This lecture will be devoted to the Maximum Principle for systems with symmetries. We will show that the symmetries can be incorporated into the perturbations of the system along an extremal trajectory resulting in a strengthened version of the Maximum Principle. According to this general principle, each extremal trajectory is the projection of an integral curve of a Hamiltonian vector field generated by a Hamiltonian which is maximal not only relative to the competing Hamiltonians of the system, but is also maximal relative to the symmetry Hamiltonians. We will then focus on two important corollaries that can be drawn from this principle. The first corollary is that the classical Noether’s theorem from the calculus of variations is equally applicable for systems with controls. The second corollary applies to the situations where a Lie group G acts as a group of symmetries on a manifold M on which a control system dx/dt = F(x(t), u(t)) is defined. We will show that the associated Moment Map is constant along each extremal curve of the system. We will then illustrate the significance of these observations for systems in quantum control.
Professor Jurdjevic is one of the founders of geometric control theory. His pioneering work with H.J. Sussmann was deemed among the most influential papers in the development of modern control theory. His book on Geometric Control theory, revealed the geometric origins of the subject and uncovered important connections to physics and geometry. It remains a major source in this subject. Jurdjevic’s expertise also extends to differential geometry, mechanics and integrable systems. His publications cover a wide range of topics including stability theory, Hamiltonian systems on Lie groups and integrable systems. His recent book titled Optimal Control and Geometry: Integrable systems, a synthesis of the calculus of variations, symplectic geometry and control theory, provides a broad basis for understanding problems of applied mathematics. Professor Jurdjevic has spent most of his career at the University of Toronto.
Bienvenue à tous!