Boundary controllability of parabolic coupled equations

This paper is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional parabolic equations are considered. We show that, in this framework, boundary controllability is not equivalent and is more complex than distributed controllability. In our main result, we provide necessary and sufficient conditions for the null controllability.     

Insensitizing controls for a class of quasilinear parabolic equations

This paper is addressed to showing the existence of insensitizing controls for a class of quasilinear parabolic equations with homogeneous Dirichlet boundary conditions. As usual, this insensitizing problem is reduced to a nonstandard null controllability problem of some nonlinear cascade system governed by a quasilinear parabolic equation and a linear parabolic equation. Nevertheless, in order to solve the later quasilinear controllability problem by the fixed point technique, we need to establish the null controllability of the linearized cascade parabolic system in the framework of classical solutions. The key point is to find the desired control function in a Hölder space for given data with certain regularities.     

A numerical study of the null boundary controllability of a convection diffusion equation

In this paper we study numerically the cost of the null controllability of a linear control parabolic 1-D equation as the diffusion coefficient tends to 0. For this linear control parabolic 1-D equation, we know from a prior work by J.-M. Coron and S. Guerrero (2005), that, when the diffusion coefficient tends to 0, for a small controllability time, the norm of the optimal control tends to infinity and that, if the controllability time is large enough, this norm tends to 0. For controllability times which are not covered by this work, we estimate numerically the norm of the optimal control as the diffusion coefficient tends to 0. To cite this article: A. Salem, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

带极大单调图的半线性抛物型方程的零能控性(英文)

本文讨论非线性项带一个极大单调图的半线性抛物型方程系统的零能控性.文中利用Kakutani不动点定理和线性抛物型方程的能控性得到该系统是零能控的,如果控制作用在内部区域上.由此,还得到该系统是零能控的,如果控制施加在一部分边界上.     

On the boundary controllability of non-scalar parabolic systems

This Note is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional linear parabolic equations are considered. We show that, with boundary controls, the situation is much more complex than for similar distributed control systems. In our main result, we provide necessary and sufficient conditions for null controllability. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Null controllability of degenerate parabolic equations

Motivated by a boundary layer problem, we are interested in the controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for a class of degenerate parabolic equation; the proof is based in particular on Hardytype inequalities. Then we deduce observability and null controllability results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact Local Controllability of a One-Control Reaction-Diffusion System

This work concerns the controllability of some parabolic systems with the control acting in one equation. The exact local controllability of a reaction-diffusion system with one control and the null controllability of a class of linear parabolic systems with one control are obtained. As an application, the existence of a time-optimal control for a reaction-diffusion system with one control is proved.Communicated by D. A. CarlsonThis work was supported by the New Century Excellent Teachers Plan of the Ministry of National Education of China, by the Key Laboratory- Optimal Control and Discrete Mathematics of Hubei Province, and by the National Science Foundation of China.     

Time optimal boundary controls for the heat equation

The fact that the time optimal controls for parabolic equations have the bang–bang property has been recently proved for controls distributed inside the considered domain (interior control). The main result in this article asserts that the boundary controls for the heat equation have the same property, at least in rectangular domains. This result is proved by combining methods from traditionally distinct fields: the Lebeau–Robbiano strategy for null controllability and estimates of the controllability cost in small time for parabolic systems, on one side, and a Remez-type inequality for Müntz spaces and a generalization of Turán?s inequality, on the other side.     

Exact Controllability of the Superlinear Heat Equation

The exact internal and boundary controllability of parabolic equations with superlinear nonlinearity is studied. Accepted 3 January 2000     

Some results on controllability of a nonlinear degenerate parabolic system by bilinear control

In this paper, we discuss the controllability of a nonlinear degenerate parabolic system with bilinear control. Based on the shrinking property of the solutions, we prove that the system is not globally approximately controllable. Furthermore, we give an approximate null controllability result. We also prove that the system is not globally exactly null controllable by a comparison principle.     

Global Approximately Controllability and Nite Dimensional Exact Controllability for Parabolic Equation

We study the globally approximate controllability and finite-dimensional exact controllability of parabolic equation where the control acts on a mobile subset of Ω,or, a curve in Q =Ω × (0, T).     

A weighted identity for partial differential operators of second order and its applications

In this Note, a weighted identity for partial differential operators of second order is established. As its applications, one may deduce all the known controllability/observability results for the parabolic, hyperbolic, Schrödinger and plate equations that are derived via Carleman estimate. Meanwhile, a new controllability/observability result is presented for the parabolic equations with a complex principal part. To cite this article: X. Fu, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Approximate controllability of a parabolic integrodifferential equation

In this paper, we obtain the approximate controllability of a parabolic integrodifferential equation with interior controls. The proof relies on the linear operator semigroup theory and the controllability theory for abstract equations. Copyright © 2013 John Wiley & Sons, Ltd.     

OPTIMAL CONTROL PROBLEM FOR PARABOLIC VARIATIONAL INEQUALITIES

1 IntroductionThroughout this paPer, we denote n an opell bounded domain in Rn with smooth boulldary0fl, let Q = fl x (0,T) and r = 0n x (0,T). Let H = L'(n).We shall study the optimal control problems governed by the following system:y'(t) Ay(t) p(y(t) -- rk(t)) 9 B1u(t) I1(t), (1.1)y(0) = u0,rk'(t) Aop(f) 7(op(f)) = B2u(t) f2(f), (1.2)op(0) = op0with the state constrchtF(y) C W' (1.3)The pay-off functional is given byJ(y, u) = l"[,(,, y) h(u)]dt. (1.4)JoNote that y' and…     

On the approximate controllability of a stochastic parabolic equation with a multiplicative noise

In this Note, we present some results concerning the approximate controllability for a stochastic parabolic equation with a multiplicative noise. For simplicity, we only consider the distributed control case.     

Controllability and speed of the process described by a parabolic equation with bounded control

Addressing the problems of controllability and speed of the processes described by a parabolic equation with distributed control in a given set, we propose some methods for solving these problems by minimizing sequences.     

一类边界控制半线性抛物型系统的逼近能控性

本文考虑一类Ｄｉｒｉｃｈｌｅｔ型边界控制半线性抛物型系统，应用半群理论和非线性分析的方法，证明了系统解映射的连续性，给出了系统逼近能控的一个充分条件．对于Ｎｅｕｍａｎｎ型边界控制或混合型边界控制的半线性抛物型系统，可以得到类似的结果．     

Internal Controllability for Parabolic Systems Involving Analytic Non-local Terms

This paper deals with the problem of internal controllability of a system of heat equations posed on a bounded domain with Dirichlet boundary conditions and perturbed with analytic non-local coupling terms. Each component of the system may be controlled in a different subdomain. Assuming that the unperturbed system is controllable—a property that has been recently characterized in terms of a Kalman-like rank condition—the authors give a necessary and sufficient condition for the controllability of the coupled system under the form of a unique continuation property for the corresponding elliptic eigenvalue system.The proof relies on a compactness-uniqueness argument, which is quite unusual in the context of parabolic systems, previously developed for scalar parabolic equations. The general result is illustrated by two simple examples.     

Carleman estimates for stochastic parabolic equations with Neumann boundary conditions and applications

By a dual method, two Carleman estimates for forward and backward stochastic parabolic equations with Neumann boundary conditions are established. Then they are used to study a null controllability problem and a state observation problem for some stochastic forward parabolic equations with Neumann boundary conditions.     

Regional controllability of semilinear degenerate parabolic equations in bounded domains

In this paper we study controllability properties of semilinear degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold in general. Thus we investigate results of ‘regional null controllability’, showing that we can drive the solution to rest at time T on a subset of the space domain, contained in the set where the equation is nondegenerate.