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Controllability properties from the exterior under positivity constraints for a 1-D fractional heat equation

Antil H, Biccari U, Ponce R, Warma M, Zamorano S. Controllability properties from the exterior under positivity constraints for a 1-D fractional heat equation

Abstract. We study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator (−∂^2_x)^s ( with 0<s<1 ) on the interval (−1,1) . Our control function is localized in an open set O in the exterior of (−1,1) , that is, O⊂(R∖(−1,1)) . We show that there exists a minimal (strictly positive) time T min such that the fractional heat dynamics can be controlled from any initial datum in L^2(−1,1) to a positive trajectory through the action of an exterior positive control, if and only if 1/2<s<1 . In addition, we prove that at this minimal controllability time, the constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. Finally, we provide several numerical illustrations that confirm our theoretical results.

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Last updated on March 17, 2022

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  • FAU MoD Lecture: Applications of AAA Rational Approximation
  • DASEL
  • Optimal actuator design via Brunovsky’s normal form
  • ERC DyCon Impact Dimension (2016-2022)
  • Spectral inequalities for pseudo-differential operators and control theory on compact manifolds
  • FAU MoD Lecture: Applications of AAA Rational Approximation
  • DASEL
  • Optimal actuator design via Brunovsky’s normal form
  • ERC DyCon Impact Dimension (2016-2022)
  • Spectral inequalities for pseudo-differential operators and control theory on compact manifolds
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