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QUADRATIC DIFFERENTIAL EQUATIONS: PARTIAL GELFAND–SHILOV SMOOTHING EFFECT AND NULL-CONTROLLABILITY

Part of: Partial differential equations Controllability, observability, and system structure

Published online by Cambridge University Press:  31 January 2020

Paul Alphonse Open the ORCID record for Paul Alphonse [Opens in a new window]
Paul Alphonse*
Affiliation:
Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000Rennes, France ([email protected])
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Abstract

We study the partial Gelfand–Shilov regularizing effect and the exponential decay for the solutions to evolution equations associated with a class of accretive non-selfadjoint quadratic operators, which fail to be globally hypoelliptic on the whole phase space. By taking advantage of the associated Gevrey regularizing effects, we study the null-controllability of parabolic equations posed on the whole Euclidean space associated with this class of possibly non-globally hypoelliptic quadratic operators. We prove that these parabolic equations are null-controllable in any positive time from thick control subsets. This thickness property is known to be a necessary and sufficient condition for the null-controllability of the heat equation posed on the whole Euclidean space. Our result shows that this geometric condition turns out to be a sufficient one for the null-controllability of a large class of quadratic differential operators.

Keywords

quadratic operatorsGelfand–Shilov regularitygevrey regularitynull-controllability

MSC classification

Primary: 35B65: Smoothness and regularity of solutions
Secondary: 93B05: Controllability
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 6 , November 2021 , pp. 1749 - 1801
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Copyright
© Cambridge University Press 2020

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