Lagrange lemma and the optimal control of diffusions II: nonlinear Lagrange functionals

In (Kosmol, 1991; Kosmol and Pavon, 1993) a new elementary approach to optimal control problems relying on the Lagrange lemma was described which appears to be technically, and conceptually, much simpler than existing methods, and, furthermore, provides a unified variational approach. In (Kosmol and Pavon, 1992) this method was further clarified and developed for linear Lagrangefunctionals. We devote this second paper to nonlinear Lagrangefunctionals. The power of this approach is here clearly demonstrated. In particular, it is shown that the nonlinear Lagrange functional induced by the value function of the problem is just one of the many functionals which may effectively be employed to solve control problems, see Examples 1 and 2 in Section 3. Moreover, in our approach, we can deal in the same framework with problems with state constraints, or nonsmooth problems. Hence, the Lagrange functionals approach provides new tools to solve control problems which may be readily applied even when existing approaches fail. Keyword~." Optimal control; Variational methods; Calculus of variations; Lagrange lemma