Time-Optimal Synthesis for Left-Invariant Control Systems on SO(3)

Abstract

Consider the control system (Σ) given byẋ = x(f + ug), where x ∈ SO(3), |u| ≤ 1 and f, g ∈ so(3) define two perpendicular left-invariant vector fields normalized so that f = cos(α) and g = sin(α), α ∈]0, π/4[. In this paper, we provide an upper bound and a lower bound for N (α), the maximum number of switchings for time-optimal trajectories of (Σ). More precisely, we show that N S (α) ≤ N (α) ≤ N S (α) + 4, where N S (α) is a suitable integer function of α such thatThe result is obtained by studying the time optimal synthesis of a projected control problem on RP 2 , where the projection is defined by an appropriate Hopf fibration. Finally, we study the projected control problem on the unit sphere S 2 . It exhibits interesting features which will be partly rigorously derived and partially described by numerical simulations.

Time-optimal universal control of two-level systems under strong driving

“…In the case of π-pulses we note the following: (a) for Ω max ω 1 the duration of the longest control sequence approaches π/ω 1 , corresponding to half a precession cycle of the spin inserted between two δ-like π/2-rotations; (b) the shortest control sequence consists of two opposite bang periods and lasts exactly 2π/ ω 2 1 + Ω 2 max , independent of the ratio of Ω max to ω 1 . These two results are consistent with previous studies of infinite-amplitude fields 23 and shortest possible spin flips 19,20 . …”

“…A similar proof can be found in Ref. 19 of the main text. We begin by parameterizing the time-evolution operator U (t) using three Euler angles,…”

“…Considering the approach of non-harmonic pulse sequences it was shown, with tools of optimal control theory, that the fastest way to steer a qubit on the Bloch sphere from one state to another, using a single driving field of finite amplitude |B (t)| ≤ B max , is a bangbang control 19,20 , i.e. rectangular pulses alternating between the extremal values of the driving field ±B max .…”

Time Minimal Trajectories for two-level Quantum Systems with Drift

“…In particular in [5] it was proved that every time optimal trajectory is a finite concatenation of bang arcs (i.e., corresponding to control a.e. constantly equal to +1 or −1) and singular arcs (i.e., corresponding to singularities of the End point mapping, see for instance [4], that in our case correspond to controls a.e.…”

“…From C. it follows that the first and the last arc on optimal trajectory connecting the north with the south pole are not singular. The following proposition (see [5] for the proof) gives more details on the optimal trajectories starting at the north pole in the case α < π/4. Proposition 2: Consider the control system (4)- (9), and assume α < π/4.…”