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Geometry of optimal control problems and Hamiltonian systems. (English) Zbl 1170.49035
Nistri, Paolo (ed.) et al., Nonlinear and optimal control theory. Lectures given at the C.I.M.E. summer school, Cetraro, Italy, June 19–29, 2004. Berlin: Springer (ISBN 978-3-540-77644-4/pbk). Lecture Notes in Mathematics 1932, 1-59 (2008).
The main idea of this survey is to explain a general variational and dynamical nature of nice and powerful concepts and results mainly known in the narrow framework of Riemannian geometry such as Jacobi fields, Morse’s index formula, Levi-Civita connection, Riemannian curvature and related topics. In the first part of the paper, the author discusses some geometric constructions and results emerged from the investigation of smooth optimal control problems. By a smooth control system with the state space $$M$$ the author understands a smooth mapping $$f: V\to TM$$, where $$V$$ is a locally trivial bundle over $$M$$ and $$f(V_ q)\subset T_ qM$$ for any fiber $$V_ q$$, $$q\in M$$. An admissible pair is a bounded measurable mapping $$v(\cdot): [t_ 0,t_ 1] \to V$$ such that $$t\mapsto \pi(v(t))=q(t)$$ is a Lipschitzian curve in $$M$$ and $$\dot q(t)=f(v(t))$$ for almost all $$t\in [t_ 0,t_ 1]$$. Integral cost is a functional $$J^{t_ 1}_{t_ 0}(v(\cdot))=\int^{t_ 1}_{t_ 0}\varphi(v(t))\,dt$$, where $$\varphi$$ is a smooth scalar function on $$V$$. The author gives a coordinate free description of Lagrange multipliers and uses them to identify a Banach submanifold of extremal pairs. The Hamiltonian of the optimal control problem and the associated Hamiltonian vector field are developed in a completely intrinsic form. Using second derivatives, the author introduces Lagrangian subspaces which leading the study of the Maslov index of continuous families of Lagrangian subspaces. A geometric version of the Legendre condition shows how the associated monotonicity simplifies the calculation of the Maslov index. The second part of the paper develops the differential geometry of Jacobi curves in the framework of general Hamiltonian systems on cotangent bundles. The author introduces the concept of Jacobi curves for general $$n$$-dimensional foliations of $$2n$$-dimensional manifolds. An intrinsic notion of curvature of curves in the Grassmannian is defined by developing an infinitesimal version of the cross-ratio of quadruples on $$n$$-planes in a $$2n$$-dimensional vector space. This theory of curves in the Grassmannian is applied to Jacobi curves which naturally give rise to a canonical connection associated to the field that generates the curve. The author shows that, for a symplectic manifold, the Hamiltonian flow generates Jacobi curves consisting of Lagrangian subspaces and that the monotonicity of these Jacobi curves is equivalent to the convexity of the Hamiltonian on each leaf of the Lagrangian foliation. Also, it is shown that the negativity of the curvature operator of a regular monotone Hamiltonian implies the hyperbolicity of the associated Hamiltonian flows.
For the entire collection see [Zbl 1137.93004].

MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting 49N60 Regularity of solutions in optimal control 58E25 Applications of variational problems to control theory 49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control