Title | Multiscale Version of the Logvinenko-Sereda Theorem |

DFG Code | Ve 253/7-1 |

GEPRIS | Projekt number 280969390 |

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: | Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports |

Authors: | Karine Beauchard, Michela Egidi, Karel Pravda-Starov |

Year: | 2019 |

Preprint | https://arxiv.org/abs/1908.10603 |

Abstract: | We study the null-controllability of some hypoelliptic quadratic parabolic equations posed on the whole Euclidean space with moving control supports, and provide necessary or sufficient geometric conditions on the moving control supports to ensure null-controllability. The first class of equations is the one associated to non-autonomous Ornstein-Uhlenbeck operators satisfying a generalized Kalman rank condition. In particular, when the moving control supports comply with the flow associated to the transport part of the Ornstein-Uhlenbeck operators, a necessary and sufficient condition for null-controllability on the moving control supports is established. The second class of equations is the class of accretive non-selfadjoint quadratic operators with zero singular spaces for which some sufficient geometric conditions on the moving control supports are also given to ensure null-controllability. |

Title: | The reflection principle in the control problem of the heat equation |

Authors: | Michela Egidi, Albrecht Seelmann |

Year: | 2019 |

Preprint | https://arxiv.org/abs/1902.08141 |

Abstract: | We consider the control problem for the generalized heat equation for a Schrödinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on its respective parts. Moreover, we show that the corresponding control cost does not exceed the one on the whole domain. |

Title: | Null-controllability and control cost estimates for the heat equation on unbounded and large bounded domains |

Authors: | Michela Egidi, Ivica Nakić, Albrecht Seelmann, Matthias Täufer, Martin Tautenhahn, Ivan Veselic |

Year: | 2018 |

Preprint | https://arxiv.org/abs/1810.11229 |

Abstract: | We survey recent results on the control problem for the heat equation on unbounded and large bounded domains. First we formulate new uncertainty relations, respectively spectral inequalities. Then we present an abstract control cost estimate which improves upon earlier results. It is particularly interesting when combined with the earlier mentioned spectral inequalities since it yields sharp control cost bounds in several asymptotic regimes. We also show that control problems on unbounded domains can be approximated by corresponding problems on a sequence of bounded domains forming an exhaustion. Our results apply also for the generalized heat equation associated with a Schrödinger semigroup. |

Title: | On null-controllability of the heat equation on infinite strips and control cost estimate |

Authors: | Michela Egidi |

Year: | 2018 |

Preprint | https://arxiv.org/abs/1809.10942 |

Abstract: | We consider an infinite strip $\Omega_L=(0,2\pi L)^{d-1}\times\mathbb{R}$, $d\geq 2$, $L>0$, and study the control problem of the heat equation on $\Omega_L$ with Dirichlet or Neumann boundary conditions, and control set $\omega\subset\Omega_L$. We provide a sufficient and necessary condition for null-controllability in any positive time $T>0$, which is a geometric condition on the control set $\omega$. This is referred to as "thickness with respect to $\Omega_L$" and implies that the set $\omega$ cannot be concentrated in a particular region of $\Omega_L$. We compare the thickness condition with a previously known necessity condition for null-controllability and give a control cost estimate which only shows dependence on the geometric parameters of $\omega$ and the time $T$. |

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 $\Bbb R^d$ |

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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