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.
<p style='text-indent:20px;'>This paper is concerned with the exact boundary controllability for a degenerate and singular wave equation in a bounded interval with a moving endpoint. By the multiplier method and using an adapted Hardy-poincaré inequality, we prove direct and inverse inequalities for the solutions of the associated adjoint equation. As a consequence, by the Hilbert Uniqueness Method, we deduce the controllability result of the considered system when the control acts on the moving boundary. Furthermore, improved estimates of the speed of the moving endpoint and the controllability time are obtained.</p>
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 subcritical/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 a new special Hardy-type inequality.
In this paper, we discuss the solvability of the nonlinear parabolic systems associated to the nonlinear parabolic equation: for $i=1,2$\begin{equation*}\mathcal{(S)} \left\{\begin{array}{lll}\displaystyle\frac{\partial u_{i}}{\partial t} -div(a(x,t,u_{i},\nabla u_{i}))+ g_{i}(x,t,u_{i},\nabla u_i)) =f_{i}(x,u_{1},u_{2})\, \quad & \mbox{in}\quad & Q_{T},\\\displaystyle u_{i}(x,t)=0 &\mbox{on}& \partial\Omega \times(0,T),\\\displaystyle u_{i}(x,(t=0))=u_{i,0}(x) & \mbox{in} & \Omega,\end{array}%\right.\end{equation*}with the source $f$ is merely integrable. The operator $\displaystyle A(u)= div \Big (a(x,t,u_{i},\nabla u_{i})\Big)$is a generalized Leray-Lions operator defined on the inhomogeneous Musielak-Orlicz spaces (the vector field $\displaystyle a(x,t,u_i,\nabla u_i)$ have a growth prescribed by a generalized N-function). The non linearity $g_{i}$ is a Carath\'{e}odory function satisfy the some condition.
In this paper, we consider a system of two degenerate wave equations coupled through the velocities, only one of them being controlled. We assume that the coupling parameter is sufficiently small and we focus on null controllability problem. To this aim, using multiplier techniques and careful energy estimates, we first establish an indirect observability estimate for the corresponding adjoint system. Then, by applying the Hilbert Uniqueness Method, we show that the indirect boundary controllability of the original system holds for a sufficiently large time.
MSC 2020: 35L80, 93B05, 93B07, 93C05, 93C20
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