DCDS-S
Motivation, analysis and control of the variable density Navier-Stokes equations
Enrique Fernández-Cara
The main objective of these Notes is to provide an introduction to variable density NS: their motivation, some of the main mathematical problems connected with them, the main techniques used to solve these problems, the main results and open questions. First, we will describe the physical origin of the equations. Then, we will be concerned with existence, uniqueness, regularity and control of initial-boundary value problems in cylindrical domains $ Ω $ $\times (0,T)$; as usual, $ Ω $ is the spatial domain, an open set in $\mathbb{R}$2 or $\mathbb{R}$3 ``filled'' by the fluid particles and (0,T) is the time observation interval. Some open problems (not all them of the same difficulty) are also recalled.
keywords: controllability. Navier-Stokes equations uniqueness and regularity existence optimal control weak and strong solutions variable density
MCRF
Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods
Enrique Fernández-Cara Arnaud Münch
This paper deals with the numerical computation of distributed null controls for semi-linear 1D heat equations, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has been obtained in [Fernandez-Cara $\&$ Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, 2000] via a fixed point reformulation; see also [Barbu, Exact controllability of the superlinear heat equation, 2000]. More precisely, Carleman estimates and Kakutani's Theorem together ensure the existence of solutions to fixed points for an equivalent fixed point reformulated problem. A nontrivial difficulty appears when we want to extract from the associated Picard iterates a convergent (sub)sequence. In this paper, we introduce and analyze a least squares reformulation of the problem; we show that this strategy leads to an effective and constructive way to compute fixed points. We also formulate and apply a Newton-Raphson algorithm in this context. Several numerical experiments that make it possible to test and compare these methods are performed.
keywords: numerical solution One-dimensional semi-linear heat equation null controllability blow up least squares method.
MCRF
On the control of some coupled systems of the Boussinesq kind with few controls
Enrique Fernández-Cara Diego A. Souza
This paper is devoted to prove the local exact controllability to the trajectories for a coupled system, of the Boussinesq kind, with a reduced number of controls. In the state system, the unknowns are the velocity field and pressure of the fluid $(\mathbf{y},p)$, the temperature $\theta$ and an additional variable $c$ that can be viewed as the concentration of a contaminant solute. We prove several results, that essentially show that it is sufficient to act locally in space on the equations satisfied by $\theta$ and $c$.
keywords: Navier-Stokes and Boussinesq-like systems control reduction. controllability
IPI
Some geometric inverse problems for the linear wave equation
Anna Doubova Enrique Fernández-Cara
In this paper we consider some geometric inverse problems for the linear wave equation. We prove uniqueness results, we present some reconstruction algorithms and we perform numerical experiments in dimensions one and two.
keywords: Inverse problems reconstruction numerical solution elastography. shape identification Freefem++ wave equation
DCDS
Null controllability of a cascade system of parabolic-hyperbolic equations
Enrique Fernández-Cara Luz de Teresa
This paper is concerned with the null controllability of a cascade linear system formed by a heat and a wave equation in a cylinder $\Omega \times (0,T)$. The control acts only on the heat equation and is supported by a set of the form $\omega \times (0,T)$, where $\omega \subset \Omega$. In the wave equation, only the restriction of the solution to the heat equation to another set $\mathcal O \times (0,T)$ appears. In the main result in this paper, we show that, under appropriate assumptions on $T$, $\omega$ and $\mathcal O$, the system is null controllable.
keywords: cascade system parabolic-hyperbolic. Null controllability
DCDS-B
A geometric inverse problem for the Boussinesq system
A. Doubov Enrique Fernández-Cara Manuel González-Burgos J. H. Ortega
In this work we present some results for the inverse problem of the identification of a single rigid body immersed in a fluid governed by the stationary Boussinesq equations. First, we establish a uniqueness result. Then, we show the way the observation depends on perturbations of the rigid body and we deduce some consequences. Finally, we present a new method for the partial identification of the body assuming that it can be deformed only through fields that, in some sense, are finite dimensional. In the proofs, we use various techniques, related to Carleman estimates, differentiation with respect to domains, data assimilation and controllability of PDEs.
keywords: inverse problem data assimilation Boussinesq system differentiation with respect to domains controllability.
CPAA
Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations
Enrique Fernández-Cara Manuel González-Burgos Luz de Teresa
This paper is concerned with the null-exact controllability of a cascade system formed by a semilinear heat and a semilinear wave equation in a cylinder $\Omega \times (0,T)$. More precisely, we intend to drive the solution of the heat equation (resp. the wave equation) exactly to zero (resp. exactly to a prescribed but arbitrary final state). The control acts only on the heat equation and is supported by a set of the form $\omega \times (0,T)$, where $\omega \subset \Omega$. In the wave equation, the restriction of the solution to the heat equation to another set $\mathcal O \times (0,T)$ appears. The nonlinear terms are assumed to be globally Lipschitz-continuous. In the main result in this paper, we show that, under appropriate assumptions on $T$, $\omega$ and $\mathcal O$, the equations are simultaneously controllable.
keywords: observability inequalities. Semilinear systems parabolic-hyperbolic equations controllability
DCDS
Analysis and optimal control of some solidification processes
Roberto C. Cabrales Gema Camacho Enrique Fernández-Cara
In this paper we consider a mathematical model that describes the solidification of a binary alloy. We prove some existence and uniqueness results for a regularized problem, depending on a small parameter $\epsilon$. We also analyze the behavior of the regularized solutions as $\epsilon \to 0$. Then, we consider some associated optimal control problems. We prove existence and optimality results and we present and discuss some iterative methods.
keywords: weak solutions Nonlinear PDEs regularization optimal control. solidification models
MCRF
Optimal control of a two-equation model of radiotherapy
Enrique Fernández-Cara Juan Límaco Laurent Prouvée

This paper deals with the optimal control of a mathematical model for the evolution of a low-grade glioma (LGG). We will consider a model of the Fischer-Kolmogorov kind for two compartments of tumor cells, using ideas from Galochkina, Bratus and Pérez-García [10] and Pérez-García [17]. The controls are of the form $(t_1, \dots, t_n; d_1, \dots, d_n)$, where $t_i$ is the $i$-th administration time and $d_i$ is the $i$-th applied radiotherapy dose. In the optimal control problem, we try to find controls that maximize, in an admissible class, the first time at which the tumor mass reaches a critical value $M_{*}$. We present an existence result and, also, some numerical experiments (in the previous paper [7], we have considered and solved a very similar control problem where tumoral cells of only one kind appear).

keywords: Fischer-Kolmogorov model optimal control tumor growth radiotherapy

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