Global exact controllability of bilinear quantum systems on compact graphs and energetic controllability

Alessandro Duca (1)

1 IF - Institut Fourier

The aim of this work is to study the controllability of the bilinear Schr\"odinger equation on compact graphs. In particular, we consider the equation (BSE) $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2(\mathscr{G},\mathbb{C})$, with $\mathscr{G}$ being a compact graph. The Laplacian $-\Delta$ is equipped with self-adjoint boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We provide a new technique leading to the global exact controllability of the (BSE) in $D(|\Delta|^{s/2})$ with $s\geq 3$. Afterwards, we introduce the "energetic controllability", a weaker notion of controllability useful when the global exact controllability fails. In conclusion, we develop some applications of the main results involving for instance star graphs.

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

Alessandro Duca. Global exact controllability of bilinear quantum systems on compact graphs and energetic controllability. SIAM Journal on Control and Optimization, 2020, ⟨10.1137/18m1212768⟩. ⟨hal-01875685v4⟩