Hamilton-Jacobi-Bellman equations on graphs— the integro-differential viewpoint

Data Science Seminar

Russell Schwab
Michigan State University

Abstract

This talk is about a class of equations on graphs— Hamilton-Jacobi-Bellman equations— that all share a common structural constraint, which we will call the global comparison property.  This constraint simply says that the operator defining the equation has the property that it preserves ordering when evaluated on any two functions that are globally ordered and agree at the point of evaluation of the operator.  Some examples that fall into this category are graph Laplacians and variants like p-Laplacians, eikonal equations and its variants, Hamilton-Jacobi equations (in whichever reasonable sense you would like), Bellman equations of optimal control, Isaacs equations from two-player games, etc… It turns out that all of these equations have the same structure as integro-differential equations in the continuum setting, and as such, ideas from that realm can be imported to the graph setting. The main mathematical goals of this talk are to explain why it is natural that all of these equations fall into one larger group with similar methods as well as to present a theorem about existence and uniqueness for solutions of these equations.  This is joint work with Nicolò Forcillo and Jun Kitagawa.