# IFIP TC7 WG7.2 – Virtual Seminar Series

Organizers: Lorena Bociu (NC State University) and George Avalos (University of Nebraska-Lincoln)

We are pleased to announce the initiation of our **IFIP TC 7.2. Virtual Seminar Series**, which will run on a monthly basis. The purpose of this series is to promote the exchange of ideas, experiences and recent developments of researchers from all nations, during these difficult times when we cannot travel and interact in person. More information on IFIP TC 7 and its working groups can be found at https://ifip-tc7.impan.pl/ .

All are welcome to join us via Zoom. If interested, please email me for the Zoom link.

**SCHEDULE:**

- April 5, 2022, at 11am EST USA

**Professor Jean-Luc Guermond**

(Texas A&M)

https://www.math.tamu.edu/~guermond/

Title: Invariant-domain preserving high-order implicit-explicit time stepping: Beyond the SSP paradigm

Abstract: Considering systems of (first-order) nonlinear partial differential equations with invariant-domain properties, the first question addressed in the talk consists of constructing explicit and high-order accurate time stepping techniques that preserve the invariant-domain property of these systems. One well-known class of methods meeting this goal consists of the so-called strong stability preserving Runge-Kutta methods. In this work we go beyond the strong stability preserving paradigm and develop a time stepping technique that makes every explicit Runge-Kutta method invariant-domain preserving. The technique is space discretization agnostic and can be combined with continuous finite elements, discontinuous finite elements, and finite volumes. The key idea is that at each stage of the Runge Kutta scheme, one computes a low-order update, a high-order update, both defined from the same intermediate Runge Kutta stage, and

then one applies the nonlinear, mass conservative limiting operator. The main advantage with respect to the strong stability preserving paradigm is more flexibility in the choice of the Runge Kutta scheme, thus allowing for less stringent restrictions on the time step. In the second part of the talk I will extend the method to partial differential equations with second or higher-order derivatives in space. I will show that the above strategy can be adapted to implicit-explicit methods. One key step is to be able to construct a linearized first-order method that is conservative and invariant-domain preserving.

- March 9, 2022, at 11am EST USA

**Professor Luz de Teresa**

(UNAM)

https://paginas.matem.unam.mx/deteresa/

Title: Simultaneous controllability of two parabolic equations

Abstract: In this talk, we present some results on the null controllability of two uncoupled parabolic equations with the same control. We give some sufficient conditions and present new Carleman inequalities for an extended adjoint system. These results were obtained in collaboration with Fagner Araruna and Felipe Chaves-Silva.

- February 8, 2022, at 11am EST USA

**Professor Michel Delfour **

(Centre de recherches mathematiques and Departement de mathematiques et de statistique, Universite de Montreal)

https://dms.umontreal.ca/~delfour/

Title: Derivative of Parametrized Minima and Minimax: Compliance and State Constrained Objective Functions

Abstract: This talk is motivated by problems where the variable at hand is a geometric object in the $n$-dimensional Euclidean space $\R^n$. It is natural to introduce a space of subsets of some fixed *holdall* $D\subset \R^n$ and to give it an appropriate structure (a metric or a topology) to deal with optimal design/control problems and sensitivity analysis (derivative, differential calculus). It is now well-established that the *shape derivative* is a differential. But the *topological derivative* is only a semidifferential, that is, a one-sided directional derivative, which is not linear with respect to the direction, and the directions are $d$-dimensional bounded measures. In that framework, the Hadamard semidifferential is the natural underlying notion. The object of this talk is to illustrate how to obtain the topological derivative in a practical mathematical setting for $d$-dimensional perturbations of a bounded open domain. The methodology applies to a wide range of problems: 1) compliance and 2) state constrained objective functions. It relies on two recent theorems on 1) the derivative of a parametrized *minimum* and 2) the derivative of the parametrized *minimax* of a *Lagrangian*. Both predict the occurence of a possibly non-zero extra term over their classical analogues. This extra term coincides with the so-called *polarization term* in Mechanics.

- December 1, 2021, at 11am EST USA

**Professor Irena Lasiecka**

(University of Memphis)

https://www.memphis.edu/msci/people/lasiecka.php

Title: Analysis, Stabilization and Control of Jordan- Moore -Gibson -Thompson -Dynamics [JMGT] Arising in HIFU Technology

Abstract: The JMGT equation is a Partial Differential Equation (PDE) model introduced to describe a nonlinear propagation of sound in an acoustic medium. The interest in studying this type of problems is motivate by a large array of applications arising in engineering and medical sciences-including high intensity focused ultrasound [HIFU] technologies. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second order in time equation referred to as Westervelt equation. Replacing Fourier’s law by Maxwell-Cattaneo’s law gives rise to the third order in time derivative scaled by a small strictly positive parameter, the latter represents the thermal relaxation time parameter and is intrinsic to the medium where the dynamics occurs. In this talk we will present several results pertinent to the model, mostly from the point of view of boundary control and stabilization. These include: (i) local and global wellposedness of the nonlinear JMGT model, (ii) asymptotic analysis of the model when the parameter of relaxation goes to zero, (iii) boundary stabillizability of JMGT in the critical and degenerate case, (iv) feedback boundary control for infinite horizon optimal control problem. This is a joint work with several collaborators to be credited during the presentation.

- November 9, at 11am EST USA

**Professor Piermarco Cannarsa**

(University of Rome “Tor Vergata”)

https://www.mat.uniroma2.it/~cannarsa/casa.html

Title: Some bilinear control problems for evolution equations

Abstract: Bilinear control systems are receiving increasing attention in recent years, as they can be used to study problems for which an additive control action would be unrealistic. For such systems, in infinite dimension, weaker controllability properties can be expected than for systems with additive controls. For instance, exact controllability is out of question due to a well-known negative result by Ball, Marsden, and Slemrod back in the 80’s. Nevertheless, one can seek to steer states to special targets either in finite or infinite time. In this talk, I will present recent results where the above problem is addressed for evolution equations of the form u'(t) = Au(t) + p(t)Bu(t), with A a self-adjoint negative operator on a Hilbert space, B a linear operator satisfying a certain spreading condition, and p(t) a single-input control. When such a condition fails for B (for instance when B=I), I will also discuss control issues that can be reduced to invariance problems.

- October 5, at 11am EST USA

**Professor Fredi Troeltzsch**

(Technische Universitat Berlin, Germany)

Title: Space-Time FEM for the Optimal Control of Reaction Diffusion Equations

Abstract: The numerical solution of forward-backward optimality systems, the necessary optimality conditions for parabolic optimal control problems, is still a challenge for spatial dimension d that are greater or equal to 2. We present and analyze a space-time finite element method on fully unstructured simplicial space-time meshes for solving such systems at once. Here, the time variable t is considered as just another spatial variable, say, the (d+1)th. For a linear state equation,

well-posedness of the system and associated discretization error estimates are shown by Babuska’s theorem. As application, we exemplarily consider the optimal control of the Schlogl model, a semilinear heat equation with non-

monotone third order polynomial nonlinearity, where traveling wave fronts occur as typical solutions. Numerical examples are presented that shed light on the control of traveling wave fronts, where space-time methods are particularly suitable. This is joint work with Ulrich Langer and Huidong Yang (RICAM Linz) and Olaf Steinbach (TU Graz).

**June 1**, at 12pm EST (noon) USA

**Professor Sarka Necasova**

(Institute of Mathematics, Czech Academy of Sciences)

http://www.math.cas.cz/homepage/main_page.php?id_membre=22

Title: A bi-fluid model for a mixture of two compressible non interacting fluids with general boundary data

Abstract: We deal with the global existence of weak solutions for a version of one velocity Baer-Nunziato system with dissipation describing a mixture of two non interacting viscous compressible fluids in a piecewise regular Lipschitz domain with general inflow/outflow boundary conditions. The geometrical setting is general enough to comply with most current domains important for applications as, for example, (curved) pipes of picewise regular and axis-dependent cross sections. Moreover, we introduce dissipative turbulent solutions and prove an existence of such solutions for all adiabatic coefficients Î³ > 1, their compatibility with classical solutions, the relative energy inequality, and the weak strong uniqueness principle in this class. The class of dissipative turbulent solutions is so far the largest class of generalized solutions which still enjoys the weak strong uniqueness property. It is a joint work with S. Kracmar, B. J. Jin, Y. Kwon and A. Novotny.

**May 4**, at 12pm EST (noon) USAÂ –**Talk was postponed to October 5, 2021.**

** Professor Fredi Troeltzsch**

(Technische Universitat Berlin, Germany)

**April 6**, at 12pm EST (noon) USA

**Professor Suncica Canic**(University of California, Berkeley, USA)

https://www.math.uh.edu/~canic/

Title: Mathematical Design of a Bioartificial Pancreas

Abstract: With the recent developments of new technologies in biomedical engineering and medicine, the need for new mathematical and numerical methods to aid these developments has never been greater. In particular, the design of an implantable bioartificial pancreas for the treatment of Type 1 diabetes hinges on the development of mathematical and computational techniques for solving nonlinear moving boundary problems. In this talk we present a complex, multi-scale model, and a recent well-posedness result in the area of fluid-poroelastic structure interaction, which have helped the design of a first implantable bioartificial pancreas without the need for immunosuppressant therapy. This is a joint work with bioengineer Shuvo Roy (UCSF), and mathematicians Yifan Wang (UCI), Lorena Bociu (NCSU), Boris Muha (University of Zagreb), and Justin Webster (University of Maryland, Baltimore County)

**March 2**, at 12pm EST (noon) USA.

**Professor Alberto Bressan**(The Pennsylvania State University, USA)

http://personal.psu.edu/axb62/

Title: Traffic flow on a network of roads

Abstract: A mathematical description of traffic flow can be provided in terms of conservation laws, describing the density of cars along each road. Additional conditions are then used, to model flow at intersections. One can also look at daily traffic patterns as the result of the decisions of a large number of drivers, trying to minimize the time spent on the road and a penalty for late arrival. This leads to the problem of finding a globally optimal solution, which minimizes the sum of the costs to all drivers, or a Nash equilibrium solution, where no driver can lower his individual cost by changing his own departure time or his route to reach destination. The talk will review some of these models, discussing main results and current research directions.

**February 2**, at 12pm EST (noon) USA.**Professor Christian Clason**(University of Duisburg-Essen, Germany)Title: Optimal control of non-smooth partial differential equations

Abstract: This talk is concerned with PDE-constrained optimization problems where the PDE constraint involves Lipschitz continuous but not classically differentiable terms. Correspondingly, the control-to-state mapping is not differentiable either, and classical approaches fail. In particular, there exists a zoo of optimality conditions of different strengths, roughly corresponding to different generalized derivatives of the control-to-state mapping. We derive such optimality conditions for model problems and discuss how they can be used for their numerical solution. This talk is based on joint work with Constantin Christof, Christian Meyer, Vu Huu Nhu, and Arnd RoÌˆsch.