F-equivalence for parabolic systems and applications to the stabilization of nonlinear PDE
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 Fequivalence 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 nonlinear parabolic systems, possibly multidimensional 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.
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BibTeX
@article{boulard2026fequivalence,
title = {{F-equivalence for parabolic systems and applications to the stabilization of nonlinear PDE}},
author = {Vincent Boulard and Amaury Hayat},
journal = {Journal of Differential Equations},
year = {2026},
month = oct,
doi = {10.1016/j.jde.2026.114828},
eprint = {2508.21605},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05340681v2},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - F-equivalence for parabolic systems and applications to the stabilization of nonlinear PDE
AU - Vincent Boulard
AU - Amaury Hayat
PY - 2026
DA - 2026/10/14
JO - Journal of Differential Equations
DO - 10.1016/j.jde.2026.114828
UR - https://hal.science/hal-05340681v2
AB - 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 Fequivalence 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 nonlinear parabolic systems, possibly multidimensional 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.
SN - 0022-0396
SN - 1090-2732
ER -