Viability Solutions to Structured Hamilton-Jacobi Equations under Constraints

• Authors: Jean-Pierre Aubin
• Venue: SIAM Journal on Control and Optimization
• Year: 2011

Structured Hamilton-Jacobi partial differential equations are Hamilton-Jacobi equations where the time variable is replaced by a vector-valued variable “structuring” the system. It could be the time-age pair (Hamilton-Jacobi-McKendrick equations) or candidates for initial or terminal conditions (Hamilton-Jacobi-Cournot equations) among a manifold of examples. Here, we define the concept of “viability solution” which always exists and can be computed by viability algorithms. This is a constructive approach allowing us to derive from the tools of viability theory (dealing with sets instead of functions) some known as well as new properties of its solutions. Regarding functions as their epigraphs, we bypass the regularity issues to arrive directly at the concept of Barron-Jensen/Frankowska viscosity solutions. Above all we take into account viability constraints and extend classical boundary conditions to other “internal” conditions. Beyond that, we use the Fenchel-Legendre transform to associate a Lagrangian with the Hamiltonian and uncover an underlying variational problem. It is proved that the viability solution to the structured Hamilton-Jacobi equation is the valuation function of this variational principle, the optimal evolutions of which are regulated by a “regulation map” which is constructed from the viability solution and can be computed by viability algorithms. In other words, the viability solution solves all at once a class of first-order partial differential equations of intertemporal optimization problems and the regulation of their optimal evolutions.