Pontryagin Maximum Principle for Control Systems on Infinite Dimensional   Manifolds
  We discuss a mathematical framework for analysis of optimal control problems on infinite-dimensional manifolds. Such problems arise in study of optimization for partial differential equations with some symmetry. It is shown that some nonsmooth analysis methods and Lagrangian charts techniques can be used for study of global variations of optimal trajectories of such control systems and derivation of maximum principle for them. 
Robert J. Kipka and Yuri S. Ledyaev
math.OC