arXiv subjects

Morgan Morancey

Publications and source records attributed to Morgan Morancey.

12 recordsLinked to original sources

arXiv2019

Minimal time for the continuity equation controlled by a localized perturbation of the velocity vector field

In this work, we study the minimal time to steer a given crowd to a desired configuration. The control is a vector field, representing a perturbation of the crowd velocity, localized on a fixed control set. We will assume that there is no interaction between the agents. We give a characterization of the minimal time both for microscopic and macroscopic descriptions of a crowd. We show that the minimal time to steer one initial configuration to another is related to the condition of having enough mass in the control region to feed the desired final configuration. The construction of the control is explicit, providing a numerical algorithm for computing it. We finally give some numerical simulations.

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arXiv2018

Minimal time problem for discrete crowd models with a localized vector field

In this work, we study the minimal time to steer a given crowd to a desired configuration. The control is a vector field, representing a perturbation of the crowd velocity, localized on a fixed control set. We characterize the minimal time for a discrete crowd model, both for exact and approximate controllability. This leads to an algorithm that computes the control and the minimal time. We finally present a numerical simulation.

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arXiv2017

Controllability and optimal control of the transport equation with a localized vector field

We study controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling such system with a control being a Lipschitz vector field on a fixed control set $ω$. We prove that, for each initial and final configuration, one can steer one to another with such class of controls only if the uncontrolled dynamics allows to cross the control set $ω$. We also prove a minimal time result for such systems. We show that the minimal time to steer one initial configuration to another is related to the condition of having enough mass in $ω$ to feed the desired final configuration.

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arXiv2017

Approximate and exact controllability of the continuity equation with a localized vector field

We study the controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling it with a control being a vector field, representing a perturbation of the velocity, localized on a fixed control set. We prove that, for each initial and final configuration, one can steer approximately one to another with Lipschitz controls when the uncontrolled dynamics allows to cross the control set. We also show that the exact controllability only holds for controls with less regularity, for which one may lose uniqueness of the associated solution.

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arXiv2017

Application of the boundary control method to partial data Borg-Levinson inverse spectral problem

We consider the multidimensional Borg-Levinson problem of determining a potential $q$, appearing in the Dirichlet realization of the Schrödinger operator $A_q=-Δ+q$ on a bounded domain $Ω\subset \mathbb{R}^n$, $n\geq2$, from the boundary spectral data of $A_q$ on an arbitrary portion of $\partialΩ$. More precisely, for $γ$ an open and non-empty subset of $\partialΩ$, we consider the boundary spectral data on $γ$ given by $\mathrm{BSD}(q,γ):=\{(λ_{k},{\partial_νϕ_{k}}_{|\overlineγ}):\ k \geq1\}$, where $\{ λ_k:\ k \geq1\}$ is the non-decreasing sequence of eigenvalues of $A_q$, $\{ ϕ_k:\ k \geq1 \}$ an associated Hilbertian basis of eigenfunctions, and $ν$ is the unit outward normal vector to $\partialΩ$. We prove that the data $\mathrm{BSD}(q,γ)$ uniquely determine a bounded potential $q\in L^\infty(Ω)$. Previous uniqueness results, with arbitrarily small $γ$, assume that $q$ is smooth. Our approach is based on the Boundary Control method, and we give a self-contained presentation of the method, focusing on the analytic rather than geometric aspects of the method.

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arXiv2016

Rapid Stabilization of a Linearized Bilinear $1-D$ Schrödinger Equation

We consider the one dimensional Schrödinger equation with a bilinear control and prove the rapid stabilization of the linearized equation around the ground state. The feedback law ensuring the rapid stabilization is obtained using a transformation mapping the solution to the linearized equation on the solution to an exponentially stable target linear equation. A suitable condition is imposed on the transformation in order to cancel the non-local terms arising in the kernel system. This conditions also insures the uniqueness of the transformation. The continuity and invertibility of the transformation follows from exact controllability of the linearized system.

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arXiv2014

Approximate controllability for a 2D Grushin equation with potential having an internal singularity

This paper is dedicated to approximate controllability for Grushin equation on the rectangle $(x,y) \in (-1,1) \times (0,1)$ with an inverse square potential. This model corresponds to the heat equation for the Laplace-Beltrami operator associated to the Grushin metric on $\mathbb{R}^2$, studied by Boscain and Laurent. The operator is both degenerate and singular on the line $\{ x=0 \}$. The approximate controllability is studied through unique continuation of the adjoint system. For the range of singularity under study, approximate controllability is proved to hold whatever the degeneracy is. Due to the internal inverse square singularity, a key point in this work is the study of well-posedness. An extension of the singular operator is designed imposing suitable transmission conditions through the singularity. Then, unique continuation relies on the Fourier decomposition of the 2D solution in one variable and Carleman estimates for the 1D heat equation solved by the Fourier components. The Carleman estimate uses a suitable Hardy inequality.

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arXiv2013

Global exact controllability of a 1D Schrödinger equations with a polarizability term

We consider a quantum particle in a 1D interval submitted to a potential. The evolution of this particle is controlled using an external electric field. Taking into account the so-called polarizability term in the model (quadratic with respect to the control), we prove global exact controllability in a suitable space for arbitrary potential and arbitrary dipole moment. This term is relevant both from the mathematical and physical points of view. The proof uses tools from the bilinear setting and a perturbation argument.

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arXiv2013

Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations

We consider a system of an arbitrary number of \textsc{1d} linear Schrödinger equations on a bounded interval with bilinear control. We prove global exact controllability in large time of these $N$ equations with a single control. This result is valid for an arbitrary potential with generic assumptions on the dipole moment of the considered particle. Thus, even in the case of a single particle, this result extends the available literature. The proof combines local exact controllability around finite sums of eigenstates, proved with Coron's return method, a global approximate controllability property, proved with Lyapunov strategy, and a compactness argument.

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arXiv2013

Simultaneous local exact controllability of 1D bilinear Schrödinger equations

We consider N independent quantum particles, in an infinite square potential well coupled to an external laser field. These particles are modelled by a system of linear Schrödinger equations on a bounded interval. This is a bilinear control system in which the state is the N-tuple of wave functions. The control is the real amplitude of the laser field. For N=1, Beauchard and Laurent proved local exact controllability around the ground state in arbitrary time. We prove, under an extra generic assumption, that their result does not hold in small time if N is greater or equal than 2. Still, for N=2, we prove using Coron's return method that local controllability holds either in arbitrary time up to a global phase or exactly up to a global delay. We also prove that for N greater or equal than 3, local controllability does not hold in small time even up to a global phase. Finally, for N=3, we prove that local controllability holds up to a global phase and a global delay.

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arXiv2013

Local controllability of 1D Schrödinger equations with bilinear control and minimal time

We consider a linear Schrödinger equation, on a bounded interval, with bilinear control. Beauchard and Laurent proved that, under an appropriate non degeneracy assumption, this system is controllable, locally around the ground state, in arbitrary time. Coron proved that a positive minimal time is required for this controllability, on a particular degenerate example. In this article, we propose a general context for the local controllability to hold in large time, but not in small time. The existence of a positive minimal time is closely related to the behaviour of the second order term, in the power series expansion of the solution.

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arXiv2012

Explicit approximate controllability of the Schrödinger equation with a polarizability term

We consider a controlled Schrödinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts non linearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak $H^2$ stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Schrödinger equation and of the averaged equation stay close on every finite time horizon provided that the control is oscillating enough. Combining these two results, we get approximate controllability to the ground state for the polarizability system.

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