The purpose of this course is to introduce the basic concepts in geometric control, from controllability to optimal control. These concepts will be then applied to the problem of controlling simple quantum mechanical systems that appear often in quantum technologies as Nuclear Magnetic Resonance and for the realization of q-bits for quantum computers.

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Lecture 1. The “control theory problem”: controllability, stabilizability, optimal control. Example of problems arising in quantum mechanics: Nuclear Magnetic Resonance, Stimulated Raman Adiabatic Passages. Systems evolving on two different scales: averaging. Systems, with an unknown parameter. The (finite-dimensional) Schroedinger equation for the wave function and for the propagator. Lecture 2. Families of vector fields. Lie groups and left invariant control systems. Lie brackets. Frobenius theorem. Non-integrable vector distributions. Lecture 3. Controllability 1. The Krener theorem, The Chow theorem. Lecture 4. Controllability 2. Convexification. Killing the drift. The recurrent drift theorem. Applications to finite dimensional quantum systems: the Lie Algebraic Rank Condition. Controlling a Spin 1/2 particle on the Bloch sphere. Lecture 5. Optimal control 1. Formulation of the problem. Existence. Lecture 6. Optimal control 2. The Pontryagin Maximum Principle (proof for minimal energy for affine systems). Lecture 7. Minimal energy for a 2-level system. Lecture 8. Minimum time for a 3-level system. Lecture 9. The adiabatic theorem: averaging. Population transfer for systems presenting conical intersections. Lecture 10. Systems with an unknown parameter. Two level systems: chirp pulses. Three level systems: the STIRAP process. [1] Jurdjevic, Velimir. Geometric control theory. Cambridge Studies in Advanced Mathematics, 52. Cambridge University Press, Cambridge, 1997. [2] D’Alessandro, Domenico Introduction to quantum control and dynamics. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, FL, 2008. [3] A. Agrachev, D. Barilari, and U. Boscain. A Comprehensive Introduction to sub-Riemannian Geometry, volume 181 of Cam- bridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2020. http://people.sissa.it/?barilari/Notes.html. xviii+746 pp

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P.D.1-1 - Dicembre