Control Of Linear Parabolic Equations Research Articles

Optimal distributed control of linear parabolic equations by spectral decomposition

AbstractWe construct an algorithm for solving a constrained optimal control problem for a first‐order evolutionary system governed by a positive self‐adjoint operator. The problem consists in identifying distributed control that minimizes a given cost functional, which comprises a cost of the control and a trajectory regulation term, while steering the final state close to a given target. The approach explores the dual problem and it generalizes the Hilbert Uniqueness Method (HUM). The practical implementation of the algorithm is based on a spectral decomposition of the operator determining the dynamics of the system. Once this decomposition is available – which can be done offline and saved for future use – the optimal control problem is solved almost instantaneously. It is practically reduced to a scalar nonlinear equation for the optimal Lagrange multiplier. The efficiency of the algorithm is demonstrated through numerical examples corresponding to different types of control operators and penalization terms.

New results on the Stackelberg–Nash exact control of linear parabolic equations

This paper is concerned with Stackelberg–Nashstrategies to control parabolic equations. We have one control, the leader, that is responsible for a null controllability property; additionally, we have a couple of controls, called the followers, that provides a Nash equilibrium for two cost functionals. This is a classical situation in many fields of science and, in mathematics, leads to a lot of interesting questions and open problems and possesses many applications. In the main result, we prove the existence of a leader such that the corresponding controlled system is driven to zero. This way, we improve some questions that were left open in previous works.

A residual based snapshot location strategy for POD in distributed optimal control of linear parabolic equations

In this paper we study the approximation of a distributed optimal control problem for linear parabolic PDEs with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the basis functions are obtained upon information contained in time snapshots of the parabolic PDE related to given input data. In the present work we show that for POD-MOR in optimal control of parabolic equations it is important to have knowledge about the controlled system at the right time instances. For the determination of the time instances (snapshot locations) we propose an a-posteriori error control concept which is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system. Finally, we present numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches.

Carleman Estimate for Elliptic Operators with Coefficients with Jumps at an Interface in Arbitrary Dimension and Application to the Null Controllability of Linear Parabolic Equations

In a bounded domain of R n+1, n ≧ 2, we consider a second-order elliptic operator, $${A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}$$ , where the (scalar) coefficient c(x) is piecewise smooth yet discontinuous across a smooth interface S. We prove a local Carleman estimate for A in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient c at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions and the Calderón projector technique. Following the method of Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) we then prove the null controllability for the linear parabolic initial problem with Dirichlet boundary conditions associated with the operator $${{\partial_t - \nabla_x \cdot (c(x) \nabla_x)}}$$ .

On the controllability of linear parabolic equations with an arbitrary control location for stratified media

We prove a null controllability result with an arbitrary control location in dimension greater than or equal to two for a class of linear parabolic operators with non-smooth coefficients. The coefficients are assumed to be smooth in all but one directions. To cite this article: A. Benabdallah et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Approximate Controllability of linear parabolic equations in perforated domains

In this paper we consider an approximate controllability problem for linear parabolic equations with rapidly oscillating coefficients in a periodically perforated domain. The holes are e -periodic and of size e . We show that, as e → 0, the approximate control and the corresponding solution converge respectively to the approximate control and to the solution of the homogenized problem. In the limit problem, the approximation of the final state is alterated by a constant which depends on the proportion of material in the perforated domain and is equal to 1 when there are no holes. We also prove that the solution of the approximate controllability problem in the perforated domain behaves, as e → 0, as that of the problem posed in the perforated domain having as rigth-hand side the (fixed) control of the limit problem.