CONTROLLABILITY OF PERIODIC BILINEAR QUANTUM SYSTEMS ON INFINITE GRAPHS

Kaïs Ammari (1) , Alessandro Duca (2)

1 Département de Mathématiques [Monastir]
2 IF - Institut Fourier

In this work, we study the controllability of the bilinear Schrödinger equation on infinite graphs for periodic quantum states. We consider the equation (BSE) $i\partial_t\psi = −\Delta \psi+ u(t)B\psi$ in the Hilbert space $L^2_p$ composed by functions defined on an infinite graph $\mathcal{G}$ verifying periodic boundary conditions on the infinite edges. The Laplacian $−\Delta$ is equipped with specific boundary conditions, $B$ is a bounded symmetric operator and $u \in L^2 ((0, T), \mathbb{R})$ with $T > 0$. We present the well-posedness of the (BSE) in suitable subspaces of $L^2_p$. In such spaces, we study the global exact controllability and we provide examples involving for instance tadpole graphs and star graphs with infinite spokes.

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

Kaïs Ammari, Alessandro Duca. CONTROLLABILITY OF PERIODIC BILINEAR QUANTUM SYSTEMS ON INFINITE GRAPHS. 2019. ⟨hal-02169344v1⟩