Mathematics Colloquium: Convex conjugate functions in stability analysis and control
4:10 p.m. Neill Hall 5W
Abstract: Convex conjugacy gives a one to one correspondence among convex (and proper and lower semicontinuous) functions. For example, for quadratic convex functions given by symmetric and positive definite matrices, convex conjugacy reduces to the relationship between such a matrix and its inverse. This, and far more involved conjugacy correspondences are at the foundations of duality theory in convex optimization. The talk will illustrate that convex functions, and the conjugacy relationship, can be also successfully employed in addressing various problems in control theory. First problem that will be discussed is that of verifying the stability of a linear differential inclusion. Considering convex Lyapunov functions is less restrictive than insisting on quadratic ones, and furthermore, through the conjugacy relationship it allows for relating the stability of a linear differential inclusion to the stability of a dual inclusion. Second problem will be that of establishing uniqueness of solutions to stationary Hamilton-Jacobi partial differential equations associated with convex optimal control problems on infinite time horizons. The said uniqueness will be related to a conjugacy relationship between optimal value functions of a primal and of a dual optimal control problem. Applications of the uniqueness result to feedback stabilization problems will be shown.