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Partial differential equation on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schr\"odinger type equations) or to the analysis of flexible structures (wave type equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life sciences such as neurobiology. This paper analyzes the controllability properties of the heat equation on a compact network under the action of a single input bilinear control. By adapting a recent method due to [F.~Alabau-Boussouira, P.~Cannarsa and C.~Urbani, {\em Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control}, arXiv:1811.08806], an exact controllability result to the eigensolutions of the uncontrolled problem is obtained in this work. A crucial step has been the construction of a suitable biorthogonal family under a non-uniform gap condition of the eigenvalues of the Laplacian on a graph. Application to star graphs and tadpole graphs are included., Comment: 20 pages, 4 fugures

International audience; In this work, we consider the bilinear Schr\"odinger equation (BSE) $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2(\mathscr{G},\mathbb{C})$ with $\mathscr{G}$ an infinite 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 study the well-posedness of the (BSE) in suitable subspaces of $D(|\Delta|^{3/2})$ preserved by the dynamics despite the dispersive behaviour of the equation. In such spaces, we study the global exact controllability and the "energetic controllability". We provide examples involving for instance infinite tadpole graphs.

In this paper, we consider an Euler-Bernoulli beam equation with time-varying internal fluid. We assume that the fluid is moving with non-constant velocity and dynamical boundary conditions are satisfied. We prove the existence and uniqueness of global solution under suitable assumptions on the tension of beam and on the parameters of the problem. Afterwards, we establish the exponential stability of the solution by introducing a suitable Lyapunov functional., 15 pages, 1 figure

We consider the Schrodinger equation where $$\Omega (t)\subset \mathbb {R}^N$$ is a moving domain depending on the time $$t\in [0,T]$$ . The aim of this work is to provide a meaning to the solutions of such an equation. We use the existence of a bounded reference domain $$\Omega _0$$ and a specific family of unitary maps $$h^\sharp (t): L^2(\Omega (t),\mathbb {C})\longrightarrow L^2(\Omega _0,\mathbb {C})$$ . We show that the conjugation by $$h^\sharp $$ provides a new equation of the form where $$h_\sharp =(h^\sharp )^{-1}$$ . The Hamiltonian H(t) is a magnetic Laplacian operator of the form $$\begin{aligned} H(t)=-({\text {div}}_x+iA)\circ (\nabla _x+iA)-|A|^2 \end{aligned}$$ where A is an explicit magnetic potential depending on the deformation of the domain $$\Omega (t)$$ . The formulation ( $$**$$ ) enables to ensure the existence of weak and strong solutions of the initial problem ( $$*$$ ) on $$\Omega (t)$$ endowed with Dirichlet boundary conditions. In addition, it also indicates that the correct Neumann-type boundary conditions for ( $$*$$ ) are not the homogeneous but the magnetic ones $$\begin{aligned} \partial _\nu u(t)+i\langle \nu | A\rangle u(t)=0, \end{aligned}$$ even though ( $$*$$ ) has no magnetic term. All the previous results are also studied in the presence of diffusion coefficients as well as magnetic and electric potentials. Finally, we prove some associated by-products as an adiabatic result for slow deformations of the domain and a time-dependent version of the so-called Moser’s trick. We use this outcome in order to simplify Eq. ( $$**$$ ) and to guarantee the well-posedness for slightly less regular deformations of $$\Omega (t)$$ .

International audience; The aim of this work is to study the controllability of infinite bilinear Schr\"odinger equations on a segment. We consider the equations (BSE) $i\partial_t\psi^{j}=-\Delta\psi^j+u(t)B\psi^j$ in the Hilbert space $L^2((0,1),\mathbb{C})$ for every $j\in\mathbb{N}^*$. The Laplacian $-\Delta$ is equipped with Dirichlet homogeneous boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We prove the simultaneous local and global exact controllability of infinite (BSE) in projection. The local controllability is guaranteed for any positive time and we provide explicit examples of $B$ for which our theory is valid. In addition, we show that the controllability of infinite (BSE) in projection ontosuitable finite dimensional spaces is equivalent to the controllability of a finite number of (BSE) (without projecting). In conclusion, we rephrase our controllability results in terms of density matrices.

International audience; 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.

In this work, we study the controllability of the bilinear Schr\"odinger equation on infinite graphs for periodic quantum states. We consider the bilinear Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2_p$ composed by functions defined on an infinite graph $\mathscr{G}$ 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 system in suitable subspaces of $D(|\Delta|^{3/2})$ . In such spaces, we study the global exact controllability and we provide examples involving for instance tadpole graphs and star graphs with infinite spokes., Comment: arXiv admin note: text overlap with arXiv:1811.04273

International audience; We study the Schrödinger equation $i\partial_t \psi = −\Delta\psi + V\psi$ on $L^2((0,1), C)$ where $V$ is a very high and localized potential wall. We aim to perform permutations of the eigenmodes and to control the solution of the equation. We consider the process where the position and the height of the potential wall change as follows. First, the potential increases from zero to a very large value, so a narrow potential wall is formed that almost splits the interval into two parts; then the wall moves to a different position, after which the height of the wall decays to zero again. We show that even though the rate of the variation of the potential's parameters can be arbitrarily slow, this process alternates adiabatic and non-adiabatic dynamics, leading to a non-trivial permutation of the eigenstates. Furthermore, we consider potentials with several narrow walls and we show how an arbitrarily slow motion of the walls can lead the system from any given state to an arbitrarily small neighborhood of any other state, thus proving the approximate controllability of the above Schrödinger equation by means of a soft, quasi-adiabatic variation of the potential.

International audience; We consider the bilinear Schrödinger equation on a bounded one-dimensional domain and we provide explicit times such that the global exact controllability is verified. In addition, we show how to construct controls for the global approximate controllability.

International audience; A major application of the mathematical concept of graph in quantum mechanics is to model networks of electrical wires or electromagnetic wave-guides. In this paper, we address the dynamics of a particle trapped on such a network in presence of an external electromagnetic field. We study the controllability of the motion when the intensity of the field changes over time and plays the role of control. From a mathematical point of view, the dynamics of the particle is modeled by the so-called bilinear Schrödinger equation defined on a graph representing the network. The main purpose of this work is to extend the existing theory for bilinear quantum systems on bounded intervals to the framework of graphs. To this end, we introduce a suitable mathematical setting where to address the controllability of the equation from a theoretical point of view. More precisely, we determine assumptions on the network and on the potential field ensuring its global exact controllability in suitable spaces. Finally, we discuss two applications of our results and their practical implications to two specific problems involving a star-shaped network and a tadpole graph.

In this work, we study the Schrödinger equation $i\partial_tψ = −\Delta\psi + \eta(t)\sum^J_{j=1}\delta_{x=a_j (t)}\psi$ on $L^2 ((0, 1), C)$ where $η : [0, T ]\rightarrow R^+$ and $a_j : [0, T ] \rightarrow (0, 1)$, $j = 1, ..., J$. We show how to permute the energy associated to different eigenmodes of the Schrödinger equation via suitable choice of the functions $η$ and $a_j$. To the purpose, we mime the control processes introduced in [17] for a very similar equation where the Dirac potential is replaced by a smooth approximation supported in a neighborhood of $x = a(t)$. We also propose a Galerkin approximation that we prove to be convergent and illustrate the control process with some numerical simulations.