On the stabilization of a vibrating equation

Dynamic boundary controls of a rotating body‐beam system with time‐varying angular velocity

“…where the series (4.9) and (4.10) converge in H 2 0 and L 2 (0,1), respectively, uniformly in t. Following the method used in [8], we will prove that a n = b n = 0 for any n = 1,2,..., and thus (ỹ(t),ỹ t (t)) = (0,0). Indeed, (ỹ(0),ỹ t (0)) = (ỹ 0 ,z 0 ) being in H 4 0 × H 2 0 (see (4.2)), one can claim that…”

“…Obviously, to deduce the desired resultφ(t) = 0, it suffices to show thatỹ = 0 is the only solution of (4.2)-(4.3). To do so, we will use the same techniques as in [8]. For simplicity, assume that ρ = EI = l = 1.…”

Energy decay estimates for a wave equation with nonlinear boundary feedback

“…Let us mention the work of B. Chentouf et al [1], where a damping model is considered and the equation tolerates a term βy, β = cte > 0, the system is asymptotically stabilized by the nonlinear feedback law depending only on the boundary velocities:…”

Saturated Boundary Feedback Stabilization of a Linear Wave Equation

“…The statements (i)--(iv) are directly proved by applying Lemma 1 and the theory of nonlinear semigroups (cf. [10], [3], [2], [13], [5]).…”

“…In the literature, the wave equation with nonlinear boundary feedback control has been extensively studied (cf. [11], [17], [5], [6], [26], and [27] to cite only a few). It has been proved in [26] and [27] that, to obtain uniform exponential stabilization of the wave equation, the nonlinear boundary velocity feedback must have a linear growth not only around zero but also at infinity.…”