Quantitative Fattorini-Hautus test and minimal null control time for parabolic problems

This paper investigates the link between the null controllability property for some abstract parabolic problems and an inequality that can be seen as a quantified Fattorini-Hautus test. Depending on the hypotheses made on the abstract setting considered we prove that this inequality either gives the exact minimal null control time or at least gives the qualitative property of existence of such a minimal time. We also prove that for many known examples of minimal time in the parabolic setting, this inequality recovers the value of this minimal time.Dans cet article nous étudions le lien entre la contrôlabilité à zéro d'un problème parabolique abstrait et la validité d'une inégalité qui est une version quantifiée du test de Fattorini–Hautus. Nous prouvons que cette inégalité permet de caractériser l'existence d'un temps minimal pour le problème de contrôlabilité à zéro et, selon les hypothèses considérées, d'obtenir la valeur de ce temps minimal. Nous prouvons aussi que dans la plupart des exemples connus de problèmes paraboliques ayant un temps minimal de contrôle à zéro, cette inégalité est une condition nécessaire et suffisante de contrôlabilité.Ministerio de Economía y Competitivida

Sharp estimates of the one-dimensional boundary control cost for parabolic systems

In this work we present new results on the cost of the boundary controllability of parabolic systems at time $T > 0$. In particular, we will study optimal estimates of the control cost at time $T$ ($T$ small enough) when the eigenvalues of the generator of the $C_0$ semigroup accumulate and do not satisfy a gap condition. The main ingredient we will use is the moment method combined with sharp estimates of the $L^2(0,T; \mathbb C)$-norm of the elements of biorthogonal families to complex exponentials.Comment: arXiv admin note: text overlap with arXiv:2401.1712

Controllability results for some nonlinear coupled parabolic systems by one control force

In this paper, we present new controllability results for some nonlinear coupled parabolic systems considered in a bounded domain Ω of IRN (with N ≥ 1 being arbitrary) when the control force acts on a unique equation of the system through an arbitrarily small open set ω ⊂ Ω. As a model example, we consider a nonlinear phase field system with certain superlinear nonlinearities and prove the null controllability, the exact controllability to the trajectories and the approximate controllability of the model. The crucial point in this paper is the new strategy developed to deal with the null controllability of linear coupled parabolic systems by a unique control force. Global Carleman estimates and the parabolic regularizing effect of the problem are used.Ministerio de Educación y Cienci

Controllability of some coupled parabolic systems by one control force

In this Note we present a new approach which allows one to prove new controllability results for some coupled parabolic systems considered in a bounded domain Ω of IRN when one controls by a unique distributed control. We analyze, as a model example, the null controllability of a linear phase field system. First, one controls the system by two controls. Then, one eliminates the introduced fictitious control. Global Carleman estimates and the parabolic regularity are used.Dans cette Note, on présente une nouvelle approche qui permet de prouver de nouveaux résultats de contrôlabilité pour quelques systèmes paraboliques couplés considerés dans un domaine borné Ω de RN et contrôlés par un seul contrôle distribué. On analyse, comme exemple modèle, la contrôlabilité nulle d'un système linéaire de champ de phases. D'abord, on contrôle le système par deux contrôles. Ensuite, on élimine le contrôle artificiel introduit. Des estimations globales de Carleman et la régularité parabolique sont employées

A result concerning controllability for the Navier-Stokes equations

The main goal of this paper is to present a new result concerning controllability of the time-dependent Navier-Stokes equations. Here, the control variable is the trace of the velocity field on a "small" part of the boundary. The main result states that the linear space spanned by final states is dense in the L space of admissible fields. For the proof, one uses a duality argument that is suggested by the linear theory. This reduces the task to an existence/regularity result for a nonlinear problem

Some recent results on controllability of coupled parabolic systems: towards a Kalman condition

Show the important differences between scalar and non scalar problems. Give necessary and sufficient conditions (Kalman condition) which characterize the controllability properties of these systems. We will only deal with “Simple" Parabolic Systems: Coupling Matrices of Constant Coefficients

Controllability of non-scalar parabolic systems: Some recent results and phenomena

In this course we will deal with non-scalar systems which in fact are coupled parabolic scalar equations. We do not present results relating to the controllability problems of systems which come from fluid mechanics as Stokes, Navier-Stokes, ..