Feedback-invariant optimal control theory and differential geometry—I. Regular extremals (original) (raw)
Related papers
Geometry of Optimal Control Problems and Hamiltonian Systems
Lecture Notes in Mathematics, 2008
These notes are based on the mini-course given in June 2004 in Cetraro, Italy, in the frame of a C.I.M.E. school. Of course, they contain much more material that I could present in the 6 hours course. The goal was to give an idea of the general variational and dynamical nature of nice and powerful concepts and results mainly known in the narrow framework of Riemannian Geometry. This concerns Jacobi fields, Morse's index formula, Levi Civita connection, Riemannian curvature and related topics. I tried to make the presentation as light as possible: gave more details in smooth regular situations and referred to the literature in more complicated cases. There is an evidence that the results described in the notes and treated in technical papers we refer to are just parts of a united beautiful subject to be discovered on the crossroads of Differential Geometry, Dynamical Systems, and Optimal Control Theory. I will be happy if the course and the notes encourage some young ambitious researchers to take part in the discovery and exploration of this subject. Contents I Lagrange multipliers' geometry 3
Jacobi fields in optimal control: Morse and Maslov indices
Nonlinear Analysis, 2022
In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in optimal control problems. Using the notion of L-derivatives we construct Jacobi curves, which represent a generalization of Jacobi elds from the classical calculus of variations. This construction includes in particular the previously known constructions for specic types of extremals. We state and prove Morse-type theorems that connect the negative inertia index of the Hessian of the problem to some symplectic invariants of Jacobi curves.
Hamiltonian systems and optimal control
2008
This lecture will highlight the contributions of optimal control theory to geometry and mechanics. The basic object of study is the reachable sets of families of vector fields parametrized by control functions. We will show how the extremal properties of the reachable sets lead to the Hamiltonians and how these Hamiltonians alter our understanding of the classical calculus of variations in which the Euler-Lagrange equation is a focal point of the subject.
Jacobi Fields in Optimal Control I: Morse and Maslov Indices
2018
In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in optimal control problems. Using the notion of L-derivatives we construct Jacobi curves, which represent a generalization of Jacobi fields from the classical calculus of variations. This construction includes in particular the previously known constructions for specific types of extremals. We state and prove Morse-type theorems that connect the negative inertia index of the Hessian of the problem to some symplectic invariants of Jacobi curves.
Optimal control and Hamiltonian dynamics
Extremals of optimal control problems are solutions to Hamiltonian systems. In my talk I am going to show how the intuition and techniques of Optimal Control Theory help to study Hamiltonian Dynamics itself; in particular, to obtain an effective test for the hyperbolicity of invariant sets and to find new systems with the hyperbolic behavior.
Geometric control theory I: mathematical foundations
2007
A geometric setup for control theory is presented. The argument is developed through the study of the extremals of action functionals defined on piecewise differentiable curves, in the presence of differentiable non-holonomic constraints. Special emphasis is put on the tensorial aspects of the theory. To start with, the kinematical foundations, culminating in the so called variational equation, are put on geometrical grounds, via the introduction of the concept of infinitesimal control . On the same basis, the usual classification of the extremals of a variational problem into normal and abnormal ones is also rationalized, showing the existence of a purely kinematical algorithm assigning to each admissible curve a corresponding abnormality index, defined in terms of a suitable linear map. The whole machinery is then applied to constrained variational calculus. The argument provides an interesting revisitation of Pontryagin maximum principle and of the Erdmann-Weierstrass corner conditions, as well as a proof of the classical Lagrange multipliers method and a local interpretation of Pontryagin's equations as dynamical equations for a free (singular) Hamiltonian system. As a final, highly non-trivial topic, a sufficient condition for the existence of finite deformations with fixed endpoints is explicitly stated and proved.
Journal of Mathematical Sciences, 2006
We construct global generating functions of the initial and of the evolution Lagrangian submanifolds, related to an Hamiltonian flow. These global parameterizations are realized by means of Amann-Conley-Zehnder reduction. In some cases we have to to face generating functions that are weakly quadratic at infinity, namely degeneracy points can occurs. Therefore, we develop a theory which allows us to treat possibly degenerate cases in order to define a Chaperon-Sikorav-Viterbo weak solution of a time-dependent Hamilton-Jacobi equation with Cauchy condition given at time t = T (T > 0). The starting motivation is to study some aspects of Mayer problems in Optimal Control Theory.
Non-holonomic equations for the normal extremals in geometric control theory
Journal of Geometry and Physics, 2021
We provide a new and simple system of equations for the normal sub-Riemannian geodesics. These use a partial connection that we show is canonically available, given a choice of complement to the distribution. We also describe conditions which, if satisfied, mean that even this choice of complement is determined canonically, and that this determines a distinguished connection on the tangent bundle. Our approach applies to sub-Riemannian geometry the point of view of nonholonomic mechanics. The geodesic equations obtained split into mutually driving horizontal and complementary parts, and the method allows for particular choices of nice coframes. We illustrate this feature on examples of contact models with non-constant symbols.
Singularities of the Hamiltonian Vectorfield in Optimal Control Problems
Journal of Mathematical Analysis and Applications, 2001
Variational problems with n degrees of freedom give rise by the Pontriaguine. n maximum principle to a hamiltonian vectorfield in T ޒ* , that presents singulari-Ž. ties non-smoothness points when the lagrangean is not convex. For the problems of the calculus of variations, the singularities that occur are points where the hamiltonian vectorfield is not C 0. For optimal control problems, we show that besides these singularities there appear other ones: points where the hamiltonian vectorfield is C 0 but not C 1 , and we classify them.