178 research outputs found
Null controllability of the 2D heat equation using flatness
International audienceWe derive in a direct and rather straightforward way the null controllability of the N -dimensional heat equation in a bounded cylinder with boundary control at one end of the cylinder. We use the so-called flatness approach, which consists in param-eterizing the solution and the control by the derivatives of a "flat output". This yields an explicit control law achieving the exact steering to zero. Replacing the involved series by partial sums we obtain a simple numerical scheme for which we give explicit error bounds. Numerical experiments demonstrate the relevance of the approach
Controllability of the 1D Schrodinger equation by the flatness approach
We derive in a straightforward way the exact controllability of the 1-D
Schrodinger equation with a Dirichlet boundary control. We use the so-called
flatness approach, which consists in parameterizing the solution and the
control by the derivatives of a "flat output". This provides an explicit
control input achieving the exact controllability in the energy space. As an
application, we derive an explicit pair of control inputs achieving the exact
steering to zero for a simply-supported beam
Null controllability of one-dimensional parabolic equations by the flatness approach
We consider linear one-dimensional parabolic equations with space dependent
coefficients that are only measurable and that may be degenerate or
singular.Considering generalized Robin-Neumann boundary conditions at both
extremities, we prove the null controllability with one boundary control by
following the flatness approach, which providesexplicitly the control and the
associated trajectory as series. Both the control and the trajectory have a
Gevrey regularity in time related to the class of the coefficient in
front of .The approach applies in particular to the (possibly degenerate
or singular) heat equation with a(x)\textgreater{}0
for a.e. and , or to the heat equation with
inverse square potential with
On the reachable set for the one-dimensional heat equation
International audienceThe goal of this article is to provide a description of the reachable set of the one-dimensional heat equation, set on the spatial domain x ∈ (−L, L) with Dirichlet boundary controls acting at both boundaries. Namely, in that case, we shall prove that for any L0 > L any function which can be extended analytically on the square {x + iy, |x| + |y| ≤ L0} belongs to the reachable set. This result is nearly sharp as one can prove that any function which belongs to the reachable set can be extended analytically on the square {x + iy, |x| + |y| < L}. Our method is based on a Carleman type estimate and on Cauchy's formula for holomorphic functions
A non-controllability result for the half-heat equation on the whole line based on the prolate spheroidal wave functions and its application to the Grushin equation
In this article, we revisit a result by A. Koenig concerning the non-controllability of the half-heat equation posed on R, with a control domain that is an open set whose exterior contains an interval. The main novelty of the present article is to disprove the corresponding observability inequality by using as an initial condition a family of prolate spheroidal wave function (PSWF) translated in the Fourier space, associated to a parameter c that goes to ∞. The proof is essentially based on the dual nature of the PSWF together with direct computations, showing that the solution "does not spread out" too much during time. As a consequence, we obtain a new non-controllability result on the Grushin equation posed on R × R
A Fredholm transformation for the rapid stabilization of a degenerate parabolic equation
This paper deals with the rapid stabilization of a degenerate parabolic equation with a right Dirich-let control. Our strategy consists in applying a backstepping strategy, which seeks to find an invertible transformation mapping the degenerate parabolic equation to stabilize into an exponentially stable system whose decay rate is known and as large as we desire. The transformation under consideration in this paper is Fredholm. It involves a kernel solving itself another PDE, at least formally. The main goal of the paper is to prove that the Fredholm transformation is well-defined, continuous and invertible in the natural energy space. It allows us to deduce the rapid stabilization
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