Optimal control problem and viscosity solutions for the Vlasov equation in Yang-Mills charged Bianchi models (Q520900)

The relativistic Vlasov equation for massive Yang-Mills particles in a plane-symmetric Bianchi spacetime are discussed and represented in the form of a Hamilton-Jacobi equation for which an initial value problem is formulated. In [\textit{A. Briani} and \textit{F. Rampazzo}, NoDEA, Nonlinear Differ. Equ. Appl. 12, No. 1, 71--91 (2005; Zbl 1075.49012)] it was shown that weak so-called viscosity solutions allow the formulation of existence and uniqueness theorems, if the Hamiltonian and the initial density probability function satisfied certain continuity and measurability conditions. These propositions are shown to be fulfilled by the Hamilton-Jacobi equation corresponding to the collision-less Vlasov equation on the background of a general (not specified to a solution of the Einstein equation) homogeneous but anisotropic Bianchi-I spacetime. This allows the formulation of an optimal control problem in the sense of an evaluation of the probability density distribution given by the solution of the Vlasov equation.