Workshop on Control Theory and Mathematical Biology

Program

Invitation and Remarks. All interested faculty, postdocs, and graduate students are invited. There is no registration fee or form to complete but it would be helpful if attendees would let the organizer know in advance that they are coming. The workshop atmosphere will be informal and relaxed with plenty of time for questions and discussions.

Confirmed Attendees. Azmy Ackleh (University of Louisiana at Lafayette), Rick Barnard (LSU), Jacob Blanton (LSU), Wei Cai (LSU), Xinjia Chen (LSU), Ross Chiquet (University of Louisiana at Lafayette), John Cleveland (University of Louisiana at Lafayette), Patrick De Leenheer (University of Florida), Marcio de Queiroz (LSU), Guillermo Ferreyra (LSU), Ameziane Harhad (LSU), Omar Galan (LSU), Feng Gao (LSU), Alvaro Guevara (LSU), Jimmie Lawson (LSU), Qingxia Li (LSU), Michael Malisoff (LSU), Laurentiu Marinovici (LSU), Frederic Mazenc (INRA, FRANCE), Rajib Mukherjee (LSU), Jeremy Thibodeaux (University of Louisiana at Lafayette), Peter Wolenski (LSU)

Patrick De Leenheer (University of Florida), Multi-Strain Virus Dynamics with Mutations: A Global Analysis. We consider within-host virus models with 2 or more strains that allow mutation between the strains. If there is no mutation, a Lyapunov function establishes global stability of the steady state corresponding to the fittest strain. For small perturbations this steady state persists, perhaps with small concentrations of some or all other strains, depending on the connectivity of the graph describing all possible mutations. Moreover, using a perturbation result due to Smith and Waltman, we show that this steady state also preserves global stability. This is joint work with Sergei Pilyugin of the Department of Mathematics of the University of Florida. Click here for the slides for this talk.

Frederic Mazenc (UMR Analyse des Systemes et Biometrie, Montpellier, FRANCE), Strict Lyapunov Functions for Nonlinear Systems and Applications. A new construction of a strict Lyapunov function for a general family of non-linear time varying system is presented. It is shown how it can be used to solve the problem of extending the backstepping approach to the case of non-linear time-varying systems and to the applied problem of the global uniform asymptotic stabilization of a model of an underactuated vessel. In a last part, the construction of a CLF for systems satisfying the conditions of the Jurdjevic-Quinn theorem is presented. Click here for the slides for this talk.

Peter Wolenski (LSU), Impulsive Dynamical Systems. This talk will describe impulsive systems that consist of dynamics driven by a measure. We will introduce solution concepts and present invariance results for these systems. Click here for the slides for this talk.

Jeremy Thibodeaux (University of Louisiana at Lafayette), Structured Population Models and Their Application to Erythropoiesis. In this talk I will present the main ideas of developing structured population models. I will then show how these ideas can be applied to develop a mathematical model that describes erythropoiesis, the process in which red blood cells are produced. Once the model is presented, I will demonstrate how a relatively simple finite difference scheme can be used to answer questions about the existence and uniqueness of solutions, long time behavior of solutions, and the sensitivity of the model to certain parameters. Click here for the slides for this talk.

Azmy Ackleh (University of Louisiana at Lafayette), Structured Population Models: Measure-Valued Solutions and Difference Approximations. A hierarchical size-structured population model is presented. The model is described by a hyperbolic first-order quasilinear initial-boundary value problem. It is shown that when the growth function is not monotone singular (measure-valued) solutions will emerge in finite time, even if the initial condition and the vital rates are very smooth. A finite difference scheme for computing these singular solutions is discussed and the convergence of this approximation is obtained. Click here for the slides for this talk.

Jimmie Lawson (LSU), Control Theory, Riccati Equations, and Contraction Semigroups. We show that the symplectic Hamiltonian operators on a Hilbert space give rise to linear fractional transformations on the open convex cone of positive definite operators that contract a natural invariant Finsler metric, the Thompson or part metric, on the convex cone. More precisely, the constants of contraction for the Hamiltonian operators satisfy the classical Birkhoff formula: the Lipschitz constant for the corresponding linear fractional transformations on the cone of positive definite operators is equal to the hyperbolic tangent of one fourth the diameter of the image. By means of the close connections between Hamilitonian operators and Riccati equations, this result and the associated machinery are applied to obtain convergence results for discrete algebraic Riccati equations and Riccati differential equations. This talk is based on the speaker's joint work with Yongdo Lim. Click here for the slides for this talk.

Marcio de Queiroz (LSU), A New Tuning Function-Based Robust Adaptive Controller for Parametric Strict-Feedback Systems. This work reports a new robust adaptive tracking controller for multi-input/multi-output nonlinear parametric strict-feedback systems in the presence of parametric uncertainty and any unknown continuous bounded additive disturbance. The proposed solution uses a new projection-like adaptation that allows the application of the standard tuning function approach, thereby avoiding overparametrization and the need for extra nonlinear damping-type terms in the control. The robust adaptive control is shown to guarantee practical tracking in the presence of the disturbance and asymptotic tracking when the disturbance disappears. Click here for the slides for this talk.

Michael Malisoff (LSU), Stabilization of a Periodic Trajectory for a Chemostat with Two Species. We study a chemostat model with two organisms using Lyapunov function methods. Using a linear feedback control of the dilution rate and an appropriate time-varying substrate input concentration, we produce a locally exponentially stable oscillatory behavior for the species concentrations, meaning all trajectories of the chemostat that stay near the oscillatory reference trajectory are actually attracted to the reference trajectory exponentially fast. We also obtain a globally stable oscillatory reference trajectory for the species concentrations, using a nonlinear feedback control depending on the dilution rate and the substrate input concentration. This guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory. We demonstrate the efficacy of our method using a numerical simulation. This talk is based on the speaker's joint work with Frederic Mazenc and Jerome Harmand. Click here for the slides for this talk.

Local Information. This workshop will take place on the main campus of Louisiana State University (LSU) in Baton Rouge. Driving directions from the nearby interstates are available on the LSU web site. The official hotel for the workshop is the Faculty Club Hotel which is located on campus. There is currently no direct public transportation between Baton Rouge Metropolitan Airport (BTR) and LSU although typically there are taxis already waiting outside the baggage claim area at BTR. A cab ride from BTR to LSU runs about $35. One popular local taxi company is Yellow Cab of Baton Rouge (225-926-6400). Participants can go to the LSU Visitor Information Center to get campus directions and parking instructions. An online campus map is also available. Some local attractions in Baton Rouge include the Rural Life Museum, the Louisiana State Museum, the Louisiana State Capitol, the Old State Capitol, and Bluebonnet Swamp. The Louisiana Office of Tourism web site has detailed information about area attractions. There is also a local events calendar in Country Roads Magazine. Some popular restaurants near the LSU campus include Chimes (which is located on Highland Road two blocks north of the Faculty Club and is open for lunch and dinner), the Faculty Club Restaurant (which is open for lunch), and Louie's Cafe (on West State St. which is one block north of and parallel to West Chimes St.) which is famous for their omelettes. Tsunami is a well known restaurant in downtown Baton Rouge. The weather in Baton Rouge in late July is typically humid with highs around 90 and lows in the 70's but the talks will be in an air conditioned building and the campus hotels are air conditioned. Questions about the workshop should be sent to Karen Cooper (kcooper@m*) or Michael Malisoff (malisoff@m*) where m* stands for math.lsu.edu.

Acknowledgements. This workshop is supported by the Louisiana Board of Regents project "Enhancing Control Theory at LSU". Frederic Mazenc's visit was made possible by the NSF project "Research in Nonlinear Control Systems Theory: Lyapunov Functions, Stabilization, and Engineering Applications" (NSF/DMS Grant 0424011). Additional funding was generously provided by the LSU College of Arts and Sciences (Guillermo Ferreyra, Dean). The organizer appreciates the expert assistance from LSU Department of Mathematics Conference Coordinator Karen Cooper.

Contact: Karen Cooper or Michael Malisoff











Click on the photos for enlargements.