Theoretical and practical aspects of heating equation

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.

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.

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.

Communications on Pure and Applied Analysis, 2008

This paper is devoted to analyze the class of initial data that can be insensitized for the heat equation. This issue has been extensively addressed in the literature both in the case of complete and approximate insensitization (see and [1], respectively).

ESAIM: Control, Optimisation and Calculus of Variations, 2000

The internal and boundary exact null controllability of nonlinear convective heat equations with homogeneous Dirichlet boundary conditions are studied. The methods we use combine Kakutani fixed point theorem, Carleman estimates for the backward adjoint linearized system, interpolation inequalities and some estimates in the theory of parabolic boundary value problems in L k .

Abstract and Applied Analysis, 2002

We study the internal exact null controllability of a nonlinear heat equation with homogeneous Dirichlet boundary condition. The method used combines the Kakutani fixed-point theorem and the Carleman estimates for the backward adjoint linearized system. The result extends to the case of boundary control.

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

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.

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.

SIAM Journal on Control and Optimization

In this paper, we study the null controllability of linear heat and wave equations with spatial nonlocal integral terms. Under some analyticity assumptions on the corresponding kernel, we show that the equations are controllable. We employ compactness-uniqueness arguments in a suitable functional setting, an argument that is harder to apply for heat equations because of its very strong time irreversibility. Some possible extensions and open problems concerning other nonlocal systems are presented.