Analysis of the controllability of bilinear closed quantum systems

Abstract : The first part of the research is dedicated to the global exact controllability of the bilinear Schrödinger equation (BSE).We show how to construct a neighborhood of some eigenfunctions of the Dirichlet Laplacian where the local exact controllability is satisfied in a specific time. Then, for any couple of those eigenfunctions, we study how to construct controls and times such that the relative dynamics of (BSE) drives the first close to the second as much desired. Third, by gathering the two previous results, we define a dynamics steering eigenstates in eigenstates and we provide an explicit time required to reach the target.In the second part, we study the simultaneous global exact controllability in projection of infinitely many (BSE) and we prove the simultaneous local exact controllability in projection up to phases for any positive time. In the proof, we use different techniques from the Coron's return method usually adopted for those types of results. The main novelty of the work is the fact that it provides a set of conditions implying the validity of the result. Given any control field, one can verify if those assumptions are satisfied.The third part of the work treats the controllability of the bilinear Schrödinger equation (BSE) on compact graph. Considering (BSE) on such a complex structure is useful when one has to study the dynamics of wave packets on graph type model. We investigate assumptions on the graph and on the control field implying the well-posedness of (BSE) in suitable spaces that we characterize by providing peculiar interpolation features.Then, we provide the global exact controllability in those spaces by studying how the structure of the graph and the boundary conditions affect the result. We also provide examples of graphs and control fields so that the spectral assumptions of the global exact controllability are satisfied, e.g. star graphs, tadpole graphs and double-ring graphs.Afterwards, when the hypothesis for the global exact controllability fail, we define a weaker notion of controllability, the so-called “energetic controllability" which ensures the existence of a set of bounded states for which the exact controllability is verified. In other words, we prove the existence of energy levels in which it is possible to change the energy of the system.This technique allows to treat a large number of interesting problems. Indeed, for complex graphs, it is not possible to verify the spectral hypothesis of the global exact controllability. However, the energetic controllability allows to obtain interesting results only by looking for particular substructure contained in the graph.
Document type :
Theses
Complete list of metadatas

Cited literature [53 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-01830104
Contributor : Abes Star <>
Submitted on : Wednesday, July 4, 2018 - 3:33:06 PM
Last modification on : Friday, July 6, 2018 - 3:18:04 PM
Long-term archiving on : Monday, October 1, 2018 - 1:03:04 PM

File

these_A_DUCA_Alessandro_2018.p...
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-01830104, version 1

Collections

Citation

Alessandro Duca. Analysis of the controllability of bilinear closed quantum systems. Analysis of PDEs [math.AP]. Université Bourgogne Franche-Comté, 2018. English. ⟨NNT : 2018UBFCD004⟩. ⟨tel-01830104⟩

Share

Metrics

Record views

289

Files downloads

75