Optimal control of a class of linear nonautonomous parabolic PDE via two-parameter semigroup representation

This paper considers the two-parameter semigroup representation of a class of parabolic partial differential equation (PDE) with time and spatially dependent coefficients. The properties of the PDE which are necessary for the initial and boundary value problem to be posed as a linear nonautonomous evolution equation on an appropriately defined infinite-dimensional function space are presented. Using these properties, the associated nonautonomous operator generates a two-parameter semigroup which yields the generalized solution of the initial and boundary value problem. The explicit expression of the two-parameter semigroup is provided and enables the application of optimal control theory for infinite dimensional systems.