1 τ (t)) + e λ τ (t) ln(1 τ (t)) 1 h(x, σ) 1 τ (t)σ e λ τ ln(1 τ (t)σ) (t) dσ, (.35) for x. Then, we have to find u. In view of the equation λv (u + av) = g, we set s = u + av and look at s. Now according to (.7), we may write v = λu f = λs f λav, or equivalently v = λ 1 + λa s 1 f. (.36) 1 + λa Hence once s will be found, we will get v by (.36) and then u by u = s av, or equivalently 1 u = 1 + λa s + a f. (.37) 1 + λa By (.36) and (.6), the function s satisfies with the boundary conditions as well as (at least formally) λ 1 + λa s s = g + λ f in, (.38) 1 + λa s = on Γ, (.39) s = µg 1 µλv 1 kz(, 1) on,

10 1 S. NICAISE, C. PIGNOTTI EJDE-11/41 which becomes due to (.36), (.37), (.3), (.34) and the requirement that v 1 = γ 1 v on : where s λ(ke λ = + µλ) s + l on, (.4) 1 + λa l = µg 1 + λ(µ kae λ ) f kz on 1 + λa if τ (t) =, otherwise where s = τ ln(1 τ (t)) (t) + µλ) λ(ke λ s λa l on, (.41) λ(µ kae λ τ l ln(1 τ (t)) (t) ) = µg1 + f kz on. 1 + λa From (.38), integrating by parts, and using (.39), (.4), (.41) we find the variational problem λ ( 1 + λa = sw + s w)dx + (g + λ 1 + λa f)wdx + lwdγ λ(ke λτ + µλ) swdγ 1 + λa w H 1 Γ () if τ (t) =, otherwise λ λ(ke ( sw + s w)dx λa Γ1 λ τ τ ln(1 τ ) + µλ) swdγ 1 + λa = (g + λ 1 + λa f)wdx + lwdγ w H 1 Γ (). (.4) (.43) As the left-hand side of (.4), (.43) is coercive on H (), the Lax-Milgram lemma guarantees the existence and uniqueness of a solution s H () of (.4), (.43). If we consider w D() in (.4), (.43), we have that s solves (.38) in D () and thus s = u + av E(, L ()). Using Green s formula (.13) in (.4) and using (.38), we obtain λ(ke λτ + µλ) swdγ + s 1 + λa ; w = lw dγ, leading to (.4) and then to the third equation of (.6) due to the definition of l and the relations between u, v and s. We find the same result if τ (t). In conclusion, we have found (u, v, v 1, z) T D(A), which verifies (.6), and thus λi A(t) is surjective for some λ > and t >. Again as κ(t) >, this proves that λi Ã(t) = (λ + κ(t))i A(t) is surjective (.44) for any λ > and t >. Then, (.19), (.4) and (.44) imply that the family à = {Ã(t) : t [, T ]} is a stable family of generators in H with stability constants independent of t, by [13,

11 EJDE-11/41 INTERIOR FEEDBACK STABILIZATION 11 Proposition 1.1]. Therefore, the assumptions (i)-(iv) of Theorem.1 are satisifed by (.14), (.19), (.4), (.5), (.44) and Lemma., and thus, the problem Ũ = Ã(t)Ũ Ũ() = U has a unique solution Ũ C([, + ), D(A())) C1 ([, + ), H) for U D(A()). The requested solution of (.5) is then given by with β(t) = t κ(s)ds, because U(t) = e β(t) Ũ(t) U (t) = κ(t)e β(t) Ũ(t) + e β(t) Ũ (t) = κ(t)e β(t) β(t) Ũ(t) + e Ã(t)Ũ(t) β(t) = e (κ(t)ũ(t) + Ã(t)Ũ(t)) = e β(t) A(t)Ũ(t) = A(t)eβ(t) Ũ(t) = A(t)U(t). This concludes the proof. Theorem.4. Assume that (1.6) (1.7) and (.) hold. Then for any initial datum U H there exists a unique solution U C([, + ), H) of problem (.11). Moreover, if U D(A()), then U C([, + ), D(A())) C 1 ([, + ), H)... Well-posedness in the case µ =. As before, we use the auxiliary unknown (.3). Then, problem (1.1) (1.5), with µ =, is equivalent to u tt (x, t) u(x, t) a u t (x, t) = in (, + ), (.45) z t (x, ρ, t) + (1 τ (t)ρ)z ρ (x, ρ, t) = in (, 1) (, + ), (.46) u(x, t) = on Γ (, + ), (.47) (u + au t ) (x, t) = kz(x, 1, t) on (, + ), (.48) z(x,, t) = u t (x, t) on (, ), (.49) u(x, ) = u (x) and u t (x, ) = u 1 (x) in, (.5) z(x, ρ, ) = f (x, ρτ()) in (, 1). (.51) If we denote U := (u, u t, z) T, then ( U := (u t, u tt, z t ) T = u t, u + a u t, τ ) T (t)ρ 1 z ρ. Therefore, problem (.45) (.51) can be rewritten as U = A (t)u, U() = (u, u 1, f (, τ())) T, (.5)

12 1 S. NICAISE, C. PIGNOTTI EJDE-11/41 where the time dependent operator A (t) is defined by A (t) u v v := (u + av), z with domain D(A (t)) := τ (t)ρ 1 z ρ { (u, v, z) T H 1 Γ () L ( ; H 1 (, 1)) : u + av E(, L ()), (u + av) = kz(, 1) on ; v = z(, ) on }. (.53) Note that for (u, v, z) T D(A (t)), (u+av) belongs to L ( ) since z(, 1) is in L ( ). Finally, as above, observe that domain of A (t) is independent of the time t; i.e., Denote by H the Hilbert space D(A (t)) = D(A ()), t >. H := H 1 Γ () L () L ( (, 1)), (.54) equipped with the scalar product ũ u v, ṽ := { u(x) ũ(x) + v(x)ṽ(x)} dx H z z 1 + z(x, ρ) z(x, ρ)dρdγ, (.55) Arguing analogously to the case µ > we can deduce an existence and uniqueness result. Theorem.5. Assume that (1.6) (1.7) and (.) hold. Then, for any initial datum U H there exists a unique solution U C([, + ), H ) of problem (.5). Moreover, if U D(A ()), then U C([, + ), D(A ())) C 1 ([, + ), H ). Remark.6. This well-posedness theorem can be also deduced from the abstract framework of [8] (see Theorem. in [8]) for second order evolution equations. On the contrary, the case µ > is not covered by this abstract result..3. Stability result. Now, we show that problem (1.1) (1.5) is uniformly exponentially stable under the assumption k < a 1 d. (.56) C P We define the energy of system (1.1) (1.5) as F (t) := 1 {u t + u }dx + ξ t e λ(s t) u t (x, s)dγds + µ u t (x, t)dγ, t (.57) where ξ, λ are suitable positive constants. We fix ξ such that k < ξ < a k. (.58) 1 d C P 1 d

13 EJDE-11/41 INTERIOR FEEDBACK STABILIZATION 13 Note that (.56) ensures that this choice is possible. Moreover, the parameter λ is fixed satisfying λ < 1 k log τ ξ. (.59) 1 d Remark that in the case of a constant delay, we can take λ = and in that case F (t) corresponds to the natural energy of (u, u t, z) (up to the factor 1 ), see [4]. Here the time dependence of the delay implies that our system is no more invariant by translation and therefore we have to replace the arguments from [4] by the use of an appropriate Lyapunov functional. We start with the following estimate. Proposition.7. Assume (1.6) (1.7) and (.56). Then, for any regular solution of problem (1.1) (1.5) the energy is decreasing and, for a suitable positive constant C, we have { F (t) C C t t u t (x, t) dx + e λ(s t) u t (x, s)dγds. } u t (x, t )dγ Proof. Differentiating (.57), we obtain F (t) = t u tt + u u t }dx + {u ξ u t (x, t)dγ + µ u t (t)u tt (t) dγ ξ e λ u t (x, t )(1 τ (t))dγ λ ξ t e λ(t s) u t (x, s)dγds, t and then, applying Green s formula, F (t) = au t (x, t) u t (x, t)dx + u t (t) u (t)dγ ξ e λ u t (x, t )(1 τ (t))dγ + ξ u t (x, t)dγ λ ξ t e λ(t s) u t (x, s)dγds + µ u t (t)u tt (t)dγ. t (.6) (.61) Integrating once more by parts and using the boundary conditions we obtain F (t) = a u t (x, t) dx k u t (t)u t (t )dγ ξ e λ u t (x, t )(1 τ (t))dγ + ξ u t (x, t)dγ (.6) λ ξ t e λ(t s) u t (x, s)dγ ds. t Now, applying Cauchy-Schwarz s inequality and recalling the assumptions (1.6) and (1.8), we obtain F (t) a u t (x, t) dx + ξ u k t (x, t)dγ + u t (x, t)dγ 1 d

14 14 S. NICAISE, C. PIGNOTTI EJDE-11/41 + k 1 d u t (t )dγ ξ (1 d)e λτ ( a ( e λτ ξ λ ξ t k C P 1 d ξ C P t t u t (x, t )dγ λ ξ ) u t (x, t) dx t ) k (1 d) 1 d u t (x, t )dγ e λ(t s) u t (x, s)dγds, e λ(t s) u t (x, s)dγ ds where in the last inequality we also use a trace estimate and Poincaré s Theorem. Therefore, (.6) immediately follows recalling (.58) and (.59). Now, let us define the Lyapunov functional [ ] ˆF (t) = F (t) + γ u(x, t)u t (x, t)dx + µ u(x, t)u t (x, t)dγ, (.63) where γ is a positive small constant that we will choose later on. Note that, from Poincaré s Theorem, the functional ˆF is equivalent to the energy F, that is, for γ small enough, there exist two positive constant β 1, β such that β 1 ˆF (t) F (t) β ˆF (t), t. (.64) Lemma.8. For any regular solution of problem (1.1) (1.5), d { dt { C } u(x, t)u t (x, t)dxdt + µ u(x, t)u t (x, t)dγ Γ 1 } u t (x, t) dx + u t (x, t )dγ 1 u(x, t) dx, (.65) for a suitable positive constant C (that is different from the one from Proposition.7). Proof. Differentiating and integrating by parts we have d uu t dx = dt = u t (x, t)dx + u t (x, t)dx u( u + a u t )dx u(x, t) dx a + u(t) (u + au t) (t)dγ. u(x, t) u t (x, t)dx (.66)

15 EJDE-11/41 INTERIOR FEEDBACK STABILIZATION 15 From (.66), using the boundary condition on, we obtain { } d uu t dx + µ u(x, t)u t (x, t)dγ dt Γ 1 = u t (x, t)dx + u( u + a u t )dx + µ u t (x, t)dγ + µ u(x, t)u tt (x, t)dγ Γ 1 = u t (x, t)dx u(x, t) dx a u(x, t) u t (x, t)dx k u(t)u t (t )dγ + µ u t (x, t)dγ. (.67) We can conclude by using Young s inequality, a trace estimate and Poincaré s Theorem. Finally using the above results we can deduce an exponential stability estimate. Theorem.9. Assume (1.6) (1.7) and (.56). Then there exist positive constants C 1, C such that for any solution of problem (1.1)-(1.5), F (t) C 1 F ()e Ct, t. (.68) Proof. From Lemma.8, taking γ sufficiently small in the definition of the Lyapunov functional ˆF, we have d dt ˆF { } (t) C u t (x, t) dx + u t (x, t )dx t C e λ(t s) u t (x, s)dγ ds γ (.69) u(x, t) dx, t for a suitable positive constant C that is different from the one in (.65). Poincaré s Theorem implying u t (x, t) dx + u t (x, t) ds C P 1 u t (x, t) dx, for some C P 1 >, we obtain d dt ˆF (t) C F (t), (.7) for a suitable positive constant C. This clearly implies the exponential estimate (.68) recalling (.64). Remark.1. Note that in the case of a constant delay the exponential stability result holds under the condition k < a/c P (corresponding to (.56) since, in this case, d = ), see [4]. On the contrary, if this condition is no more valid, then some instabilities may occur, we refer to [4] for some illustrations. 3. Internal delay feedback 3.1. Well-posedness. First of all we formulate a well-posedness result under the assumption a 1 a 1 d. (3.1) Let us set z(x, ρ, t) = u t (x, t ρ), x, ρ (, 1), t >. (3.)

16 16 S. NICAISE, C. PIGNOTTI EJDE-11/41 Then, problem (1.9) (1.1) is equivalent to u tt (x, t) u(x, t) + a u t (x, t) + a 1 z(x, 1, t) = in (, + ) (3.3) z t (x, ρ, t) + (1 τ (t)ρ)z ρ (x, ρ, t) = in (, 1) (, + ) (3.4) u(x, t) = on (, + ) (3.5) z(x,, t) = u t (x, t) on (, ) (3.6) u(x, ) = u (x) and u t (x, ) = u 1 (x) in (3.7) z(x, ρ, ) = g (x, ρτ()) in (, 1). (3.8) Let us denote U := (u, u t, z) T, then ( U := (u t, u tt, z t ) T = u t, u a u t a 1 z(, 1, ), τ (t)ρ 1 ) T z ρ. Therefore, problem (3.3) (3.8) can be rewritten as U = A 1 (t)u U() = (u, u 1, g (, τ())) T (3.9) where the time dependent operator A 1 (t) is defined by u v A 1 (t) v := u a v a 1 z(, 1), z τ (t)ρ 1 z ρ with domain { D(A 1 (t)) := (u, v, z) T ( H () H 1 () ) H 1 () L (; H 1 (, 1)) : } v = z(, ) in. Note that the domain of A 1 (t) is independent of the time t; i.e., Let us introduce the Hilbert space D(A 1 (t)) = D(A 1 ()), t >. (3.1) H 1 := H 1 () L () L ( (, 1)), (3.11) equipped with the inner product ũ u v, ṽ H z z 1 := { u(x) ũ(x) + v(x)ṽ(x)}dx + 1 z(x, ρ) z(x, ρ)dρdx. (3.1) Next we state the well-posedness result then follows from [8, Theorem.] that extends the well-posedness result of [3] for wave equations with constant delays to an abstract second order evolution equation with time-varying delay. Theorem 3.1. Assume (1.6) (1.7) and (3.1). Then, for any initial datum U H 1 there exists a unique solution U C([, + ), H 1 ) of problem (3.9). Moreover, if U D(A 1 ()), then U C([, + ), D(A 1 ())) C 1 ([, + ), H 1 ).

17 EJDE-11/41 INTERIOR FEEDBACK STABILIZATION 17 Remark 3.. In [5] the authors considered a wave equation with boundary timevarying delay feedback without the assumption > τ > of non degeneracy of τ, but in less general spaces. We expect that a well-posedness result holds also for problem (1.9) (1.1) without this restriction on τ. However, we preferred to consider non degenerate τ in order to avoid technicalities. 3.. Stability result. We will give an exponential stability result for problem (1.9) (1.1) under assumption (1.14). We define the energy of system (1.9) (1.1) as E(t) := 1 {u t + u }dx + ξ t e λ(s t) u t (x, s)dx ds, (3.13) t where ξ, λ are suitable positive constants. We will fix ξ such that a a 1 1 d ξ >, ξ a 1 1 d >, (3.14) λ < 1 a 1 log τ ξ. (3.15) 1 d Note that assumption (1.14) guarantees the existence of such a constant ξ. We have the following estimate. Proposition 3.3. Assume (1.6)-(1.7) and (1.14). Then, for any regular solution of problem (1.9)-(1.1) the energy decays and there exists a positive constant C such that t E (t) C {u t (x, t)+u t (x, t )} C e λ(s t) u t (x, s)dxds. (3.16) t Proof. Differentiating (3.13), we obtain E (t) = t u tt + u u t }dx + {u ξ u t (x, t) dx ξ e λ u t (x, t )(1 τ (t))dx λ ξ t e λ(t s) u t (x, s)dxds, t and then, applying Green s formula, E (t) = a u t (x, t)dx a 1 u t (t)u t (x, t )dx ξ e λ u t (x, t )(1 τ (t))dx + ξ λ ξ t e λ(t s) u t (x, s)dxds. t u t (x, t)dx (3.17) Now, applying Cauchy-Schwarz s inequality and recalling the assumptions (1.6) and (1.8), we obtain E (t) a u t (x, t)dx a 1 u t (t)u t (t )dx + ξ u t (x, t)dx ξ (1 d)e λτ u t (x, t )dx λ ξ t e λ(t s) u t (x, s)dxds t

18 18 S. NICAISE, C. PIGNOTTI EJDE-11/41 ( a a 1 1 d ξ ) u t (x, t)dx ( e λτ ξ (1 d) a ) 1 1 d u t (x, t ) dx λ ξ t e λ(t s) u t (x, s) dx ds t from which easily follows (3.16) recalling (3.14) and (3.15). Now, let us introduce the Lyapunov functional Ê(t) = E(t) + γ u(x, t)u t (x, t)dx, (3.18) where γ is a suitable small positive constant. Note that, from Poincaré s Theorem, the functional Ê is equivalent to the energy E, that is, for γ small enough, there exist two positive constant β 1, β such that β 1 Ê(t) E(t) β Ê(t), t. (3.19) Lemma 3.4. For any regular solution of problem (1.9) (1.1), d u(x, t)u t (x, t) dx dt dt C [u t (x, t) + u t (x, t )]dx 1 u(x, t) dx, for a suitable positive constants C. Proof. Differentiating and integrating by parts d uu t dx = u t (x, t)dx + u( u a u t (t) a 1 u t (t ) dx dt = u t (x, t)dx u(x, t) dx a u(t)u t (t)dx + a 1 u(t)u t (t ) dx. We can conclude by using Young s inequality and Poincaré s Theorem. (3.) (3.1) Therefore, analogously to the case of boundary delay feedback, we can now obtain a uniform exponential decay estimate. Theorem 3.5. Assume (1.6) (1.7) and (1.14). Then there exist positive constants C 1, C such that for any solution of problem (1.9) (1.1), E(t) C 1 E()e Ct, t. (3.) Remark 3.6. Using Lemma 3.4 this stability result can be deduced with the help of Theorem 4.3 of [8]. The difference with [8] relies on the choice of a simpler Lyapunov functional that renders the proof of the exponential decay more simple. Remark 3.7. Note that in [3] we have assumed that the coefficient a 1 of the delay term is positive. But this assumption is not necessary. The results of [3] are valid, with analogous proofs, also for a 1 of arbitrary sign satisfying a 1 < a.

19 EJDE-11/41 INTERIOR FEEDBACK STABILIZATION 19 Remark 3.8. Note that in the proof of the stability estimate we did not use the condition of non degeneracy of the delay τ >. So, if u is a regular solution of problem (1.9) (1.1) the exponential stability result holds for u also in presence of a possibly degenerate τ, (cf. Remark 3.). The same is true for solutions to problem (1.1) (1.5). References [1] K. T. Andrews, K. L. Kuttler and M. Shillor. Second order evolution equations with dynamic boundary conditions. J. Math. Anal. Appl., 197(3): , [] G. Chen. Control and stabilization for the wave equation in a bounded domain I-II. SIAM J. Control Optim., 17:66 81, 1979; 19:114 1, [3] F. Conrad and Ö. Morgül. On the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim., 36(6): , [4] R. Datko. Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim., 6: , [5] R. Datko. Two questions concerning the boundary control of certain elastic systems. J. Differential Equations, 9:7 44, [6] R. Datko, J. Lagnese and M. P. Polis. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim., 4:15 156, [7] G. G. Doronin and N. A. Larkin. Global solvability for the quasilinear damped wave equation with nonlinear second-order boundary conditions. Nonlinear Anal., 5: ,. [8] E. Fridman, S. Nicaise and J. Valein. Stabilization of second order evolution equations with unbounded feedback with time-dependent delay. SIAM J. Control and Optim., 48:58-55, 1. [9] S. Gerbi and B. Said-Houari. Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions. Adv. Differential Equations, 13: , 8. [1] P. Grisvard. Elliptic problems in nonsmooth domains, volume 1 of Monographs and Studies in Mathematics. Pitman, Boston-London-Melbourne, [11] M. Grobbelaar-Van Dalsen. On fractional powers of a closed pair of operators and a damped wave equation with dynamic boundary conditions. Appl. Anal., 53:41 54, [1] T. Kato. Nonlinear semigroups and evolution equations. J. Math. Soc. Japan, 19:58 5, [13] T. Kato. Linear and quasilinear equations of evolution of hyperbolic type. C.I.M.E., II ciclo:15 191, [14] V. Komornik. Rapid boundary stabilization of the wave equation. SIAM J. Control Optim., 9:197 8, [15] V. Komornik. Exact controllability and stabilization, the multiplier method, volume 36 of RMA. Masson, Paris, [16] V. Komornik and E. Zuazua. A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl., 69:33 54, 199. [17] J. Lagnese. Decay of solutions of the wave equation in a bounded region with boundary dissipation. J. Differential Equations, 5:163 18, [18] J. Lagnese. Note on boundary stabilization of wave equations. SIAM J. Control and Optim., 6:15 156, [19] I. Lasiecka and R. Triggiani. Uniform exponential decay in a bounded region with L (, T ; L (Σ))-feedback control in the Dirichlet boundary conditions. J. Differential Equations, 66:34 39, [] I. Lasiecka, R. Triggiani and P. F. Yao. Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J. Math. Anal. Appl., 35:13 57, [1] W. Littman and L. Markus. Stabilization of a hybrid system of elasticity by feedback boundary damping. Ann. Mat. Pura Appl. (4), 15:81 33, [] Ö. Mörgul. On the stabilization and stability robustness against small delays of some damped wave equations. IEEE Trans. Automat. Control., 4: , [3] S. Nicaise and C. Pignotti. Stability and instability results of the wave equation with a delay term in the boundary or internal feedback. SIAM J. Control and Optim., 45: , 6.

20 S. NICAISE, C. PIGNOTTI EJDE-11/41 [4] S. Nicaise and C. Pignotti. Exponential stability of second order evolution equations with structural damping and dynamic boundary delay feedback. preprint 1, submitted. [5] S. Nicaise, C. Pignotti and J. Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete Contin. Dyn. Syst. Ser. S, 4:693 7, 11. [6] S. Nicaise, J. Valein and E. Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete Contin. Dyn. Syst. Ser. S, : , 9. [7] A. Pazy. Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Math. Sciences. Springer-Verlag, New York, [8] M. Pellicer and J. Sòla-Morales. Analysis of a viscoelastic spring-mass model. J. Math. Anal. Appl., 94: , 4. [9] G. Q. Xu, S. P. Yung and L. K. Li. Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var., 1(4):77 785, 6. [3] E. Zuazua. Exponential decay for the semi-linear wave equation with locally distributed damping. Comm. Partial Differential Equations, 15:5 35, 199. Serge Nicaise Université de Valenciennes et du Hainaut Cambrésis, MACS, ISTV, Valenciennes Cedex 9, France address: serge.nicaise@univ-valenciennes.fr Cristina Pignotti Dipartimento di Matematica Pura e Applicata, Università di L Aquila, Via Vetoio, Loc. Coppito, 671 L Aquila, Italy address: pignotti@univaq.it