Title | Multiscale Version of the Logvinenko-Sereda Theorem |

DFG Code | Ve 253/7-1 |

Principal Investigator | Prof. Dr. Ivan Veselić |

Researcher | Dr. Michela Egidi |

The validity of a multiscale version of the Logvinenko-Sereda Theorem is suggested by recent scale-free unique continuation estimates or uncertainty principles for eigenfunctions and spectral projections of Schrödinger operators. Here the functions are considered on intervals of length L, ranging over the positive reals. While the intended estimate appears at first sight simpler than in the case of the whole axis, one has the additional task to control effectively the dependence of the estimate on the additional size parameter L. Ideally, one would like to show that the estimates hold uniformly in L.

While the conjectured bound is a Harmonic Analysis result, it immediately triggers consequences in the theory of Inverse Problems, in particular under appropriate sparsity assumptions. Thus it can be seen as a continuum relative of compressed sensing and sparse recovery. Moreover, the conjectured estimates have applications in the spectral theory of Schrödinger operators and in the control theory of the heat equation. A multidimensional extension of this estimate will have even wider relevance.

Title: | Sharp geometric condition for null-controllability of the heat equation on R d and
consistent estimates on the control cost |

Authors: | Michela Egidi, Ivan Veselic |

Year: | 2017 |

Preprint | https://arxiv.org/abs/1711.06088 |

Abstract: | In this note we study the control problem for the heat equation on $R^d$, $d>= 1$, with control set a given subset of $R^d$. We provide a necessary and sufficient condition on the control set such that the heat equation is null-controllable in any positive time. We give an estimate of the control cost with explicit dependency on the characteristic geometric parameters of the control set. Finally, we derive a control cost estimate for the heat equation on cubes with periodic, Dirichlet, or Neumann boundary conditions, where the control sets are again assumed to be thick. We show that the control cost estimate is consistent with the $R^d$ case. |

Title: | Scale-free unique continuation estimates and Logvinenko-Sereda Theorems on the torus |

Authors: | Michela Egidi, Ivan Veselic |

Year: | 2016 |

Preprint | https://arxiv.org/abs/1609.07020 |

Abstract: | We study uncertainty principles or observability estimates for function classes on the torus. The classes are defined in terms of spectral subspaces of the energy or the momentum, respectively. In our main theorems, the support of the Fourier Transform of the functions is allowed to be supported in a (finite number of) parallelepipeds. The estimates we obtain do not depend on the size of the torus and the position of the parallelepipeds, but only on their size and number, and the size and scale of the observability set. Our results are on the one hand closely related to unique continuation and observability estimates which can be obtained by Carleman estimates and on the other hand to the Logvinenko and Sereda theorem. In fact, we rely on the methods used by Kovrijkine to refine and generalize the results of Logvinenko and Sereda. |

Title: | Sharp geometric condition for null-controllability of the heat equation on the whole space |

Occasion: | 3rd GAMM AGUQ Workshop on Uncertainty Quantification |

Institution/Location: | TU Dortmund |

Link: | PDF file |

Date: | 10th-14th March 2018 |

Title: | Null-controllability of the heat equation on rectangular regions |

Occasion: | 5th Najman Conference on Spetral Theory and Differential Equations |

Institution/Location: | Opatjia, Croatia |

Link: | PDF file |

Date: | September 2017 |

Title: | Unique continuation and Logvinenko-Sereda Theorem on T^d_L |

Occasion: | Workshop Mathematical Physics and Dynamical Systems 2017 |

Institution/Location: | TU Dortmund |

Link: | PDF file |

Date: | 20th-22nd March 2017 |

Title: | Necessary and sufficient geometric condition for null-controllability of the heat equation on R^d |

Speaker: | Ivan Veselic |

Occasion: | Worskhop on Control theory of infinite-dimensional systems |

Institution: | FernUniversität Hagen |

Date: | 10th-12th January 2018 |

Title: | Longvinenko-Sereda Theorems: from complex analysis to application in Control Theory |

Speaker: | Michela Egidi |

Occasion: | Worskhop on Control theory of infinite-dimensional systems |

Institution: | FernUniversität Hagen |

Date: | 10th-12th January 2018 |

Title: | Logvinenko-Sereda Theorems and application to control theory of the heat equation |

Speaker: | Michela Egidi |

Occasion: | Group de travail |

Institution: | ENS Rennes |

Date: | 15th November 2017 |

Title: | Unique continuation estimates and the Logvinenko Sereda Theorem |

Speaker: | Ivan Veselic |

Occasion: | Oberseminars Stochastik/Mathematische Physik |

Institution: | Fernuni Hagen |

Date: | 15th February 2017 |

Title: | Logvinenko-Sereda Theorems for periodic functions |

Speaker: | Michela Egidi |

Occasion: | Summer school “Spectral Theory, Differential Equations and Probability“ |

Institution: | Johannes Gutenberg Universität Mainz |

Date: | 4th-15th September 2016 |

Title: | Uncertainty relations and applications to the Schrödinger
and heat conduction equations |

Speaker: | Ivan Veselic |

Occasion: | Summer school “Spectral Theory, Differential Equations and Probability“ |

Institution: | Johannes Gutenberg Universität Mainz |

Date: | 4th-15th September 2016 |

Title: | Quantitative uncertainty principle on the torus |

Speaker: | Michela Egidi |

Occasion: | GGA Seminar |

Institution: | Durham University |

Date: | 23rd May 2016 |

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