Geometry and Optimal Control

Optimal control has strongly influenced geometry since the early days of both subjects. In particular, it played a crucial role in the birth of differential geometry in the nineteenth century through the revolutionary ideas of redefining the notion of “straight line” (now renamed “geodesic”) by means of a curve minimization problem, and of emphasizing general invariance and covariance conditions. More recently, modern control theory has been heavily influenced by geometry. One aspect of this influence is the geometrization of the necessary conditions for optimality, which are recast as geometric conditions about reachable sets, thus becoming special cases of the broader question of the structure and properties of these sets. Recently, this has led to a new general version of the finite-dimensional maximum principle, stated here in full detail for the first time. A second aspect—in which Roger Brockett’s ideas have played a crucial role—is the use in control theory of concepts and techniques from differential geometry. In particular, this leads to regarding a control system as a collection of vector fields, and exploiting the algebraic structure given by the Lie bracket operation. This approach has led to new important developments on various nonlinear control problems. In optimal control, the vector-field view has produced invariant formulations of the maximum principle on manifolds—using either Poisson brackets or connections along curves—which, besides being more general and mathematically natural, actually have real advantages for the solution of concrete problems.