Null controllability of degenerate heat equations

ESAIM: Control, Optimisation and Calculus of Variations, 1997

This paper is concerned with the null controllability of systems governed by semilinear parabolic equations. The control is exerted either on a small subdomain or on a portion of the boundary. W e p r o ve that the system is null controllable when the nonlinear term f (s) grows slower than s log jsj as jsj ! +1. 1. In the linear case (f(s) = as for some a), (1.1) is null controllable with no restriction on y 0 , T or O.

Journal of Dynamical and Control Systems, 2012

The purpose of this paper is to provide a full analysis of the null controllability problem for the one dimensional degenerate/singular parabolic equation ut − (a(x)ux)x − λ x β u = 0, (t, x) ∈ (0, T) × (0, 1), where the diffusion coefficient a(•) is degenerate at x = 0. Also the boundary conditions are considered to be Dirichlet or Neumann type related to the degeneracy rate of a(•). Under some conditions on the function a(•) and parameters β, λ, we prove global Carleman estimates. The proof is based on an improved Hardy-type inequality.

Evolution Equations and Control Theory, 2020

We consider the heat equation in a bounded domain of R N with distributed control (supported on a small open subset) subject to dynamic boundary conditions of surface diffusion type and involving drift terms on the bulk and on the boundary. We prove that the system is null controllable at any time. The result is based on new Carleman estimates for this type of boundary conditions.

Facta Universitatis, 2019

In this paper we are interested in the study of the null controllability for the one dimensional degenerate nonautonomous parabolic equation ut − M (t)(a(x)ux)x = hχω, (x, t) ∈ Q = (0, 1) × (0, T), where ω = (x1, x2) is a small nonempty open subset in (0, 1), h ∈ L 2 (ω × (0, T)), the diffusion coefficients a(•) is degenerate at x = 0 and M (•) is nondegenerate on [0, T ]. Also, the boundary conditions are considered to be Dirichlet-or Neumann-type related to the degeneracy rate of a(•). Under some conditions on the functions a(•) and M (•), we prove some global Carleman estimates which will yield the observability inequality of the associated adjoint system and, equivalently, the null controllability of our parabolic equation.

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.

Evolution Equations and Control Theory

We prove null controllability for linear and semilinear heat equations with dynamic boundary conditions of surface diffusion type. The results are based on a new Carleman estimate for this type of boundary conditions.

Journal of Differential Equations, 2004

In this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form @y @n þ f ðyÞ ¼ 0 and distributed controls, with support in a small set. We show that, when the function f is globally Lipschitzcontinuous, the system is approximately controllable. We also show that the system is locally null controllable and null controllable for large time when f is regular enough and f ð0Þ ¼ 0: For the proofs of these assertions, we use controllability results for similar linear problems and appropriate fixed point arguments. In the case of the local and large time null controllability results, the arguments are rather technical, since they need (among other things) Ho¨lder estimates for the control and the state. r On the other hand, it will be said that system (1) is null controllable at time T if, for each y 0 AL 2 ðOÞ; there exist vAL 2 ðO Â ð0; TÞÞ and an associated solution yAC 0 ð½0; T; L 2 ðOÞÞ such that yðx; TÞ ¼ 0 in O: ð3Þ ARTICLE IN PRESS A. Doubova et al. / J. Differential Equations 196 (2004) 385-417 386

ESAIM: Control, Optimisation and Calculus of Variations, 2010

This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general case consists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost. In the special case of the heat equation in rectangular domains, we provide an alternative way to check the Lebeau-Robbiano spectral condition. We then show that the sophisticated Carleman and interpolation inequalities used in previous literature may be replaced by a simple result of Turán. In this case, we provide explicit values for the constants involved in the above mentioned spectral condition. As far as we are aware, this is the first proof of the null-controllability of the heat equation with arbitrary control domain in a n-dimensional open set which avoids Carleman estimates.

Systems & Control Letters, 2016

This paper deals with the local null control of a free-boundary problem for the classical 1D heat equation with distributed controls, locally supported in space. In the main result we prove that, if the final time T is fixed and the initial state is sufficiently small, there exist controls that drive the state exactly to rest at time t = T .

We prove null controllability for linear and semilinear heat equations with dynamic boundary conditions of surface diffusion type. The results are based on a new Carleman estimate for this type of boundary conditions.

ESAIM: Control, Optimisation and Calculus of Variations, 2004

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically R+ or R N. Considering an unbounded and disconnected control region of the form ω := ∪nωn, we prove two null controllability results: under some technical assumption on the control parts ωn, we prove that every initial datum in some weighted L 2 space can be controlled to zero by usual control functions, and every initial datum in L 2 (Ω) can be controlled to zero using control functions in a weighted L 2 space. At last we give several examples in which the control region has a finite measure and our null controllability results apply.

ESAIM: Control, Optimisation and Calculus of Variations, 2014

The liner parabolic equation ∂y ∂t − 1 2 Δy +F •∇y = ½O 0 u with Neumann boundary condition on a convex open domain O ⊂ R d with smooth boundary is exactly null controllable on each finite interval if O0 is an open subset of O which contains a suitable neighbourhood of the recession cone of O. Here, F : R d → R d is a bounded, C 1-continuous function, and F = ∇g, where g is convex and coercive.

Journal of Mathematical Physics, 2018

In this paper we consider a linear hybrid system which composed by two nonhomogeneous rods connected by a point mass and generated by the equations        ρ 1 (x)u t = (σ 1 (x)u x) x − q 1 (x)u, x ∈ (−1, 0), t > 0, ρ 2 (x)v t = (σ 2 (x)v x) x − q 2 (x)v, x ∈ (0, 1), t > 0, u(0, t) = v(0, t) = z(t), t > 0, M z t (t) = σ 2 (0)v x (0, t) − σ 1 (0)u x (0, t), t > 0, with Dirichlet boundary condition on the left end x = −1 and a boundary control acts on the right end x = 1. We prove that this system is null controllable with Dirichlet or Neumann boundary controls. Our approach is mainly based on a detailed spectral analysis together with the moment method. In particular, we show that the associated spectral gap in both cases (Dirichlet or Neumann boundary controls) are positive without further conditions on the coefficients ρ i , σ i and q i (i = 1, 2) other than the regularities.

52nd IEEE Conference on Decision and Control, 2013

We derive in a direct and rather straightforward way the null controllability of a 2-D heat equation with 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 law achieving the exact steering to zero. Numerical experiments demonstrate the relevance of the approach.

Le Centre pour la Communication Scientifique Directe - HAL - Université de Nantes, 2022

We consider linear one-dimensional strongly degenerate parabolic equations with measurable coefficients that may be degenerate or singular. Taking 0 as the point where the strong degeneracy occurs, we assume that the coefficient a = a(x) in the principal part of the parabolic equation is such that the function x → x/a(x) is in L p (0, 1) for some p > 1. After establishing some spectral estimates for the corresponding elliptic problem, we prove that the parabolic equation is null controllable in the energy space by using one boundary control.