Geometric control theory I: mathematical foundations

TL;DR

This paper develops a geometric framework for control theory, focusing on the extremals of action functionals with non-holonomic constraints, and revisits classical principles like Pontryagin's maximum principle through a tensorial, variational approach.

Contribution

It introduces a geometric, tensorial approach to control theory, providing new insights into extremals, abnormality indices, and classical variational principles within a unified framework.

Findings

Revisits Pontryagin maximum principle geometrically

Provides a tensorial foundation for control extremals

Establishes conditions for finite deformations with fixed endpoints

Abstract

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.