Stability result of a suspension bridge Problem with time-varying delay and time-varying weight

In this paper, we study a plate equation as a model for a suspension bridge with time-varying delay and time-varying weights. Under some conditions on the delay and weight functions, we establish a stability result for the associated energy functional. The present work extends and generalizes some similar results in the case of wave or plate equations.

Motivated by the result in [3], where an energy decay estimate is established for a wave equation with constant feedback (τ ≡ constant) and h ≡ 0, we establish a stability result for Problem (1.1)-(1.3). The present paper improved and extended the results in [6,17] for plate equations and the results in [3,18] for wave equations. This paper is organized as follows: in Sect. 2, we recall some fundamental materials for the clarity of the reader. In Sect. 3, we state and prove our main stability result for Problem (1.1)-(1.3).

Functional setting and hypotheses
Here, we assume the hangers restoring force h satisfies the following, as given in [17]: (A1) h : R → R is a locally Lipschitz non-decreasing function such that h(0) = 0 and the positive function Using the ideas of [3], we assume the time varying weights δ 1 (t) and δ 2 (t) satisfy and As in [19], we assume the time varying internal feedback τ (t) satisfies (A3) There exist τ 0 , τ 1 > 0 such that As in [7], we introduce the Hilbert space endowed with the inner product and denote by H( ) the dual of H 2 * ( ). We now state some needed Lemmas: Then, for any w ∈ H 2 * ( ), there exists an embedding constant C e = C e ( , p) > 0 such that Similarly, as in Nicaise and Pignotti [18], we introduce the following change of variable: (2.11) Then simple calculations gives for (x, y, t) ∈ × (0, T ) (2.14) and initial data Letξ be a positive constant satisfying The energy function of Problem (2.13)-(2.15) is defined by where ξ(t) =ξδ 1 (t) is a non-increasing C 1 − function. The main existence and uniqueness result of (2.13)-(2.15) is the following.
Using Young's inequality, the fact that ξ(t) is non-increasing and recalling (2.16), it follows that

Stability result
The main stability result is the following. Proof We start by defining the function ψ as follows: It follows from (A2) that ψ is non-decreasing and bounded C 1 -function such that Hence, ψ satisfies the conditions of Lemma 2.3. Now, multiplying (2.13) 1

by E q (t)ψ (t)u and integrating over × (s, T ), T > s, we obtain
Using Lemma 2.2 and some calculations, we obtain Next, we multiple (2.13) 2 by E q (t)ψ (t)ξ(t)e −2τ (t)ρ z and integrate over This gives us Thus, we obtain Recalling conditions (A1) and setting k = min 1, e −2τ 0 , d τ 0 , the summation of (3.3) and (3.4) leads to Now, we estimate the terms on the right-hand side of (3.5). Using (A2)-(A3), Cauchy-Schwarz inequality, the embedding Lemma 2.1, (2.19) and assuming that ψ is a non-negative and bounded function on [0, +∞), we obtain Thus, using (3.7) and the definition of ψ in (3.2), we get where C e is the embedding constant in Lemma 2.1. By assumption (A3) and recalling that ξ(t) is a nonincreasing function, we obtain ≤ C E q+1 (s) (3.11) and using Cauchy-Schwarz inequality, we get (3.14) By selecting small enough so that k − C e > 0, we get Letting T −→ +∞ and applying Lemma 2.3, we obtain the result.

Examples
Let where a and b are constants. Then and Therefore, δ 1 and δ 2 satisfy the assumptions in (A2). It follows from (3.1) and (3.2) that