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Register a new userNull controllability of one dimensional degenerate parabolic equations with first order terms
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In this paper we present a null controllability result for a degenerate semilinear parabolic equation with first order terms. The main result is obtained after the proof of a new Carleman inequality for a degenerate linear parabolic equation with first order terms.
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The aim of this paper is to prove the superexponential stabilizability to the ground state solution of a degenerate parabolic equation of the form begin{equation*} u_t(t,x)+(x^{alpha}u_x(t,x))_x+p(t)x^{2-alpha}u(t,x)=0,qquad tgeq0,xin(0,1) end{equation*} via bilinear control $pin L_{loc}^2(0,+infty)$. More precisely, we provide a control function $p$ that steers the solution of the equation, $u$, to the ground state solution in small time with doubly-exponential rate of convergence. The parameter $alpha$ describes the degeneracy magnitude. In particular, for $alphain[0,1)$ the problem is called weakly degenerate, while for $alphain[1,2)$ strong degeneracy occurs. We are able to prove the aforementioned stabilization property for $alphain [0,3/2)$. The proof relies on the application of an abstract result on rapid stabilizability of parabolic evolution equations by the action of bilinear control. A crucial role is also played by Bessels functions.
The aim of this paper is to perform a Stackelberg strategy to control parabolic equations. We have one control, textit{the leader}, that is responsible for a null controllability property; additionally, we have a control textit{the follower} that solves a robust control objective. That means, that we seek for a saddle point of a cost functional. In this way, the follower control is not sensitive to a broad class of external disturbances. As far as we know, the idea of combining robustness with a Stackelberg strategy is new in literature
In a separable Hilbert space $X$, we study the linear evolution equation begin{equation*} u(t)+Au(t)+p(t)Bu(t)=0, end{equation*} where $A$ is an accretive self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $pin L^2_{loc}(0,+infty)$ is a bilinear control. We give sufficient conditions in order for the above control system to be locally controllable to the ground state solution, that is, the solution of the free equation ($pequiv0$) starting from the ground state of $A$. We also derive global controllability results in large time and discuss applications to parabolic equations in low space dimension.
We are concerned about the controllability of a general linear hyperbolic system of the form $partial_t w (t, x) = Sigma(x) partial_x w (t, x) + gamma C(x) w(t, x) $ ($gamma in mR$) in one space dimension using boundary controls on one side. More precisely, we establish the optimal time for the null and exact controllability of the hyperbolic system for generic $gamma$. We also present examples which yield that the generic requirement is necessary. In the case of constant $Sigma$ and of two positive directions, we prove that the null-controllability is attained for any time greater than the optimal time for all $gamma in mR$ and for all $C$ which is analytic if the slowest negative direction can be alerted by {it both} positive directions. We also show that the null-controllability is attained at the optimal time by a feedback law when $C equiv 0$. Our approach is based on the backstepping method paying a special attention on the construction of the kernel and the selection of controls.
We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order version is known to converge to an entropy solution of the problem. A particular feature of the method is the storage of the multiresolution representation of the solution in a graded tree, whose leaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the $L^1$ contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate the computational efficiency together with the convergence properties.
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