March 01, 2012, ESB 2001

The Linear Convex Regulator (LCR) is a problem which generalizes the Linear Quadratic Regulator (LQR) by considering convex and positive definite, but not necessarily quadratic, costs. LCR can model LQR with control and state constraints, and can be used to stabilize linear systems with saturation. The presentation will focus on the continuous-time LCR problem on an infinite time horizon. Through an as elementary and as self-contained as possible approach I will motivate a convex duality framework for studying the LCR, obtaining along the way conditions for optimality, Hamilton-Jacobi equations, and underlining the role of Hamiltonian dynamics. My goal is to provide a framework and tools for further study rather than state-of-the-art results. Some familiarity with convex analysis and a previous encounter with the Pontryagin maximum principle will be helpful for this presentation, but not essential. A key notion in convex analysis, the conjugate function, will be (re)introduced to the audience.

Rafal Goebel received his M.Sc. degree in mathematics in 1994 from the University of Maria Curie Sklodowska in Lublin, Poland, and his Ph.D. degree in mathematics in 2000 from University of Washington, Seattle. He held a postdoctoral position at the Departments of Mathematics at the University of British Columbia and Simon Fraser University in Vancouver, Canada, 2000 -- 2002; a postdoctoral and part-time research positions at the Electrical and Computer Engineering Department at the University of California, Santa Barbara, 2002 -- 2005; and a part-time teaching position at the Department of Mathematics at the University of Washington, 2004 -- 2007 In 2008, he joined the Department of Mathematics and Statistics at Loyola University Chicago. His interests include convex, nonsmooth, and set-valued analysis; control theory, including optimal control; hybrid dynamical systems; and optimization.