arXiv subjects

Ugo Boscain

Publications and source records attributed to Ugo Boscain.

77 records · Page 5Linked to original sources

arXiv2005

Time Minimal Trajectories for two-level Quantum Systems with Drift

On a two-level quantum system driven by an external field, we consider the population transfer problem from the first to the second level, minimizing the time of transfer, with bounded field amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds. Let $(-E,E)$ be the two energy levels, and $|Ω(t)|\leq M$ the bound on the field amplitude. For each values of $E$ and $M$, we provide the explicit expression of the time optimal trajectory steering the state one to the state two in terms of a parameter that should be computed numerically. For $M< E$ the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. Fixed $E$ we also prove that for $M\to\infty$ the time needed to reach the state two tends to zero. Finally we compare these results with some known results of Khaneja, Brockett and Glaser and with those obtained in the Rotating Wave Approximation.

quant-ph↗

arXiv2005

Stability of Planar Nonlinear Switched Systems

We consider the time-dependent nonlinear system $\dot q(t)=u(t)X(q(t))+(1-u(t))Y(q(t))$, where $q\in\R^2$, $X$ and $Y$ are two %$C^\infty$ smooth vector fields, globally asymptotically stable at the origin and $u:[0,\infty)\to\{0,1\}$ is an arbitrary measurable function. Analysing the topology of the set where $X$ and $Y$ are parallel, we give some sufficient and some necessary conditions for global asymptotic stability, uniform with respect to $u(.)$. Such conditions can be verified without any integration or construction of a Lyapunov function, and they are robust under small perturbations of the vector fields.

math.OC↗

arXiv2004

Common Polynomial Lyapunov Functions for Linear Switched Systems

In this paper, we consider linear switched systems $\dot x(t)=A_{u(t)} x(t)$, $x\in\R^n$, $u\in U$, and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching ({\bf UAS} for short). We first prove that, given a {\bf UAS} system, it is always possible to build a common polynomial Lyapunov function. Then our main result is that the degree of that common polynomial Lyapunov function is not uniformly bounded over all the {\bf UAS} systems. This result answers a question raised by Dayawansa and Martin. A generalization to a class of piecewise-polynomial Lyapunov functions is given.

math.OC↗

arXiv2004

Resonance of Minimizers for $n$-level Quantum Systems with an Arbitrary Cost

We consider an optimal control problem describing a laser-induced population transfer on a $n$-level quantum system. For a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for $n=2$ and $n=3$): instead of looking for minimizers on the sphere $S^{2n-1}\subset\C^n$ one is reduced to look just for minimizers on the sphere $S^{n-1}\subset \R^n$. Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.

quant-ph↗

arXiv2002

On the $K+P$ Problem for a Three-level Quantum System: Optimality Implies Resonance

We apply techniques of subriemannian geometry on Lie groups to laser-induced population transfer in a three-level quantum system. The aim is to induce transitions by two laser pulses, of arbitrary shape and frequency, minimizing the pulse energy. We prove that the Hamiltonian system given by the Pontryagin Maximum Principle is completely integrable, since this problem can be stated as a ``$\k\oplus\p$ problem'' on a simple Lie group. Optimal trajectories and controls are exhausted. The main result is that optimal controls correspond to lasers that are ``in resonance''.

math.OC↗