F-equivalence for parabolic systems and applications to the stabilization of nonlinear PDE
Vincent Boulard, Amaury Hayat
TL;DR
The paper develops a comprehensive framework for F-equivalence in parabolic systems, proving that, under $oldsymbol{ extlambda}$-approximate controllability and with a finite-dimensional control, there exists an explicit parabolic F-equivalence $(T,K)$ to a $oldsymbol{ extlambda}$-target $D$, yielding exponential stabilization at rate $oldsymbol{ extlambda}$. This both handles unbounded control operators and multidimensional spatial domains, without requiring eigenvalue multiplicity limitations, and extends to nonlinear perturbations via Assumption (F1), ensuring local well-posedness and rapid stabilization of semilinear parabolic PDEs. The approach delivers constructive feedback laws by solving finite-dimensional algebraic systems and applies to classical PDEs such as the heat equation on manifolds, Kuramoto–Sivashinsky, Navier–Stokes, and a quasilinear heat equation. By clarifying a link between FEQ uniqueness and approximate controllability, the work provides a robust criterion for both stability design and theoretical analysis in infinite-dimensional parabolic control. These results have practical implications for rapid stabilization in higher-dimensional PDE models where traditional controllability conditions are restrictive or inapplicable.
Abstract
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 arbitrarily exponentially stable evolution system 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.