A Hopf-Lax formula for Hamilton-Jacobi equations with Caputo time derivative
This work provides a theoretical advance for mathematicians studying fractional PDEs, but it is an incremental extension of classical Hopf-Lax theory to a specific fractional derivative case.
The authors derive a Hopf-Lax representation formula for solutions to Hamilton-Jacobi equations with Caputo time-fractional derivatives, which arise in optimal control problems with random trapping effects. The formula provides an explicit solution method for this class of equations.
We prove a representation formula of Hopf-Lax type for the solution of a Hamilton-Jacobi equation involving Caputo time-fractional derivative. Equations of these type are associated with optimal control problems where the controlled dynamics is replaced by a time-changed stochastic process describing the trajectory of a particle subject to random trapping effects.