Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities

Abstract

In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities (see (1.1)). If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish an optimal polynomial decay rate. Finally, we provide some illustrative examples.1 γ and that this decay rate is in some sense optimal. Regarding System (1.1) when α = 0, it reduces to System (1.2) with B = A γ and a = 1. In this case, we recover the results of [29].When the coupling acts through displacements, Loreti and Rao studied in [31] the stability of the following abstract system of coupled equations (1.3) u tt + Au + A γ u t + αy = 0, y tt + Ay − αu = 0.They proved that System (1.3) is not exponentially stable and an optimal polynomial energy decay rate of type t −τ (γ) is obtained where

Exponential stability and exact controllability of a system of coupled wave equations by second‐order terms (via Laplacian) with only one non‐smooth local damping

“…Russell [5] first presented the concept of indirect stabilization, which means that just one equation of the coupled system is damped, and since that, it has been an interest of many authors (please see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and references therein).…”

Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay

“…The Figure 1 System (1.1) consists of two wave equations with only one singular viscoelastic damping acting on the first equation, the second one is indirectly damped via a singular coupling between the two equations. The notion of indirect damping mechanisms has been introduced by Russell in [48] and since then, it has attracted the attention of many authors (see for instance [4,5,6,9,16,1,36,53]). The study of such systems is also motivated by several physical considerations like Timoshenko and Bresse systems (see for instance [2,3,39,41]).…”

Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay

“…The Figure 1 System (1.1) consists of two wave equations with only one singular viscoelastic damping acting on the first equation, the second one is indirectly damped via a singular coupling between the two equations. The notion of indirect damping mechanisms has been introduced by Russell in [49] and since then, it has attracted the attention of many authors (see for instance [3], [5], [6], [9], [16], [24], [37] and [54] ). The study of such systems is also motivated by several physical considerations like Timoshenko and Bresse systems (see for instance [1], [2], [40] and [42]).…”