Brief Introduction to Control Theory

Members of the control group include G. Ferreyra, J. Lawson, M. Malisoff, P. Sundar, and P. Wolenski. The group cooperates with the larger LSU Systems and Control Group which includes faculty from the LSU departments of electrical and mechanical engineering. The concept of control can be summarized by saying that one seeks to influence a dynamical system in order to achieve some desired goal. The dynamical system in mathematical control theory is usually a system of differential or difference equations that depends on a set of parameters, where the parameters are the "control" variables. The idea then is to find these control variables so as to minimize (or maximize) a given objective function, to stabilize the system, or to move the system to a desired destination.

Faculty members at LSU are primarily interested in deterministic and stochastic optimal control theory in discrete and continuous time. Research interests include topics from systems theory, optimization, partial differential equations, and stochastic processes. The control group is also closely involved in the applied and interdisciplinary initiatives being developed in the department.

More specifically, research methods include (a) non-smooth and convex analysis (see "Non-smooth Analysis and Control Theory" by Clarke, Ledyaev, Stern, and Wolenski) (b) partial differential equations and Hamilton-Jacobi theory (see "Controlled Markov Processes and Viscosity Solutions" by Fleming and Soner), (c) probability theory (see "Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, W.M. McEneaney, G. Yin, and Q. Zhang (Eds.), (d) geometric methods (see "Differential Geometry and Control", G. Ferreyra et al.), (e) Lie semigroup theory, and (f) stabilization (see "Mathematical Control Theory, Second Edition", by E. Sontag).

Members of the group have been involved with developing interdisciplinary and industrial programs. These include (1) instituting the new Master's Program in Mathematics with a concentration in Finance, (2) developing a Master's Degree in Mathematics with a concentration in Applied Mathematics, (3) organizing projects for the Mathematics Consultation Clinic (which brings real world mathematical problems from industry to the Math Department), and (4) running an internship program with a local company. A new PC lab has been created by several members of the group to support research projects and the Math Clinic.