Hilbert manifolds with corners of finite codimension and the theory of optimal control

This survey presents a version of Palais-Smale theory for Hilbert manifolds that is convenient for investigation of optimal control problems associated with smooth control systems of constant rank. Results obtained with A. A. Agrachev concerning the finite-dimensional case — the analog of Morse theory for (finite-dimensional) manifolds with corners are given. Simple applications of the theory are discussed. A necessary condition for global controllability of systems of constant rank is obtained, as well as a dual result on the multiplicity of solutions for the corresponding optimization problems.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 28, pp. 96–171, 1990.