CMX Lunch Seminar

For nonlinear ODEs and PDEs that cannot be solved exactly, various properties can be inferred by constructing functions that satisfy suitable inequalities. Although the most familiar example is proving nonlinear stability of an equilibrium by constructing Lyapunov functions, similar approaches can produce many other types of mathematical statements, including for systems with chaotic behavior. Such statements include bounds on attractor properties or on transient behavior, estimates of basins of attraction, design of nonlinear controls, and more. Analytical results of these types often give overly conservative results in order to remain tractable. Much stronger results can be achieved by using computational methods of polynomial optimization to construct functions that satisfy the desired inequalities, especially via sum-of-squares constraints that imply nonnegativity of polynomials. This talk will provide an introduction to how polynomial optimization can be used to study ODEs, including some theoretical guarantees of arbitrarily sharp results, followed by some extensions to PDEs.