LAGRANGIAN SUBMANIFOLD LANDSCAPES OF NECESSARY CONDITIONS FOR MAXIMA IN OPTIMAL CONTROL: GLOBAL PARAMETERIZATIONS AND GENERALIZED SOLUTIONS

摘要

We construct global generating functions of the initial and of the evolution Lagrangian sub-manifolds related to a Hamiltonian flow. These global parameterizations are realized by means of Amann-Conley-Zehnder reduction. In some cases, we have to to face generating functions that are weakly quadratic at infinity; more precisely, degeneracy points can occurs. Therefore, we develop a theory which allows us to treat possibly degenerate cases in order to define a Chaperon-Sikorav-Viterbo weak solution of a time-dependent Hamilton-Jacobi equation with a Cauchy condition given at time t = T (T > 0). The starting motivation is to study some aspects of Mayer problems in optimal control theory.