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STRONG OPTIMALITY FOR A BANG-BANG TRAJECTORY ANDREI A. AGRACHEV, GIANNA STEFANI, AND PIERLUIGI ZEZZA
 

Summary: STRONG OPTIMALITY FOR A BANG-BANG TRAJECTORY
ANDREI A. AGRACHEV, GIANNA STEFANI, AND PIERLUIGI ZEZZA§
SIAM J. CONTROL OPTIM. c 2002 Society for Industrial and Applied Mathematics
Vol. 41, No. 4, pp. 991­1014
Abstract. In this paper we give sufficient conditions for a bang-bang regular extremal to be
a strong local optimum for a control problem in the Mayer form; strong means that we consider
the C0 topology in the state space. The controls appear linearly and take values in a polyhedron,
and the state space and the end point constraints are finite-dimensional smooth manifolds. In the
case of bang-bang extremals, the kernel of the first variation of the problem is trivial, and hence the
usual second variation, which is defined on the kernel of the first one, does not give any information.
We consider the finite-dimensional subproblem generated by perturbing the switching times, and we
prove that the sufficient second order optimality conditions for this finite-dimensional subproblem
yield local strong optimality. We give an explicit algorithm to check the positivity of the second
variation which is based on the properties of the Hamiltonian fields.
Key words. optimal control, bang-bang controls, sufficient optimality condition, strong local
optima
AMS subject classifications. Primary, 49K15; Secondary, 49K30, 58E25
PII. S036301290138866X
1. Introduction. This paper is part of a general research program whose aim
is to further extend the use of Hamiltonian methods in the study of optimal control

  

Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)

 

Collections: Engineering; Mathematics