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

and several summer schools, in the period 2008-2018. It contains material for an introductory course in sub-Remannan geometry at master or PhD level, as well as material for a more advanced course. The book attempts to be as elementary as possible but, although the main concepts are recalled, it requires a certain ability in managing object in differential geometry (vector fields, differential forms, symplectic manifolds, etc.). We try to avoid as much as possible the use of functional analysis (some is required starting from Chapter 6). We do not require any knowledge in Riemannian geometry. Actually from the book one can extract an introductory course in Riemannian geometry as a special case of sub-Riemannian one, starting from the geometry of surfaces in Chapter 1. There are few other books of sub-Riemannian geometry available. Besides the pioneering book edited by A. Bellaïche and J.-J. Risler [BR96], a nowadays classical reference is the book of R. Montgomery [Mon02], that inspired several of our chapters. More recent books, written in a language similar to the one we use, are those of F. Jean [Jea14] and L. Rifford [Rif14]; see also the collection of lectures notes [BBS16a, BBS16b]. Other related books, although with a different approach, are the monographs [BLU07] and [CDPT07]. Example of an introductory course of sub-Riemannan geometry. Chapters 2, 3 (without the appendices), 4, 7 (without 7.1), 9, 13, 21. Example of an advanced course of sub-Riemannan geometry.