Exponential decay rate for the energy of a flexible structure with dynamic delayed boundary conditions and a local interior damping

On the stabilization of a flexible structure via a nonlinear delayed boundary control

“…Our contribution considerably improves earlier results of [10,11,12,22,20,23,26,27,28,29,37,38,41,42,44], where the time-delay is disregarded and also of [6,8,17,18,19,45], where both delayed and non-delayed terms in the proposed controls are linear. It is also worth mentioning that unlike the previous works [41] and [20], we are able to treat more general nonlinearities thanks to the general results in [1].…”

Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control

“…Such a finding is obtained via Lyapunov method despite the presence of a time-dependent boundary delay but under some conditions. Unlike the previous papers [2,3,9,8], the advantage of such an approach lies in its simplicity and more importantly the fact that an estimate of the decay rate is provided. Obviously, the frequency-domain (resolvent) method applied in [2,3,9,8] cannot be used in our case as the system operator is time-dependent.…”

“…Note that γ and C 0 are appropriate positive constants that will be fixed later on, while ε is an arbitrary positive constant. Moreover, F is a negative function defined by (see [34] and [2,3,9,10,12] for similar situations)…”

“…It turns out that in order to improve the convergence rate from polynomial to exponential, one has to add another global distributed damping y t (x, t) [3]. Later, the authors in [9] proved that such a desirable exponential convergence rate can be achieved by means of a locally distributed damping b(x)y t (x, t), where b(x) ∈ W 1,∞ (I) has a local support on I and Another interesting result has been obtained in [8], where the boundary control F(t) involves the angular velocity term y xt (0, t) instead of the velocity term y t (0, t). More precisely, the proposed control in [8] take the form…”

“…Needless to mention that such a result improves those mentioned above where the delay does not depend on time t. Obviously, in practice the time-delay involved in systems control is often time-dependent. Furthermore, inspired by [30,31], our stability result is obtained via the Lyapunov technique, which is direct and also provides an estimate of the convergence rate in contrast to [2,3,9,8], where the frequency-domain method is applied. The latter permits to reach the exponential stability result for only autonomous systems and does not provide any estimate of the exponential rate.…”