Optimal control for a mixed flow of Hamiltonian and gradient type in space of probability measures (with Appendix B by Atanas Stefanov) (Q2847027)

Motivated by a stochastic interacting particle model which gives the two-dimensional Navier-Stokes equations, the authors study an optimal control problem in the space of measures on \(\mathbb{R}^{2}\). More precisely, they consider a system of controlled partial differential equations of the form NEWLINE\[NEWLINE \partial_{t} \rho +\text{div}(\rho u)=\nu \Delta \rho +m, \qquad u=-K^{\perp}\star\rho,\tag{1}NEWLINE\]NEWLINE where \(m\) is the control. Using the tools of calculus on the space of probability measures, the authors show that \((1)\) can be rewritten as an abstract evolution equation which is a mixture of a Hamiltonian flow, a gradient flow and the control variable \(m\), i.e., the Hamiltonian-Jacobi-Bellman equation \((2) \,(I-\alpha H)f=h\) for a sufficiently large class of \(h\). The main result of the paper resides essentially in giving a well posedness result for \((2)\), in an appropriately defined viscosity solution sense applied to discontinuous functions.