Schedule for: 17w5123 - Women in Control: New Trends in Infinite Dimensions

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

First, I will present work on optimal control of parabolic PDEs applied to the ecological issue of population movement and its distribution in reaction to resources and to competition. We present the choices of directed movement through controlling the advective coefficients in a parabolic system, modeling two competing species. Then, optimal control of parabolic PDEs will presented as an approximation to control agent-based models. Besides presenting some control analysis setup, some numerical examples will be given for illustration.

In this talk, we consider the controllability of the Gear-Grimshaw system with different combination of the control inputs. The study is made through the duality-compactness approach together with some hidden regularity properties of the boundary terms by extending the results of a type of Boundary-Value-Problem for the KdV equation.

This talk is concerned with the optimal control of a 1D cubic nonlinear Schr\"odinger equation that describes the propagation of optical pulses. We consider the noise on an optical transmission systems as a control variable and study the existence of a minimum norm control such that the pulse is degraded at the end of the transmission (integral restriction on the state). We also give first order necessary conditions for an optimal solution.

In this talk, we will discuss decay properties of solutions to wave equations in a bounded domain with two types of dissipative mechanisms, i.e. either with a small boundary or an internal damping. When the Geometric Control Condition on the dissipative region is not satisfied, we show that sufficiently smooth solutions to the equations decay logarithmically, under sharp regularity assumptions on the coefficients, the damping and the boundary of the domain involved in the equations.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We discuss the asymptotic stability as well as the well-posedness of a transmission problem in bounded domains with dissipative internal conditions. The results presented in this talk have been obtained in collaboration with M. Cavalcanti and E. R. de Sousa Coelho.

One of the main goal in control theory is to drive the state of the system to a given configuration using a control that act through a source term located inside the domain or through a boundary condition. The reference works for the control of linear parabolic problems are due to H.O.~Fattorini and D.L.~Russell in the 70's, \cite{FR1} for the one dimensional case and to A.V.~Fursikov, O.Yu.~Imanuvilov, \cite{FI} on one side and G.~Lebeau, L.~Robbiano, \cite{LR} on the other side both in the 90's for the multi-dimensional case. They established null-controllability of heat equations with distributed or boundary controls in any time and for any control domain. The aim of this talk is to give an overview on the recent results on the controllability of parabolic {\bf{systems}}. Through simple examples, I will show that new phenomena appear as minimal time of control, dependance on the location of the control. \subsection{Results} In this talk, I will focus on controllability of two simple examples in order to illustrate theses unexpected behavior in control of parabolic systems. \subsection{Boundary control for parabolic systems} $$ \label{Pbi} \left\{ \begin{array}{ll} \displaystyle \partial_t y - \left( D \partial^2_{xx} + A\right) y=0 & \hbox{in }Q=(0,\pi )\times (0,T), \\ \noalign{\smallskip} y(0,\cdot ) = Bv, \quad y(\pi ,\cdot )=0 & \hbox{on }(0,T), \\ \noalign{\smallskip} y(\cdot ,0)=y^{0}\ & \hbox{in }(0,\pi ), \end{array}% \right. $$ where $T>0$ is a given time, \begin{equation*} D = \left( \begin{array}{cc} 1 & 0 \\ 0 & d \end{array} \right) \ (\hbox{with }d>0), \quad A=\left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right) ,\quad B=\left(\begin{array}{cc}0\\1\end{array}\right)\in \mathcal M_{2,1}(\R) \end{equation*} \begin{theorem}[F.~Ammar Khodja, A.~Benabdallah, M.~Gonz\'alez-Burgos, L.~de Teresa, \cite{JFA}] \vskip 0.3cm Let $d\neq 1$ \begin{enumerate} \item $ \forall T>0:$ Approximate controllability if and only if $\sqrt{d}\not\in \Q $. \item $\exists T_0=c(\Lambda)\in [0,+\infty]$ such that \begin{enumerate} \item The system is null controllable at time $T$ if $\sqrt{d}\not\in \Q $ and $T>T_0$. \item Even if $\sqrt{d}\not\in \Q $, if $T0$, one has: \begin{enumerate} \item Assume that $T>T_0(q)$. Then, system~\eqref{s2x2} is null controllable at time $T$. \item If $T (TCPL 201)

A regulatory genes network plays an important role in the whole life process, including cellular differentiation, metabolism, cell cycle, signal transduction (i.e., when a cell converts one type of signal or stimulus into another), etc. In this work, it is considered a simplified biological model incorporating the genes FlbA, FlbB, FlbC, FlbD, brlA and PkaA, which act in the cell division of some types of fungi, and three external controls. This model is given by a system of initial and boundary conditions containing coupled nonlinear parabolic partial differential equations. The main result is, under certain conditions, the exact local controllability to a constant suitable trajectory by using less control functions than equations and these functions having arbitrarily small support. The main difficulties of this work are to deal with the nonlinearities of the equations and of the fact that we have less control functions than equations. The idea to deal with this problem is: to use Kalman rank condition to obtain the exact controllability for a homogeneous linearized system; the ideas of~\cite{CoronLissy} to obtain the exact controllability for a non-homogeneous linearized system; and the Liusternik's Inverse Mapping Theorem to obtain the local exact controllability for the non linear system. Work in collaboration with: Enrique Fern\'andez-Cara and Andr\'e R. Lopes. \begin{thebibliography}{00} \bibitem{AmmarKhodja} F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. Gonz\'alez-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems, {\it Journal of Evolution Equations}, volume~9 (2), 2009, pp 267-291.%DOI: 10.1007/s00028-009-0008-8. \bibitem{CoronLissy} J. M. Coron, P. Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, {\it Inventiones Mathematicae}, volume~198, Issue~3, pp 833-880. \end{thebibliography}

We consider optimization and optimal control problems subject to free and moving boundary nonlinear fluid-elasticity interactions. As the coupled fluid-structure state is the solution of a system of partial differential equations that are coupled through continuity relations on velocities and normal stress tensors, defined on the free and moving interface, the investigation (existence of optimal controls, sensitivity equations, necessary optimality conditions, etc.) is heavily dependent on the geometry of the problem, and falls into moving shape analysis framework.

We study well-posedness and asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and a classical (nonlinear) full von Karman plate equations that accounts for both transversal and lateral displacements on a flexible part of the boundary. Rotational inertia of filaments of the shell is not taken into account. We also do not assume any mechanical damping acting on the palte equations. Our main result shows well-posedness of strong solutions to the problem, thus the problem generates a semiflow in an appropriate phase space. We also prove uniform stability of strong solutions to homogeneous problem in the norm of strong phase space. Thus, viscous energy dissipation in the fluid component is sufficient to stabilize the whole system plate+fluid.

The Jordan-Moore-Gibson-Thompson equation is a nonlinear Partial Differential Equation (PDE) model which describes the acoustic velocity potential in ultrasound wave propagation; the use of the constitutive Cattaneo law for the heat flux, in place of the Fourier law, accounts for its being of third order in time. Aiming at the understanding of the fully quasilinear PDE, a great deal of attention has been recently devoted to its linearization -- referred to in the literature as the Moore-Gibson-Thompson (MGT) equation --, whose mathematical analysis is also of independent interest, posing already several questions and challenges. Semigroup well-posedness, a threshold for uniform stability (depending on physical parameters), a careful spectral analysis for the MGT equation, supplemented with Dirichlet or Neumann boundary conditions, are obtained in the original papers by Kaltenbacher, Lasiecka and Marchand (2012) and by Marchand, McDevitt and Triggiani (2011). In this lecture we will report recently obtained results concerning (i) an appropriate optimal control problem associated with the MGT equation, and (ii) the regularity of solutions corresponding to non-homogeneous boundary data. As for (i), by following a similar approach than the one used in the nineties by Lasiecka, Lukes and Pandolfi for the strongly damped wave equation with boundary control -- that brings about the very same pattern -- we attain a feedback synthesis of the optimal control as well as well-posed operator Riccati equations, in spite of the unfavourable hyperbolic character of the dynamics. (In this connection, if useful to the audience, a brief overview of the (by now, classical) linear-quadratic theories for single PDE of either parabolic or hyperbolic type, yet not applicable in the present case, will be provided.) As for (ii), the perspective of viscoelasticity -- intrinsic to the MGT equation --, brings about a method of proof for the regularity of solutions to the MGT equation. (The talk is based on ongoing joint work with Irena Lasiecka (University of Memphis, USA) for the former part and with Luciano Pandolfi (Politecnico di Torino, Italy) for the latter.)

The conference focuses on how coupling different evolution equations with integro-differential terms can bring about different reachability results. The presence of integral terms, the so-called memory, is justified by the classical constitutive law introduced to yield a representation of viscoelastic materials. Another physical property for viscoelastic materials consists in assuming that memory fades in time. For this reason in our study, we consider a class of integral kernels that satisfy the principle of fading memory. The talk is based on findings obtained in collaboration with Paola Loreti.

We will deal with the numerical approximation of some geometric inverse problems for the wave and the Lam\'e equations motivated by Elastography. We present several recent results and open questions concerning the numerical reconstruction of the unknown domain where the equations evolve. In the numerical experiments, we solve appropriate optimization problems. Two different numerical techniques will be proposed. Firstly, the finite element method for the numerical solution of the PDE's, that will be performed with \texttt{FreeFem++}. The routines on the \texttt{ff-NLopt} package, that provide an interface to a free/open-source library for nonlinear optimization, are also required. On the other hand, we will consider the numerical approximation based on the method of fundamental solutions. We present some numerical results in the 2D and 3D cases. The first part is joint work with E. Fern\'andez-Cara (Dpto.\ E.D.A.N., Universidad de Sevilla, [email protected]) and the second part is joint work with E. Fern\'andez-Cara, Jairo Rocha de Faria (Universidade Federal da Paraiba, [email protected]) y Pitágoras P. de Carvalho (Universidad Federal Fluminense (Niteroi), [email protected]).

We apply the optimal control theory to Elliptic and Parabolic partial differential equations. We study the control problem of maximizing the net benefit in the conservation of a single species with a fixed amount of resources. We also consider an optimal control problem of a system of parabolic partial differential equations modeling the competition between an invasive and a native species. The existence of an optimal control is established and the uniqueness and characterization of the optimal control are investigated for both applications. Numerical simulations illustrate several cases. Some open problems are discussed.

In this talk we study the notions of input to state stability (ISS) and integral input to state stability (iISS) for boundary control systems, which are stronger notions than exponential stability of the corresponding semigroup and include stability with respect to input functions as well. It will be shown that if the semigroup is exponentially stable, then ISS is equivalent to admissibility of the input operator with respect to $L^\infty$ . Further, under the assumption of exponential stability iISS is just admissibility of the input operator with respect to an Orlicz space. Further, we prove that for parabolic systems ISS and iISS are equivalent notions.

We discuss the problem of optimal mixing of an inhomogeneous distribution of a scalar field via an active control of the flow velocity, governed by Stokes or Navier-Stokes equations, in a two dimensional open bounded and connected domain. The problem is motivated by mixing the fluids within a cavity or vessel by moving the walls or stirring at the boundaries. It is natural to consider the velocity field that is induced by a control input tangentially acting on the boundary of the domain through the Navier slip boundary conditions. Our main objective is to design an optimal Navier slip boundary control that optimizes mixing at a given final time. This essentially leads to a finite time optimal control problem of a bilinear system. A rigorous proof of the existence of an optimal controller and the first-order necessary conditions for optimality are presented.

Authors: IRENA LASIECKA, KATARZYNA SZULC, ANTONI ZOCHOWSKI We consider a coupled system of the linearly elastic body immersed in the flowing fluid which is modeled by means of incompressible Navier-Stokes equations. For this system we formulate an optimal control problem which amounts to a minimization of a hydro-elastic pressure on the interface between the two environments. The corresponding functional lacks convexity and radial coercivity. The approach taken is based on transforming the variable domain occupied by the fluid to the fixed one corresponding to the undeformed elastic inclusion. This leads to a free boundary elliptic problem. Mathematical challenge also results from the fact that the corresponding quasillinear elliptic model is equipped with mixed (Zaremba type) boundary conditions, which intrinsincly lead to compromised regularity of elliptic solutions. It is shown that under the assumption of small strains the controlled structure is wellposed in suit- able Sobolev?s spaces and the nonlinear control to state map is well defined and continuous. The obtained wellposedness result provides a foundation for proving an existence of optimal control where the latter is based on compensated compactness methods. Acknowledgment: This research was founded by the Polish National Science Center (NCN), grant Opus. Agreement UMO-2014/15/B/ST1/00067.

We consider a full Von Karman system accounting for in-plane acceleration and nonlinear thermal effects. Our model features free boundary conditions and an internal damping term on the in-plane displacement. We will discuss wellposedness of regular and weak solutions, and obtain uniform decay rates for the energy function.

In this talk we explore the methodology of model order reduction based on singular perturbations for a fexible-joint robot within the port-Hamiltonian framework. The model is an ode model that is obtained after discretisation. We show that a fexible-joint robot has a port-Hamiltonian representation which is also a singularly perturbed ordinary differential equation. Moreover, the associated reduced slow subsystem corresponds to a port-Hamiltonian model of a rigid-joint robot. To exploit the usefulness of the reduced models, we provide a numerical example where an existing controller for a rigid robot is implemented. In addition, we provide ideas on how to expand this to planar slow-fast systems at a non-hyperbolic point. At these type of points, the classical theory of singular perturbations is not applicable and new techniques need to be introduced in order to design a controller that stabilizes such a point. We show for some class of nonlinear systems that using geometric desingularization (also known as blow up), it is possible to design, in a simple way, controllers that stabilize non-hyperbolic equilibrium points of slow-fast systems. Furthermore, we include controller design in the development.

Given an evolutionary system on some bounded domain, we seek to drive the system to rest by acting on the boundary. Among the different approaches to achieve this goal, we consider here a spectral method, based on trigonometrical inequalities. These inequalities lead to observability estimates for the solutions of the associated systems, and then a duality relation yields controllability results for the original problem.The method is illustrated by applications to square and circular membranes.

In this paper, we study a fluid-structure interaction model of Stokes-wave equation coupling system with Kelvin-Voigt type of damping. We show that this damped coupling system generates an analyticity semigroup and thus the semigroup solution, which also satisfies variational framework of weak solution, decays to zero at exponential rate.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.