A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems (original) (raw)

Recent results on the controllability of linear coupled parabolic problems: A survey

2011

This paper tries to summarize recent results on the controllability of systems of (several) parabolic equations. The emphasis is placed on the extension of the Kalman rank condition (for finite dimensional systems of differential equations) to parabolic systems. This question is itself tied with the proof of global Carleman estimates for systems and leads to a wide field of open problems.

On Algebraic condition for null controllability of some coupled degenerate systems

Mathematical Control & Related Fields

In this paper we will generalize the Kalman rank condition for the null controllability to n-coupled linear degenerate parabolic systems with constant coefficients, diagonalizable diffusion matrix, and m-controls. For that we prove a global Carleman estimate for the solutions of a scalar 2n-order parabolic equation then we infer from it an observability inequality for the corresponding adjoint system, and thus the null controllability.

A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups

Mathematics of Control, Signals, and Systems, 2017

In this article, we give a necessary and sufficient condition of Kalman type for the indirect controllability of systems of groups of linear operators, under some "regularity and locality" conditions on the control operator that will be made precise later and fit very well the case of distributed controls. Moreover, in the case of first order in time systems, when the Kalman rank condition is not satisfied, we characterize exactly the initial conditions that can be controlled. Some applications to the control of systems of Schrödinger or wave equations are provided. The main tool used here is the fictitious control method coupled with the proof of an algebraic solvability property for some related underdetermined system and some regularity results.

Null controllability of \varvec{n}n−coupleddegenerateparabolicsystemswithn -coupled degenerate parabolic systems withncoupleddegenerateparabolicsystemswith\varvec{m}$$ m -controls

Journal of Evolution Equations, 2017

In this paper we will analyze the null controllability properties of a linear coupled degenerate parabolic system of n equations when m distributed controls are exerted on the system. First we start with the case when the coupling matrix A is cascade, and then when A is a full matrix, we will prove that the Kalman rank condition on the coupling and the control matrices A and B characterizes the controllability properties of the system. 1 i n y i (0, x) = y 0,i (x) 1 i n in (0, 1).

Controllability of linear and semilinear non-diagonalizable parabolic systems

ESAIM: Control, Optimisation and Calculus of Variations, 2015

This paper is concerned with the controllability of some (linear and semilinear) nondiagonalizable parabolic systems of PDEs. We will show that the well known null controllability properties of the classical heat equation are also satisfied by these systems at least when there are as many scalar controls as equations and some (maybe technical) conditions are satisfied. We will also show that, in some particular situations, the number of controls can be reduced. The minimal amount is then determined by a Kalman rank condition.

Controllability for a class of reaction–diffusion systems: the generalized Kalman's condition

Comptes Rendus Mathematique, 2007

In this article, we study the controllability of a class of parabolic systems of the form Yt = (D∆+A)Y +Bχωu with Dirichlet conditions on the boundary of a bounded domain Ω, where ω ⊂ Ω is a subdomain. Here D, A ∈ L(R n ), B ∈ L(R m ; R n ) and we prove that the algebraic Kalman condition extends to such systems. To cite this article: A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser.

Controllability results for cascade systems of m coupled parabolic PDEs by one control force

Portugaliae Mathematica, 2000

In this paper we will analyze the controllability properties of a linear coupled parabolic system of m equations when a unique distributed control is exerted on the system. We will see that, when a cascade system is considered, we can prove a global Carleman inequality for the adjoint system which bounds the global integrals of the variable ϕ = (ϕ1, . . . , ϕm) * in terms of a unique localized variable. As a consequence, we will obtain the null controllability property for the system with one control force. . 93B05, 93B07, 35K50.