Georgia Institute of Technology College of Sciences

Using metric derivative and local Lipschitz constant, we define action integral and Hamiltonian operator for a class of optimal control problem on curves in metric spaces. Main requirement on the space is a geodesic property (or more generally, length space property). Examples of such space includes space of probability measures in R^d, general Banach spaces, among others. A well-posedness theory is developed for first order Hamilton-Jacobi equation in this context. The main motivation for considering the above problem comes from variational formulation of compressible Euler type equations. Value function of the variation problem is described through a Hamilton-Jacobi equation in space of probability measures. Through the use of geometric tangent cone and other properties of mass transportation theory, we illustrate how the current approach uniquely describes the problem (and also why previous approaches missed). This is joint work with Luigi Ambrosio at Scuola Normale Superiore di Pisa.