A Panorama of Geometrical Optimal Control Theory

1982

A general formalism is introduced for the optimal control problem on manifolds. It is based on a general formulation of Lagrange's multiplier theorem and recent definitions of nonlinear control systems. It is shown that we can give Pontryagin's maximum principle in this formalism. We expect that the problem formulation given in this paper is particularly suitable for application of modern results about controllability etc. in nonlinear control systems.

1998

This is the second article in the series that began in . Jacobi curves were defined, computed, and studied in that paper for regular extremals of smooth control systems. Here we do the same for singular extremals. The last section contains a feedback classification and normal forms of generic single-input affine in control systems on a 3-dimensional manifold.

Springer INdAM Series, 2015

This paper is devoted to second-order necessary optimality conditions for the Mayer optimal control problem with an arbitrary closed control set U ⊂ R m. Admissible controls are supposed to be measurable and essentially bounded. Using second order tangents to U , we first show that ifū(•) is an optimal control, then an associated quadratic functional should be nonnegative for all elements in the second order jets to U alongū(•). Then we specify the obtained results in the case when U is given by a finite number of C 2-smooth inequalities with positively independent gradients of active constraints. The novelty of our approach is due, on one hand, to the arbitrariness of U. On the other hand, the proofs we propose are quite straightforward and do not use embedding of the problem into a class of infinite dimensional mathematical programming type problems. As an application we derive new second-order necessary conditions for a free end-time optimal control problem in the case when an optimal control is piecewise Lipschitz.

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.

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