A panorama of geometrical optimal control theory

Abstract

Control theory is a young branch of mathematics that has developed mostly in the realm of engineering problems. It is splitted in two major branches; control theory of problems described by partial differential equations where control are exercized either by boundary terms and/or inhomogeneous terms and where the objective functionals are mostly quadratic forms; and control theory of problems described by parameter dependent ordinary differential equations. In this case it is more frequent to deal with non-linear systems and non-quadratic objective functionals. In spite that control theory can be consider part of the general theory of differential equations, the problems that inspires it and some of the results obtained so far, have configured a theory with a strong and definite personality that is already of- fering interesting returns to its ancestors. For instance, the geometrization of nonlinear affine-input control theory problems by introducing Lie-geometrical methods into its analysis started already by R. Brockett is inspiring classical Riemannian geometry and creating what is called today subriemannian geometry.