Bilinear quantum systems on compact graphs: well-posedness and global exact controllability

Alessandro Duca (1)

1 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization

In the present work, we study the well-posedness and the controllability of the bilinear Schr\"odinger equation (BSE) on compact graphs. We study interpolation properties of the spaces $D(|-\Delta|^{s/2})$ for $s>0$, which lead to the well-posedness of the equation in $D(|-\Delta|^{s/2})$ with suitable $s\geq 3$. In such spaces, we attain the global exact controllability of the (BSE) and we provide examples of the main results involving star graphs and tadpole graphs.

Mots clés: perturbation theory; moment problems; quantum graphs; global exact controllability; Schrödinger equation; Bilinear Schrödinger equation; quantum compact graphs; star graphs; tadpole graphs; moments problems

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]; Physique mathématique [math-ph]; Optimisation et contrôle [math.OC]; Théorie spectrale [math.SP]

Alessandro Duca. Bilinear quantum systems on compact graphs: well-posedness and global exact controllability. 2019. ⟨hal-01830297v3⟩