Null controllability of nonlinear convective heat equations

ESAIM: Control, Optimisation and Calculus of Variations, 2006

In this paper, we prove the global null controllability of the linear heat equation completed with linear Fourier boundary conditions of the form ∂y ∂n + β y = 0. We consider distributed controls with support in a small set and nonregular coefficients β = β(x, t). For the proof of null controllability, a crucial tool will be a new Carleman estimate for the weak solutions of the classical heat equation with nonhomogeneous Neumann boundary conditions.

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.

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, 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.

Advances in Differential Equations

We prove null controllability results for the degenerate onedimensional heat equation ut − (x α ux)x = fχω, x ∈ (0, 1), t ∈ (0, T). As a consequence, we obtain null controllability results for a Croccotype equation that describes the velocity field of a laminar flow on a flat plate.

Comptes Rendus Mathematique, 2011

This note deals with the computation of distributed null controls for a semi-linear 1D heat equation, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has been obtained in [Fernández-Cara & Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, 2000] via a fixed point reformulation; see also [Barbu, Exact controllability of the superlinear heat equation, 2000]. More precisely, Carleman estimates and Kakutani's theorem together ensure the existence of fixed points for a corresponding linearized control mapping. In practice, the difficulty is to extract from the Picard iterates a convergent (sub)sequence. We introduce and analyze a least squares reformulation of the problem; we show that this strategy leads to an effective and constructive way to compute fixed points. To cite this article: E.

Siam Journal on Control and Optimization, 2004

In this paper we present a local result on the existence of insensitizing controls for a semilinear heat equation when nonlinear boundary conditions of the form ∂ny + f (y) = 0 are considered. The problem leads to analyze a special type of nonlinear null controllability problem. A sharp study of the linear case and a later application of an appropriate fixed point argument constitutes the scheme of the proof of the main result. The boundary conditions we are dealing with lead to seek a fixed point, thus also control functions, in certain Hölder spaces. The main clue in this paper is the construction of controls with hölderian regularity starting from L 2 -controls in the linear case. Enough regularity on the data and appropriate assumptions on the right hand side term ξ of the equation are required.

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.

Evolution Equations and Control Theory, 2022

This paper deals with the hierarchical control of the anisotropic heat equation with dynamic boundary conditions and drift terms. We use the Stackelberg-Nash strategy with one leader and two followers. To each fixed leader, we find a Nash equilibrium corresponding to a bi-objective optimal control problem for the followers. Then, by some new Carleman estimates, we prove a null controllability result.

ESAIM: Control, Optimisation and Calculus of Variations, 2008

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 provides explicitly the control and the associated trajectory as series. Both the control and the trajectory have a Gevrey regularity in time related to the L p class of the coefficient in front of ut. The approach applies in particular to the (possibly degenerate or singular) heat equation (a(x)ux)x − ut = 0 with a(x) > 0 for a.e. x ∈ (0, 1) and a + 1/a ∈ L 1 (0, 1), or to the heat equation with inverse square potential uxx + (µ/|x| 2 )u − ut = 0 with µ ≥ 1/4.

Nonlinear Analysis-theory Methods & Applications, 2004

In this paper we present two results on the existence of insensitizing controls for a heat equation in a bounded domain of IR N . We first consider a semilinear heat equation involving gradient terms with homogeneous Dirichlet boundary conditions. Then a heat equation with a nonlinear term F (y) and linear boundary conditions of Fourier type is considered. The nonlinearities are assumed to be globally Lipschitz-continuous. In both cases, we prove the existence of controls insensitizing the L 2 −norm of the observation of the solution in an open subset O of the domain, under suitable assumptions on the data. Each problem boils down to a special type of null controllability problem. General observability inequalities are proved for linear systems similar to the linearized problem. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.

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.

Nonlinear Analysis-theory Methods & Applications, 2004

In this paper we present two results on the existence of insensitizing controls for a heat equation in a bounded domain of IR N . We first consider a semilinear heat equation involving gradient terms with homogeneous Dirichlet boundary conditions. Then a heat equation with a nonlinear term F (y) and linear boundary conditions of Fourier type is considered. The nonlinearities are assumed to be globally Lipschitz-continuous. In both cases, we prove the existence of controls insensitizing the L 2 −norm of the observation of the solution in an open subset O of the domain, under suitable assumptions on the data. Each problem boils down to a special type of null controllability problem. General observability inequalities are proved for linear systems similar to the linearized problem. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.

Journal of Mathematical Analysis and Applications, 1997

Using Browder-Minty's surjective theorem from the theory of monotone operators, We consider the exact internal controllability for the semilinear heat equation. We show that the system is exactly controllable in L 2 (Ω) if the nonlinearities are globally Lipschitz continuous. Furthermore, we prove that the controls depend Lipschitz continuously on the terminal states, and discuss the behaviour of the controls as the nonlinear terms tend to zero in some sense. A variant of the Hilbert Uniqueness Method is presented to cope with the nonlinear nature of the problem.

We consider a parabolic problem with degeneracy in the interior of the spatial domain, and we focus on controllability results through Carleman estimates for the associated adjoint problem. The novelty of the present paper is that the degeneracy is also in the interior of the control region, so that no previous result can be adapted to this situation.