Exact Observability of a 1-Dimensional Wave Equation on a Noncylindrical Domain

Abstract

We discuss admissibility and exact observability estimates of boundary observation and interior point observation of a one-dimensional wave equation on a time dependent domain for sufficiently regular boundary functions. We also discuss moving observers inside the noncylindrical domain and simultaneous observability results.Since we suppose f, g ∈ D((0, 1)), h satisfies the periodicity condition h (α) (−1)=h (α) (1) for all derivative orders α ≥ 0. As a consequence, the series of F , g and h above may be differentiated term by term. We let u(x, t) := n∈Z A n e 2πin ϕ(t+x)) − e 2πin ϕ(t−x)) † In particular, (bn) is a Riesz basis in L 2 ([−1, 1]).

Optimization of non-cylindrical domains for the exact null controllability of the 1D wave equation

“…By the way of the d'Alembert formula, a uniform observability inequality is proved for a precise set of curves {(γ(t), t)} t∈[0,T ] leading to moving controls in H −1 (∪ t∈(0,T ) γ(t) × t). Still in the 1D setting, Ciu, Liu and Gao [11] and Haak and Hoang [13] analyze the boundary and moving interior point observability for wave equations posed on a sufficiently regular time-dependent domain. In the N -dimensional case, Liu and Yong [19] employ the multiplier method to prove that the wave equation is controllable under the hypothesis that the distributed control region q covers the whole space domain Ω before the time T .…”

Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains

“…From what preceed, it is not difficult to see that the solution (p, q) to system (7) satisfies the regularity given in (11). 48)-(51) that the solution (p, q) to system (7) satisfies the regularity…”

“…An observability inequality has been established for the dual of system (1) with β ≡ 1 for sufficiently large time under the assumption that the boundary function α must be periodic and satisfies α L ∞ (0,∞) < 1. More general boundary functions are considered in [11] with boundary conditions y(t, 0) = 0, y(t, s(t)) = u(t), t ∈ (0, ∞), where s : [0, ∞) → (0, ∞) is assumed to be a C 1 function satisfying s L ∞ (0,∞) < 1. Furthermore, it has been assumed that s must be in some admissible class of curves (see [11] for more details).…”

“…More general boundary functions are considered in [11] with boundary conditions y(t, 0) = 0, y(t, s(t)) = u(t), t ∈ (0, ∞), where s : [0, ∞) → (0, ∞) is assumed to be a C 1 function satisfying s L ∞ (0,∞) < 1. Furthermore, it has been assumed that s must be in some admissible class of curves (see [11] for more details). Under these assumptions, the authors proved that exact controllability holds if, and only if T ≥ s + • (s − ) −1 (0), where s ± (t) = t ± s(t).…”

Exact controllability for a degenerate and singular wave equation with moving boundary

“…Such situations, where the spacial domain is time-dependent, appear in several fields of the applied sciences such as fluid dynamics, astrophysics, fluidstructure interaction and quantum mechanics, see [14,24,29,36]. We refer to [3,6,11,12,19,27,28,31,32] for some known results in this direction. It is worth mentioning that controllability results, in these cited works, are obtained using different approaches such as domain transform, multiplier techniques or the non-harmonic Fourier analysis.…”