F-equivalence for parabolic systems and applications to the stabilization of nonlinear PDE

We consider the $F$-equivalence problem for parabolic systems: under which conditions a control system, governed by a parabolic operator $A$ and a control operator $B$, can be made equivalent to an exponentially stable system with arbitrarily large decay rate through an appropriate control feedback law? While this problem has been resolved for finite-dimensional systems fifty years ago, good conditions for infinite-dimensional systems remain a challenge, especially for systems in spatial dimension larger than one. Our main result establishes optimal conditions for the existence of an $F$-equivalence pair $(T,K)$ for a given parabolic control system $(A,B)$. We introduce an extended framework for $F$-equivalence of parabolic operators, addressing key limitations of existing approaches, and we prove that the pair $(T,K)$ is unique if and only if $(A,B)$ is approximately controllable. As a consequence, this provides a method to construct feedback operators for the rapid stabilization of semilinear parabolic systems, possibly multi-dimensional in space. We provide several illustrative examples, including the rapid stabilization of the heat equation, the Kuramoto-Sivashinsky equation, the Navier-Stokes equations and the quasilinear heat equation.