# Control and Stabilization for Partial Differential Equations

Organizadores: Fágner D. Araruna (fagner@mat.ufpb.br), Eduardo Cerpa (eduardo.cerpa@usm.cl), Luz de Teresa (deteresa@matem.unam.mx)

• #### Lunes 13

15:00 - 15:45

#### Small-time global exact controllability to the trajectories of the Boussinesq system

Felipe Chaves (Universidade Federal de Paraiba, Brasil)

In this talk, we consider the global exact controllability problem to the trajectories of the Boussinesq system. We show that it is possible to drive the solution to the prescribed trajectory in small time by acting on the system through the velocity and the temperature on an arbitrary small part of the boundary. The proof relies on three main arguments. First, we transform the problem into a distributed controllability problem by using a domain extension procedure. Then, we prove a global approximate controllability result by following a strategy of Coron and collaborators, which deals with the Navier-Stokes equations. This part relies on the controllability of the inviscid Boussinesq system and asymptotic boundary layer expansions. Finally, we conclude with a local controllability result that we establish with the help of a linearization argument and appropriate Carleman estimates.

15:45 - 16:30

#### Stabilization Aspects of the Boussinesq System on a Bounded Interval

Fernando A. Gallego (Universidad Nacional de Colombia, Colombia)

In this talk, we present some recent results related to the rapid boundary stabilization for the Boussinesq System of the KdV-KdV Type on a Bounded interval introduced by J. Bona, M. Chen and J.-C. Saut: $$\left\{ \begin{array} [c]{l}% \eta_{t}+v_{x}+( \eta v) _{x}+av_{xxx}-b\eta_{xxt}=0\text{,}\\ v_{t}+\eta_{x}+vv_{x}+c\eta_{xxx}-dv_{xxt}=0. \end{array} \right.$$

This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet--Neumann boundary controls acting at the right end point of the interval. Firstly, we build suitable integral transformations to get a feedback control law that leads to the stabilization of the system. More precisely, we will prove that the solution of the nonlinear closed-loop system decays exponentially to zero in the $$L^2(0,L)$$--norm and the decay rate can be tuned to be as large as desired if the initial data is small enough under the effects of two boundary feedback. Moreover, by using a Gramian-based method introduced by Urquiza to design our feedback control, we show that the solutions of the linearized system decay uniformly to zero when the feedback control is applied. The decay rate can be chosen as large as we want. The main novelty of our work is that we can exponentially stabilize this system of two coupled equations using only one scalar input.

16:45 - 17:30

#### On the controllability of a model system for long waves in nonlinear dispersive media

Considered here is a higher order generalization of the classical Boussinesq system which models the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal or the propagation of long-crested waves on large lakes and oceans. Our aim is to investigate the controllability properties of this nonlinear model in terms of the values of the parameters involved in the system. We give general conditions which ensure both the well-posedness and the local exact controllability of the nonlinear problem in some well chosen Sobolev spaces.

Joint work with G. J. Bautista Sanchez (Universidad Privada del Norte, Peru) and S. Micu (University of Craiova, Romania).

17:30 - 18:15

#### A unified strategy for observability of waves with several boundary conditions

We study the controllability of the wave equation acting on an annulus $$\Omega \subset \mathbb{R}^2$$, i.e. $$\Omega = B_{R_1} \setminus B_{R_0}$$, where $$0 \lt R_0 \lt R_1$$ and $$B_R$$ denotes the ball in $$\mathbb{R}^2$$ with center at the origin and radius $$R >0$$. We are interested in the case of a single control $$h$$ acting on the exterior part of the boundary, with a given boundary condition imposed on the interior boundary: $$\left\{ \begin{array}{ll} \partial_{tt} y - \Delta y = 0, & \hbox{ in } (0,T) \times \Omega, \\ {\mathcal B}y = 0, & (0,T) \times \partial B_{R_0}. \\ y(t,x) = {h(t,x)}, & \hbox{ on } (0,T) \times \partial B_{R_1}. \end{array} \right.$$

We provide a robust strategy to prove the observability of this system with any of several boundary conditions on the internal boundary $$(0,T) \times \partial B_{R_0}$$, as Fourier conditions, dynamic conditions, conditions with the fractional Laplacian or fluid-structure models.

The main tool we use is given by resolvent characterizations, and the use of adequate estimates. We prove observability estimates with observations performed on the whole external boundary, which are valid for all the mentioned cases for boundary conditions at $$\partial B_{R_0}$$. We will mention also recent related results concerning wave and Schrodinger equations in more general domains with the same topological structure.

• #### Martes 14

15:00 - 15:45

#### Controllability from the exterior of fractional heat equation

Sebastián Zamorano Aliaga (Universidad de Santaigo de Chile, Chile)

In this talk, we consider the controllability problem from the exterior for the one dimensional heat equation on the interval $$(-1,1)$$ associated with the fractional Laplace operator $$(-\partial_x^2)^s$$, where $$0\lt s\lt1$$.

In the first part, we will show that there is a control function which is localized in a nonempty open set $$\mathcal O\subset \left(\mathbb{R}\setminus(-1,1)\right)$$, that is, at the exterior of the interval $$(-1,1)$$, such that the system is null controllable at any time $$T \gt 0$$ if and only if $$\frac 12 \lt s \lt1$$.

Then, in the second part, we study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator on the interval $$(-1,1)$$. Our control function is localized in an open set $$\mathcal O$$ in the exterior of $$(-1,1)$$. We will show that there exists a minimal (strictly positive) time $$T_{\rm min}$$ such that the fractional heat dynamics can be controlled from any initial datum in $$L^2(-1,1)$$ to a positive trajectory through the action of an exterior positive control, if and only if $$\frac 12 \lt s \lt 1$$. In addition, we prove that at this minimal controllability time, the constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. Finally, we provide several numerical illustrations that confirm our theoretical results.

15:45 - 16:30

#### Nematic liquid crystals: well posedness, optical solitons and control

Constanza Sánchez de la Vega (Universidad de Buenos Aires, Argentina)

In this talk we present results on well-posedness, decay, soliton solutions and control of the coupled nonlinear Schr\"odinger (NLS) equation \begin{align*} & \partial_{z}u= \frac{1}{2} \mathrm{i} \nabla^{2} u+ \mathrm{i} \gamma (\sin^2(\psi+\theta_0)-\sin^2(\theta_0)) u,\\ &\nu \nabla^{2} \psi= \frac{1}{2}E_0^2\sin(2\theta_0)-\frac{1}{2}(E_0^2+|u|^{2})\sin(2(\psi+\theta_0)), \end{align*} where $$u$$ and $$\psi$$ depend on the optical axis'' coordinate $$z \in \mathbb{R}$$, and the transverse coordinates'' $$(x, y) \in \mathbb{R}^2$$. Also $$\nabla^{2} = \partial_x^2 + \partial_y^2$$ is the Laplacian in the transverse directions, $$E_0$$, $$\nu$$ and $$\gamma$$ are positive constants, and $$\theta_0$$ is a constant satisfying $$\theta_0 \in (\pi/4, \pi/2)$$. The model arises in the study of optical beam propagation in nematic liquid crystals, and in particular a set of experiments by Assanto and collaborators since 2000. The complex field $$u$$ represents the electric field amplitude of a linearly polarized laser beam that propagates through a nematic liquid crystal along the optical axis $$z$$. The elliptic equation describes the effects of the beam electric field on the local orientation(director field) of the nematic liquid crystal and has an important regularizing effect, seen experimentally and understood theoretically in related models. The director field'' $$\psi + \theta_0$$ is a field of angles that describes the macroscopic orientation of the nematic liquid crystal molecules. The laser beam causes an additional deviation $$\psi$$ in the orientation of the liquid crystal molecules.

In this talk we will show well posedness of the coupled system, existence of stationary solutions and a saturation'' effect consistent with a bound $$\psi+\theta_0 < \pi/2$$ on the total angle. Finally, we will discuss some recent results concerning an optimal control problem where the external electric field varying in time is the control.

Work in collaboration with: Juan Pablo Borgna, Panayotis Panayotaros, Diego Rial.

16:45 - 17:30

#### Internal controllability of the Kadomtsev-Petviashvili II equatio

Ivonne Rivas (Universidad del Valle, Colombia), joint with Chenmin Sun

In this talk, we present the internal control problem for the Kadomstev-Petviashvili II equation, better known as KP-II. The problem is studied first when the equation is set in a vertical strip proving by the Hilbert Unique Method and semiclassical techniques proving the internal controllability and second, in a horizontal strip where the controllability in $$L^2(\mathbb{T})$$ cannot be reached.

17:30 - 18:15

#### Identification of a boundary obstacle in a Stokes fluid with Navier–slip boundary conditions: an exterior approach

Pedro González Casanova (Universidad Nacional Autónoma de México, México)

The problem of identifying an obstruction into a fluid duct has several major applications, for example in medicine the presence of a stenosis in a coronary vessels is a life threaten disease. In this talk we formulate a continuous setting and study from a numerical perspective the inverse problem of identifying an obstruction contained in a 2D elastic duct where a Stokes flow becomes turbulent after hitting the boundary (Navier--slip boundary conditions), generating an acoustic waves. To be precise, by using acoustic wave measurements at certain points at the exterior to the duct, we are able to identify the location; extension and height of the obstruction. Thus, our framework constitutes an external approach for solving this obstacle inverse problem. Synthetic examples are used in order to verify the effectiveness of the proposed numerical formulation.

In collaboration with: L. Breton, J. López Estrada and C. Montoya