, Volume 135, Issue 4, pp 3125-3144

Lagrangian submanifold landscapes of necessary conditions for maxima in optimal control: Global parameterizations and generalized solutions

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Abstract

We construct global generating functions of the initial and of the evolution Lagrangian submanifolds 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.

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.
An erratum to this article is available at http://dx.doi.org/10.1007/s10958-006-0177-8.