Regularity of hyperbolic PDEs governing backstepping gain kernels for parabolic PDEs

In this paper a problem of boundary stabilization of a general class of linear parabolic PDEs is considered. Unlike in previous work in this field, strictly infinite dimensional backstepping is used, independent of any spatial discretization. The problem is formulated as a design of an integral operator whose kernel is shown to satisfy a well posed hyperbolic PDE. This PDE is then converted to an equivalent integral equation and by applying the method of successive approximations a unique solution to this equation is found and its properties are established. For important special cases feedback laws are constructed explicitly.