Stability results for a class of second-order evolution equations - PowerPoint PPT Presentation

Stability results for a class of second-order evolution equations with intermittent delay Cristina PIGNOTTI Universit` a di L’Aquila Italy Joint with Serge Nicaise Universit ´ e de V alenciennes, France Chambery, June 15-18, 2015 1 / 43

wave without delay Let Ω be an open bounded domain of IR n , n ≥ 1 , with a boundary ∂ Ω of class C 2 . It is well-known that the problem  u tt ( x, t ) − ∆ u ( x, t ) + au t ( x, t ) = 0 , in Ω × (0 , + ∞ ) ,  u ( x, t ) = 0 , on ∂ Ω × (0 , + ∞ ) u ( x, 0) = u 0 ( x ) , u t ( x, 0) = u 1 ( x ) , in Ω ,  whith a > 0 and initial data ( u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω) , is exponentially stable, that is the energy E ( t ) = E ( u, t ) := 1 ∫ [ u 2 t ( x, t ) + |∇ u ( x, t ) | 2 ] dx, 2 Ω 2 / 43

wave without delay satisfies the uniform estimate, E ( t ) ≤ Ce − C ′ t E (0) , t > 0 , for all initial data. • [Rauch and Taylor, 1974], [Bardos,Lebeau and Rauch, 1992]. Exponential stability is not conserved in general in presence of a TIME DELAY! Delay effects arise in many applications and practical problems and it is well-known that a delay arbitrarily small may destabilize a system which is uniformly asymptotically stable in absence of delay (see e.g. [Datko,Lagnese and Polis, 1986], [Datko, 1988]). 3 / 43

Time delay effects Let us consider the problem u tt ( x, t ) − ∆ u ( x, t ) + au t ( x, t − τ ) = 0 , in Ω × (0 , + ∞ ) ,    on ∂ Ω × (0 , + ∞ ) , u ( x, t ) = 0 ,  u ( x, 0) = u 0 ( x ) , u t ( x, 0) = u 1 ( x ) , in Ω ,    u t ( x, t ) = f ( x, t ) , in Ω × ( − τ, 0) , where τ > 0 is the time delay and the initial data are taken in suitable spaces. In this case exponential stability FAILS! Indeed, as shown in [Nicaise and P., 2006] it is possible to find for the above problem a sequence { τ k } k of delays with τ k → 0 ( τ k → ∞ ) for which the corresponding solutions u k have an increasing energy. 4 / 43

Simultaneous dampings In [Nicaise and P., 2006] (cfr. [Xu, Yung and Li, 2006] for boundary delay in 1-d) in order to contrast the destabilizing effect of the time delay a “good” (not delayed) damping term is introduced in the first equation. More precisely the problem there considered is  u tt ( x, t ) − ∆ u ( x, t ) + au t ( x, t ) + ku t ( x, t − τ ) = 0 , x ∈ Ω , t > 0 ,  u ( x, t ) = 0 , x ∈ ∂ Ω , t > 0 , ( P0 )  + I.C. with k, a > 0 and initial data in suitable spaces. If a > k the system is uniformly exponentially stable. On the contrary, if a ≤ k the are instability phenomena. Energy functional: ∫ 1 E ( t ) = E ( u, t ) := 1 ∫ t ( x, t ) + |∇ u ( x, t ) | 2 ] dx + ξ ∫ [ u 2 u 2 t ( x, t − τρ ) dρdx , 2 2 Ω Ω 0 with τk < ξ < τ (2 a − k ) . 5 / 43

The problem Now, we consider second–order evolution equations with intermittent delay, this means that the standard damping and the delayed one act in different time intervals. Let H be a real Hilbert space and let A : D ( A ) → H be a positive self–adjoint operator with a compact inverse in H. 1 1 2 ) the domain of A 2 . Denote by V := D ( A Moreover, for i = 1 , 2 , let U i be real Hilbert spaces with norm and inner product denoted respectively by ∥ · ∥ U i and ⟨· , ·⟩ U i and let B i ( t ) : U i → V ′ , be time–dependent linear operators satisfying B ∗ 1 ( t ) B ∗ 2 ( t ) = 0 , ∀ t > 0 . 6 / 43

The problem Let us consider the problem { u tt ( t ) + Au ( t ) + B 1 ( t ) B ∗ 1 ( t ) u t ( t ) + B 2 ( t ) B ∗ 2 ( t ) u t ( t − τ ) = 0 , t > 0 , ( P ) u (0) = u 0 and u t (0) = u 1 , where the constant τ > 0 is the time delay. We assume that the delay feedback operator B 2 is bounded, that is B 2 ∈ L ( U 2 , H ) , while the standard one B 1 ∈ L ( U 1 , V ′ ) may be unbounded. AIM: We are interested in giving stability results for such a problem under suitable assumptions on the feedback operators B 1 and B 2 . 7 / 43

Comments REMARK This is related to the stabilization problem of second–order evolution equations with positive–negative dampings. We refer for this subject to [Haraux, Martinez, Vancostenoble, 2005] . See [P., 2012] for the link between wave equation with time delay in the damping and wave equation with indefinite damping. Similar problem has been considered in [Ammari, Nicaise, P., 2013] for 1-d models for the wave equation but with a different approach. Indeed, here we give stability results under conditions that allow to compensate the destabilizing delay effect with the good behaviour of the system in the time intervals without delay. On the contrary, there we obtain stability results for particular values of the time delays, related to the length of the domain, by using the D’Alembert formula. 8 / 43

Well–posedness We assume that for all n ∈ IN , there exists t n > 0 with t n < t n +1 and such that B 2 ( t ) = 0 ∀ t ∈ I 2 n = [ t 2 n , t 2 n +1 ) , B 1 ( t ) = 0 ∀ t ∈ I 2 n +1 = [ t 2 n +1 , t 2 n +2 ) , with B 2 ∈ C ([ t 2 n +1 , t 2 n +2 ]; L ( U 2 , H )) ; for the operators B 1 , we assume either B 1 ∈ C 1 ([ t 2 n , t 2 n +1 ]; L ( U 1 , H )) or √ B 1 ( t ) = b 1 ( t ) C n , with C n ∈ L ( U 1 , V ′ ) and b 1 ∈ W 2 , ∞ ( t 2 n , t 2 n +1 ) such that b 1 ( t ) > 0 , ∀ t ∈ I 2 n . 9 / 43

Well–posedness We further assume that τ ≤ T 2 n for all n ∈ IN , where T n denotes the lenght of the interval I n , that is T n = t n +1 − t n , n ∈ IN . Using semigroup theory we prove THEOREM Under the above assumptions, for any u 0 ∈ V and u 1 ∈ H , the problem ( P ) has a unique solution u ∈ C ([0 , ∞ ); V ) ∩ C 1 ([0 , ∞ ); H ) . 10 / 43

Stability result - B 1 bounded We first consider the case B 1 bounded. Assume that there exist Hilbert spaces W i , i = 1 , 2 , such that H is continuously embedded into W i , i.e. ∥ u ∥ 2 W i ≤ C i ∥ u ∥ 2 H , ∀ u ∈ H with C i > 0 independent of u. Moreover, we assume that for all n ∈ IN , there exist three positive constants m 2 n , M 2 n and M 2 n +1 with m 2 n ≤ M 2 n and such that for all u ∈ H we have i) m 2 n ∥ u ∥ 2 W 1 ≤ ∥ B ∗ 1 ( t ) u ∥ 2 U 1 ≤ M 2 n ∥ u ∥ 2 W 1 for t ∈ I 2 n = [ t 2 n , t 2 n +1 ) , ∀ n ∈ IN; ii) ∥ B ∗ 2 ( t ) u ∥ 2 U 2 ≤ M 2 n +1 ∥ u ∥ 2 W 2 for t ∈ I 2 n +1 = [ t 2 n +1 , t 2 n +2 ) , ∀ n ∈ IN . We now assume W 1 = W 2 (Later, we will drop this assumption by introducing a restriction on the size of delay intervals.) and we use the notation W := W 1 = W 2 . 11 / 43

Stability result - B 1 bounded Moreover, let us assume m 2 n inf > 0 . ( C ) M 2 n +1 n ∈ I N Energy functional: ∫ t E ( t ) := 1 + ξ ( ) ∥ u ( t ) ∥ 2 V + ∥ u t ( t ) ∥ 2 ∥ B ∗ 2 ( s + τ ) u t ( s ) ∥ 2 U 2 ds, H 2 2 t − τ where ξ is a positive number satisfying m 2 n ξ < inf . M 2 n +1 n ∈ I N PROPOSITION Assume i), ii) and ( C ) . For any regular solution of problem ( P ) the energy is decreasing on the intervals I 2 n , n ∈ IN , and E ′ ( t ) ≤ − m 2 n 2 ∥ u t ∥ 2 W . ( S1 ) 12 / 43

Stability result - B 1 bounded Moreover, on the intervals I 2 n +1 , n ∈ IN , E ′ ( t ) ≤ M 2 n +1 ( ξ + 1 ξ ) ∥ u t ∥ 2 W . ( S2 ) 2 Proof: Differentiating E ( t ) and using the definition of A and the equation, we get E ′ ( t ) = −∥ B ∗ 1 ( t ) u t ( t ) ∥ 2 U 1 − ( B ∗ 2 ( t ) u t , B ∗ 2 ( t ) u t ( t − τ )) U 2 + ξ 2 ∥ B ∗ 2 ( t + τ ) u t ( t ) ∥ 2 U 2 − ξ 2 ∥ B ∗ 2 ( t ) u t ( t − τ ) ∥ 2 U 2 . If t ∈ I 2 n , then B 2 ( t ) = 0 and then U 1 + ξ E ′ ( t ) = −∥ B ∗ 1 ( t ) u t ( t ) ∥ 2 2 ∥ B ∗ 2 ( t + τ ) u t ( t ) ∥ 2 U 2 . Since T 2 n = | I 2 n | ≥ τ, it results that t + τ ∈ I 2 n ∪ I 2 n +1 ∪ I 2 n +2 . Now, if t + τ ∈ I 2 n ∪ I 2 n +2 , then B 2 ( t + τ ) = 0 . Therefore, B 2 ( t + τ ) ̸ = 0 only if t + τ ∈ I 2 n +1 . In both cases, by our assumptions i) and ii), we get ( S1 ). 13 / 43

Stability result - B 1 bounded For t ∈ I 2 n +1 , as B 1 ( t ) = 0 , the previous identity becomes 2 ( t ) u t ( t − τ )) U 2 + ξ E ′ ( t ) = − ( B ∗ 2 ( t ) u t , B ∗ 2 ∥ B ∗ 2 ( t + τ ) u t ( t ) ∥ 2 U 2 − ξ 2 ∥ B ∗ 2 ( t ) u t ( t − τ ) ∥ 2 U 2 . By Young’s inequality we get E ′ ( t ) ≤ 1 U 2 + ξ 2 ξ ∥ B ∗ 2 ( t ) u t ( t ) ∥ 2 2 ∥ B ∗ 2 ( t + τ ) u t ( t ) ∥ 2 U 2 . This proves ( S2 ) using ii) because t + τ is either in I 2 n +1 , or in I 2 n +2 and in that last case B ∗ 2 ( t + τ ) = 0 . 14 / 43

The related conservative system Let us consider the conservative system associated with ( P ), w tt ( t ) + Aw ( t ) = 0 t > 0 ( PH ) w (0) = w 0 and w t (0) = w 1 with ( w 0 , w 1 ) ∈ V × H. For our stability result we need that an appropriate observability inequality holds. Namely we assume that there exists a time T > 0 such that for every time T > T there is a constant c, depending on T but independent of the initial data, such that ∫ T ∥ w t ( s ) ∥ 2 E S (0) ≤ c ( OI ) W ds, 0 E S ( t ) := 1 ( ) 2( ∥ u ( t ) ∥ 2 V + ∥ u t ( t ) ∥ 2 H ) for every weak solution of problem ( PH ) with initial data ( w 0 , w 1 ) ∈ V × H. 15 / 43