Dissipative control system for the stochastic nonlinear H∞ problems

We characterize functions satisfying a dissipative inequality associated with a stochastic control problem. Such a characterization is provided in terms of an upper generalized Gaussian solution, or a viscosity supersolution to a partial differential equation called Hamilton–Jacobi equation (H–J). Links between upper generalized Gaussian solutions and viscosity supersolutions to Hamilton–Jacobi equation are studied. Finally it shows that generalized Gaussian solutions is identical to viscosity solutions to Hamilton–Jacobi equation.