Control of Coupled Systems

9:00--9:45: E. Crépeau (Univ. Versailles), Some results on approximate controllability of a reaction-diffusion system.

In this talk, we present some results obtained with C. Prieur on an approximate control for a linear reaction-diﬀusion system, coupling an ODE and a PDE. To design this open-loop control, we use a ﬂatness-like property, namely we express the solution in terms of an inﬁnite series depending on a ﬂat output, its derivatives and integrals. This series is shown to be convergent provided that the output is Gevrey of order 1 < a ≤ 2. Then we get the approximate control.

9:45--10:30: L. Tébou (FIU Miami), Desensitizing controls for some hyperbolic systems.

The notion of desensitizing control was introduced by J.-L. Lions in the late eighties in the framework of parabolic equations. A desensitizing control is a control built so that a certain norm of the state is insensitive to small variations of the data of the system under consideration. Many works followed that pioneering work of Lions in the same framework. As for hyperbolic equations, that concept is still at a very early stage. In this talk, we will discuss what is known at the present time (to the best of my knowledge) as far as hyperbolic equations are concerned, and raise some problems that we hope will help move forward.

10:30--11:00: Coffee Break

11:00--11:45: E. Fernandez-Cara (Univ. Seville), Reducing the number of controls in systems of Stokes and Navier-Stokes kind.

The general question in this talk is the following: Consider a controlled system of the Stokes or Navier-Stokes kind, where the state is given by the velocity ﬁeld and the pressure and maybe some additional variables like the temperature, a solute concentration, etc.; then, how many controls do we need to ensure a controllability property? We will ﬁrst deal with the approximate controllability of the 2D and 3D classical Stokes equations with distributed or boundary controls. We will then analyze the null controllability of these and other more complex systems with few controls. In particular, it will be seen that the problem is far from trivial when the control is exerted on (a part) the boundary.

11:45--12:30: H. Zwart (Univ. Twente), From hyperbolic to parabolic partial differential equations.

In many textbooks on partial diﬀerential equations hyperbolic and parabolic diﬀerential equations are treated diﬀerently. Since their behavior is diﬀerent, there is a good reason for doing so. In this presentation, we show that many (linear) parabolic diﬀerential equations can be seen as a (linear) hyperbolic diﬀerential equation together with a closure relation. This structure enables us to show existence of solutions for the parabolic equation by using existence results for the hyperbolic one. Furthermore, in some situations systems properties like controllability carries over. Finally, we show that the technique can be extended to a class of non-linear parabolic equations.

14:30--15:15: M. Gonzales-Burgos (Univ. Seville), Some recent results on controllability of coupled parabolic systems: towards a Kalman condition.

In the last ten years, the study of the controllability properties of coupled parabolic systems has had an increasing interest. It is well known that the controllability of the ordinary diﬀerential system z'=Az+Bv (n,m≥1, A in L(R^n) and B in L(R^m ;R^n) are given) is equivalent to the Kalman rank condition
(1) rank [A | B] = rank (B | AB | · · · | A^{n−1}B) = n.
In this talk we will present some recent results on controllability of coupled parabolic systems when the control is exerted in a part of the domain (distributed control) or on a part of the boundary of the domain (boundary control). In both cases, we will give a generalization of the algebraic condition (1) which characterizes the controllability properties of a class of parabolic systems. As a consequence of the previous result, we will also see that the distributed and boundary controllability properties of coupled parabolic systems are, in general, not equivalent.
References:
[1] F. Ammar-Khodja, A. Benabdallah, C. Dupaix, M. Gonzalez-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Diﬀer. Equ. Appl. 1 (2009), no. 3, 139–151.
[2] F. Ammar-Khodja, A. Benabdallah, C. Dupaix, M. Gonzalez-Burgos, A Kalman rank condition for the localized distributed control lability of a class of linear parabolic systems, J. Evol. Equ. 9 (2009), no. 2, 267–291, http://hal.archives-
ouvertes.fr/hal-00290867/fr/.
[3] F. Ammar-Khodja, A. Benabdallah, C. Dupaix, M. Gonzalez-Burgos, The Kalman condition for the boundary control lability of coupled parabolic systems. Bounds on biorthogonal families to matrix complex exponentials, In preparation.
[4] E. Fernandez-Cara, M. Gonzalez-Burgos, L. de Teresa, Boundary controllability of parabolic coupled equations, J. Funct. Anal. 259 (2010), no. 7, 1720–1758.

15:15--16:00: F. Alabau (Univ. Metz), Stabilization, control and observability of coupled systems of PDE's by a reduced number of controls or observations.

The purpose of this talk is to present several aspects and results on the stabilization, exact controllability and observability of coupled systems of PDE’s. We will focus on the indirect control and stabilization of such systems. This means that only a reduced number of components of the state equation are directly stabilized, controlled or observed, whereas we want to stabilize, control or observe the full state.
We will show that some general mathematical tools can be introduced to study these questions. In particular, we will show how stabilization –controllability or observability– properties are transferred from the damped – respectively controlled or observed– component to the undamped – resp. uncontrolled or unobserved– one through the coupling. Moreover, a key step is also to identify which type of stability –exponential, polynomial– properties can be expected, as well as in which functional spaces exact controllability or observability may hold. Such analysis may also depend in a complex way on compatibility properties of the involved PDE’s, as well as on properties of
the coupling and of the damping –resp. observability– operators. We will illustrate this complexity through several examples and classes of results.
The importance of reducing the number of controls or of actuators in applications in engineering for the control of complex mechanical structures for instance, or for applications in pollution with the null controllability of parabolic systems is one of the motivations for these studies.