Stochastic representation of partial differential inclusions
Partial differential inclusions of the form \(u_t^\prime \in \mathbb L _{FG}u+cu\) and \(\psi \in \mathbb L_{FG}u+cu\), where \(\mathbb L_{FG}\) is a uniformly parabolic second order set-valued operator, are considered. In particular, based on diffusions properties of weak solutions to stochastic differential inclusions, some existence and representation theorems for solutions of such type partial differential inclusions are given. Partial differential inclusions are investigated together with some initial and boundary conditions. The presented approach leads to natural methods of solving some optimal control problems for systems described by partial differential equations depending on control parameters.
scientific article; zbMATH DE number 5553639