Observability and stabilization of degenerate wave equations with singular potential on time-varying domains

Neumann boundary control for degenerate wave equations in time-varying domains

We study a wave equation in one space dimension with a general diffusion coefficient which degenerates on part of the boundary. Degeneracy is measured by a real parameter $μ_a>0$. We establish observability inequalities for weakly (when $μ_a \in [0,1[$) as well as strongly (when $μ_a \in [1,2[$) degenerate equations. We also prove a negative result when the diffusion coefficient degenerates too violently (i.e. when $μ_a>2$) and the blow-up of the observability time when $μ_a$ converges to $2$ from below. Thus, using the HUM method we deduce the exact controllability of the corresponding degenerate control problem when $μ_a \in [0,2[$. We conclude the paper by studying the boundary stabilization of the degenerate linearly damped wave equation and show that a suitable boundary feedback stabilizes the system exponentially. We extend this stability analysis to the degenerate nonlinearly boundary damped wave equation, for an arbitrarily growing nonlinear feedback close to the origin. This analysis proves that the degeneracy does not affect the optimal energy decay rates at large time. We apply the optimal-weight convexity method of \cite{alaamo2005, alajde2010} together with the results of the previous section, to perform this stability analysis.

We discuss exact boundary observability of a 1-dimensional degenerate wave equation on a time-varying domain. We deduce the exact boundary observability at the moving endpoint. Moreover, a more optimal exact boundary observability time is given.

Stability for degenerate wave equations with drift under simultaneous degenerate damping

This paper is devoted to a study of the null controllability problems for one-dimensional linear degenerate wave equations through a boundary controller. First, the well-posedness of linear degenerate wave equations is discussed. Then the null controllability of some degenerate wave equations is established, when a control acts on the non-degenerate boundary. Different from the known controllability results in the case that a control acts on the degenerate boundary, any initial value in state space is controllable in this case. Also, an explicit expression for the controllability time is given. Furthermore, a counterexample on the controllability is given for some other degenerate wave equations.

This paper is addressed to a study of the persistent regional null controllability problems for one‐dimensional linear degenerate wave equations through a distributed controller. Different from non‐degenerate wave equations, the classical null controllability results do not hold for some degenerate wave equations. Thus, persistent regional null controllability is introduced, which means finding a control such that the corresponding state of the degenerate wave equation may vanish in a suitable subset of the space domain in a period of time. In order to solve this problem, we need to establish the regional null controllability for degenerate wave equations. This problem is reduced to a suitable observability problem of a linear degenerate wave equation. The key point is to choose a suitable multiplier in order to establish this observability inequality. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we investigate the stability of a degenerate/singular wave equation featuring localized singular damping, along with a drift term and a leading operator in non-divergence form. We establish exponential stability results in this context under suitable conditions on the degeneracy and singularity coefficients.

Existence and regularity of multiple solutions for infinitely degenerate nonlinear elliptic equations with singular potential

Motivated by the broad applications of wave propagation in non-uniform and time-varying environments, such as in acoustics, elasticity, and seismology, we investigate the controllability of degenerate wave equations with time-dependent wave speeds. In this work, we examine non-autonomous degenerate wave equations in a one-dimensional spatial domain, addressing both divergence and non-divergence forms. A control function is applied at the non-degeneracy boundary point, while Dirichlet or Neumann conditions are imposed at the degeneracy one. Using the generalized energy conservation law, we first establish boundary observability for the homogeneous problem, a key step in our analysis. Building on this, we prove the null-controllability of the non-autonomous degenerate wave systems. To achieve this, we construct solutions using the transposition method, which accommodates low regularity requirements, enabling us to handle weaker smoothness assumptions while maintaining rigorous control over the system’s behavior. We conclude by presenting some insightful observations and potential avenues for future work, which could further advance the understanding of this problem.

A degenerate wave equation with time‐varying delay in the boundary control input is considered. The well‐posedness of the system is established by applying the semigroup theory. The boundary stabilization of the degenerate wave equation is concerned and the uniform exponential decay of solutions is obtained by combining the energy estimates with suitable Lyapunov functionals and an integral inequality under suitable conditions.

<p>This paper explores the boundary stabilization of a degenerate wave equation in the non-divergence form, which includes a drift term and a singular potential term. Additionally, we introduce boundary fractional derivative damping at the endpoint where divergence is absent. Using semi-group theory and the multiplier method, we establish polynomial stability, with a decay rate depending upon the order of the fractional derivative.</p>

In this article, we study exact controllability for degenerate and singular wave equations with a general coefficient. We estimate the observability inequality by the multiplier method and determine the observability time. We also deduce the exact controllability of the corresponding degenerate and singular control problem at a sufficiently large time, employing the Hilbert uniqueness method. For more information see https://ejde.math.txstate.edu/Volumes/2025/04/abstr.html

We consider the global existence and asymptotic stability of solutions to the Cauchy problem for degenerate nonlinear wave equations of Kirchhoff type with a dissipative term in unbounded domain. We derive the sharp decay estimates of the solution and its derivatives. Moreover, we show that the solution has a lower decay estimate of some algebraic rate.

We consider the initial data boundary value problem for the degenerate dissipative wave equations of Kirchhoff type ρu′′ + ∥A1/2u∥2γAu+ u′ = 0. When either the coefficient ρ or the initial data are appropriately small at least, we show the global existence theorem by using suitable identities together with the energy. Moreover, under the same assumption for ρ and the initial data, we derive the sharp decay estimates of the solutions and their second derivatives. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, we deal with the boundary controllability of a one‐dimensional degenerate and singular wave equation with degeneracy and singularity occurring at the boundary of the spatial domain. Exact boundary controllability is proved in the range of both subcritical and critical potentials and for sufficiently large time, through a boundary controller acting away from the degenerate/singular point. By duality argument, we reduce the problem to an observability estimate for the corresponding adjoint system, which is proved by means of the multiplier method and new Hardy‐type inequalities.