Introductory material
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manifolds, vector fields, tangent spaces;
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orbits of vector fields and Frobenius Theorem;
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controllablity and Chow Theorem;
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drift versus driftless systems, accessibility versus controllability
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Bilinear control systems
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bilinear systems and matrix transition Lie groups;
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structure of matrix Lie groups (homogeneous spaces, transitivity, exponential
map and canonical coordinates)
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Lie algebras (Levi decomposition, semisimpicity, solvability, nilpotency,
Cartan criteria);
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controllability properties for bilinear control systems on matrix Lie groups;
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series expansions of control systems;
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nonholonomic behavior in robotic systems: examples systems with of first
order nonholonomic constraints (trailer systems, chained form);
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examples of control systems on Lie groups (rigid bodies on SO(3) and SE(3);
system on a sphere)
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Open-loop control methods for bilinear systems
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piecewise-constant controls
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feedback linearization
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system inversion
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averaging and oscillatory control
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Application of bilinear systems: finite dimensional Schrodinger
equation
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matrix transition Lie group of a Schordinger equation: SU(N);
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controllability on semisimple Lie algebras;
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control by piecewise constant inputs;
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functional series expansions in the controls: Magnus expansion and product
of exponentials expansions;
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application to elementary unitary gates synthesis for "quantum circuits";
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example two-level system: qbit.
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