DCDS-S

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.

MCRF

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.

MCRF

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 $c$.

IPI

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.

DCDS

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.

DCDS-B

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.

CPAA

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.

DCDS

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.

MCRF

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).