In this article one discusses the controllability of a semi-discrete system obtained by discretizing in space the linear 1-D wave equation with a boundary control at one extremity. It is known that the semi-discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero (see [8]).Here we introduce a new semi-discrete model based on a mixed finite element method with two different basis functions for the position and velocity. We show that the controls obtained with these semi-discrete systems can be chosen uniformly bounded in L 2 (0, T ) and in such a way that they converge to the HUM control of the continuous wave equation, i.e. the minimal L 2 -norm control. We illustrate the mathematical results with several numerical experiments.
Abstract. We consider the linear heat equation on the half-line with a Dirichlet boundary control. We analyze the null-controllability problem. More precisely, we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the L 2 boundary control. We rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems. Next, the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials {e jt } j≥1 in which the usual summability condition on the inverses of the eigenvalues does not hold. Roughly speaking, we prove that controllable data have Fourier coefficients that grow exponentially for large frequencies. This result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a Sobolev space of negative order may be driven to zero in an arbitrarily small time.
A numerical scheme for the controlled semi-discrete 1-D wave equation is considered. We analyze the convergence of the boundary controls of the semi-discrete equations to a control of the continuous wave equation when the mesh size tends to zero. We prove that, if the high modes of the discrete initial data have been filtered out, there exists a sequence of uniformly bounded controls and any weak limit of this sequence is a control for the continuous problem. The number of the eliminated frequencies depends on the mesh size and the regularity of the continuous initial data. The case of the HUM controls is also discussed.
The null-controllability property of a 1 − d parabolic equation involving a fractional power of the Laplace operator, (−Δ) α , is studied. The control is a scalar time-dependent function g = g(t) acting on the system through a given space-profile f = f (x) on the interior of the domain. Thus, the control g determines the intensity of the space control f applied to the system, the latter being given a priori. We show that, if α ≤ 1/2 and the shape function f is, say, in L 2 , no initial datum belonging to any Sobolev space of negative order may be driven to zero in any time. This is in contrast with the existing positive results for the case α > 1/2 and, in particular, for the heat equation that corresponds to α = 1. This negative result exhibits a new phenomenon that does not arise either for finite-dimensional systems or in the context of the heat equation. On the contrary, if more regularity of the shape function f is assumed, then we show that there are initial data in any Sobolev space H m that may be controlled. Once again this is precisely the opposite behavior with respect to the control properties of the heat equation in which, when increasing the regularity of the control profile, the space of controllable data decreases. These results show that, in order for the control properties of the heat equation to be true, the dynamical system under consideration has to have a sufficiently strong smoothing effect that is critical when α = 1/2 for the fractional powers of the Dirichlet Laplacian in 1 − d. The results we present here are, in nature and with respect to techniques of proof, similar to those on the control of the heat equation in unbounded domains in [S.
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