Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions

In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, discuss their optimality, and calculate the metric spheres, proving their Euclidean rectifiability.Comment: 24 pages, 17 figure

L1L^1-Minimization for Mechanical Systems

Second order systems whose drift is defined by the gradient of a given potential are considered, and minimization of the $L^1$-norm of the control is addressed. An analysis of the extremal flow emphasizes the role of singular trajectories of order two [25,29]; the case of the two-body potential is treated in detail. In $L^1$-minimization, regular extremals are associated with controls whose norm is bang-bang; in order to assess their optimality properties, sufficient conditions are given for broken extremals and related to the no-fold conditions of [20]. An example of numerical verification of these conditions is proposed on a problem coming from space mechanics

Total variation regularization of multi-material topology optimization

This work is concerned with the determination of the diffusion coefficient from distributed data of the state. This problem is related to homogenization theory on the one hand and to regularization theory on the other hand. An approach is proposed which involves total variation regularization combined with a suitably chosen cost functional that promotes the diffusion coefficient assuming prespecified values at each point of the domain. The main difficulty lies in the delicate functional-analytic structure of the resulting nondifferentiable optimization problem with pointwise constraints for functions of bounded variation, which makes the derivation of useful pointwise optimality conditions challenging. To cope with this difficulty, a novel reparametrization technique is introduced. Numerical examples using a regularized semismooth Newton method illustrate the structure of the obtained diffusion coefficient.

Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field

In this paper we consider the minimum time population transfer problem for the $z$-component of the spin of a (spin 1/2) particle driven by a magnetic field, controlled along the x axis, with bounded 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 $|\Omega(t)|\leq M$ the bound on the field amplitude. For each couple of values $E$ and $M$, we determine the time optimal synthesis starting from the level $-E$ and we provide the explicit expression of the time optimal trajectories steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For $M/E<<1$, every time optimal trajectory is bang-bang and in particular the corresponding control is periodic with frequency of the order of the resonance frequency $\omega_R=2E$. On the other side, for $M/E>1$, 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. In the case $M/E>1$ there are time optimal trajectories containing a singular arc. Finally we compare these results with some known results of Khaneja, Brockett and Glaser and with those obtained by controlling the magnetic field both on the $x$ and $y$ directions (or with one external field, but in the rotating wave approximation). As byproduct we prove that the qualitative shape of the time optimal synthesis presents different patterns, that cyclically alternate as $M/E\to0$, giving a partial proof of a conjecture formulated in a previous paper.Comment: 31 pages, 10 figures, typos correcte

Planar tilting maneuver of a spacecraft: singular arcs in the minimum time problem and chattering

In this paper, we study the minimum time planar tilting maneuver of a spacecraft, from the theoretical as well as from the numerical point of view, with a particular focus on the chattering phenomenon. We prove that there exist optimal chattering arcs when a singular junction occurs. Our study is based on the Pontryagin Maximum Principle and on results by M.I. Zelikin and V.F. Borisov. We give sufficient conditions on the initial values under which the optimal solutions do not contain any singular arc, and are bang-bang with a finite number of switchings. Moreover, we implement sub-optimal strategies by replacing the chattering control with a fixed number of piecewise constant controls. Numerical simulations illustrate our results.Comment: 43 pages, 18 figure