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We consider a quantum particle in a 1D infinite square potential well with variable length. It is a nonlinear control system in which the state is the wave function $\phi$ of the particle and the control is the length $l\left(t\right)$ of the potential well. We prove the following controllability result : given ${\phi }_0$ close enough to an eigenstate corresponding to the length $l=1$ and ${\phi }_f$ close enough to another eigenstate corresponding to the length $l=1$, there exists a continuous function $l:[0,T]\to {ℝ}_{+}^{*}$ with $T\>0$, such that $l\left(0\right)=1$ and $l\left(T\right)=1$, and which...

One considers a quantum particle in a 1D moving infinite square potential well. It is a nonlinear control system in which the state is the wave function of the particle and the control is the acceleration of the potential well. One proves the local controllability around any eigenstate, and the steady state controllability (controllability between eigenstates) of this control system. In particular, the wave function can be moved from one eigenstate to another one, exactly and in finite time, by...

The goal of this note is to present the results of the references [5] and [4]. We study the null controllability of the parabolic equations associated with the Grushin-type operator ${\partial }_x^2+{\left|x\right|}^{2\gamma }{\partial }_y^2$ ($\gamma \>0$) in the rectangle $(x,y)\in (-1,1)\times (0,1)$ or with the Kolmogorov-type operator $v^{\gamma }{\partial }_{x}f+{\partial }_v^{2}f$ ($\gamma \in \{1,2\}$) in the rectangle $(x,v)\in 𝕋\times (-1,1)$, under an additive control supported in an open subset $\omega$ of the space domain. We prove that the Grushin-type equation...

We consider a quantum particle in a 1D infinite square potential well with variable length. It is a nonlinear control system in which the state is the wave function  of the particle and the control is the length of the potential well. We prove the following controllability result : given ${\phi }_0$ close enough to an eigenstate corresponding to the length and ${\phi }_f$ close enough to another eigenstate corresponding to the length , there exists a continuous function $l:[0,T]\to {ℝ}_{+}^{*}$ with , such that and , and which moves...

The study of small magnetic particles has become a very important topic, in particular for the development of technological devices such as those used for magnetic recording. In this field, switching the magnetization inside the magnetic sample is of particular relevance. We here investigate mathematically this problem by considering the full partial differential model of Landau-Lifschitz equations triggered by a uniform (in space) external magnetic field.

The study of small magnetic particles has become a very important topic, in particular for the development of technological devices such as those used for magnetic recording. In this field, switching the magnetization inside the magnetic sample is of particular relevance. We here investigate mathematically this problem by considering the full partial differential model of Landau-Lifschitz equations triggered by a uniform (in space) external magnetic field.

We study the null controllability of the parabolic equation associated with the Grushin-type operator $A={\partial }_x^2+{\left|x\right|}^{2\gamma }{\partial }_{\gamma }^2,(\gamma >0)$, in the rectangle $\Omega =(-1,1)\times (0,1)$, under an additive control supported in an open subset $\omega$ of $\Omega$. We prove that the equation is null controllable in any positive time for $\gamma <1$ and that there is no time for which it is null controllable for $\gamma >1$. In the transition regime $\gamma =1$ and when $\omega$ is a strip $\omega =(a,b)\times (0,1)(0<a,b\le 1)$), a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular...