Synthesis Theory in Optimal Control

Boscain, Ugo; Piccoli, Benedetto

doi:10.1007/978-1-4471-5058-9_50

Cited by 4 publications

(9 citation statements)

References 40 publications

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“…We refer to [1] for a general version of PMP and also [12] for a version of PMP on two-dimensional manifolds for single input control affine systems. The PMP is a first order necessary condition for optimality.…”

Section: Basic Definitions and Pontryagin Maximum Principlementioning

confidence: 99%

“…In this section we briefly recall the theory of time-optimal syntheses on 2D-manifolds for systems of the form (3.3), developed by Sussmann, Bressan, Piccoli and Boscain in [11,13,36,46] and rewritten in [12]. This section is written to be as much self-consistent as possible.…”

Section: An Overview On Optimal Synthesis On 2d-manifoldsmentioning

confidence: 99%

“…These loci are fundamental in the construction of the optimal synthesis. In fact, following [12], assuming that they are smooth embedded one dimensional submanifolds of M we have the following:…”

Section: Basic Definitionsmentioning

confidence: 99%

“…the number of switchings of extremal trajectories, canceling at each step the non optimal trajectories (see [12,Chapter 1]). In the optimal syntheses some special curves appears, namely:…”

Section: Basic Definitionsmentioning

confidence: 99%

“…For the problem (R), under generic conditions on the vector fields F and G, one can make the complete classification of synthesis singularities, stable synthesis, singularities of the minimum time wave fronts. We refer to [12] for the general theory. For some extensions to higher dimension, see [2,41].…”

Section: More On Singular Extremals and Predicting Switchings For 2d-mentioning

confidence: 99%

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Lyapunov and Minimum-Time Path Planning for Drones

et al. 2014

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In this paper, we study the problem of controlling an unmanned aerial vehicle (UAV) to provide a target supervision and/or to provide convoy protection to ground vehicles.We first present a control strategy based upon a Lyapunov-LaSalle stabilization method to provide supervision of a stationary target. The UAV is expected to join a pre-designed admissible circular trajectory around the target which is itself a fixed point in the space.Our strategy is presented for both HALE (High Altitude Long Endurance) and MALE (Medium Altitude Long Endurance) types UAVs. A UAV flying at a constant altitude (HALE type) is modeled as a Dubins vehicle (i.e. a planar vehicle with constrained turning radius and constant forward velocity). For a UAV that might change its altitude (MALE type), we use the general kinematic model of a rigid body evolving in R 3 . Both control strategies presented are smooth and unlike what is usually proposed in the literature these strategies asymptotically track a circular trajectory of exact minimum turning radius.We also present the time-optimal control synthesis for tracking a circle by a Dubins vehicle. This optimal strategy, although much simpler than the point-to-point time-optimal strategy obtained by P. Souères and J.-P. Laumond in the 1990s (see [45]), is very rich.Finally, we propose control strategies to provide supervision of a moving target, that are based upon the previous ones.

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Section: Basic Definitions and Pontryagin Maximum Principlementioning

confidence: 99%

Section: An Overview On Optimal Synthesis On 2d-manifoldsmentioning

confidence: 99%

Section: Basic Definitionsmentioning

confidence: 99%

“…the number of switchings of extremal trajectories, canceling at each step the non optimal trajectories (see [12,Chapter 1]). In the optimal syntheses some special curves appears, namely:…”

Section: Basic Definitionsmentioning

confidence: 99%

Section: More On Singular Extremals and Predicting Switchings For 2d-mentioning

confidence: 99%

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Lyapunov and Minimum-Time Path Planning for Drones

et al. 2014

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Time Minimal Trajectories for two-level Quantum Systems with Drift

Boscain

Mason

Proceedings of the 44th IEEE Conference on Decision and Control

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Abstract-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)| ≤ 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, every time optimal trajectory is periodic (and in particular bang-bang) with frequency of the order of the resonance frequency ωR = 2E.On the other side, 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 → ∞ 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.

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Training Schrödinger’s cat: quantum optimal control

et al. 2015

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Abstract. It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a dynamical system from a given initial state into a desired target state with minimized expenditure of energy and resources. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantumenhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems. We address key challenges and sketch a roadmap for future developments. ForewordThe authors of this paper represent the QUAINT consortium, a European Coordination Action on Optimal Control of Quantum Systems, funded by the European Commission Framework Programme 7, Future Emerging Technologies FET-OPEN programme and the Virtual Facility for Quantum Control (VF-QC). This consortium has considerable expertise in optimal control theory and its applications to quantum systems, both in existing areas, such as spectroscopy and imaging, and in emerging quantum technologies, such as quantum information processing, quantum communication, quantum simulation a e-mail: fwm@lusi.uni-sb.de and quantum sensing. The list of challenges for quantum control has been gathered by a broad poll of leading researchers across the communities of general and mathematical control theory, atomic, molecular-, and chemical physics, electron and nuclear magnetic resonance spectroscopy, as well as medical imaging, quantum information, communication and simulation. 144 experts in these fields have provided feedback and specific input on the state of the art, mid-term and long-term goals. Those have been summarized in this document, which can be viewed as a perspectives paper, providing a roadmap for the future development of quantum control. Because such an endeavour can hardly ever be complete (there are many additional areas of quantum control applications, such as spintronics, nano-optomechanical technologies etc.), this roadmap

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Synthesis Theory in Optimal Control

Cited by 4 publications

References 40 publications

Lyapunov and Minimum-Time Path Planning for Drones

Lyapunov and Minimum-Time Path Planning for Drones

Time Minimal Trajectories for two-level Quantum Systems with Drift

Training Schrödinger’s cat: quantum optimal control

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