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On local state optimality of bang-bang extremal. (English) Zbl 1169.49019
Summary: Free horizon optimal control problems are studied where the cost functional is given by
\[ C(T,\xi,u)=c_0(\xi(0))+c_f(\xi(T))+\int^T_0f^0(\xi(t),u(t))\,dt. \]
Sufficient second-order conditions are given for the trajectory \(\widehat\xi\) of a bang-bang regular Pontryagin extremal \((\widehat T,\widehat\xi,\widehat u)\) to be state locally optimal.
The control system is control-affine and the controls take values in a polyhedron. The state space and the end points constraints are smooth finite-dimensional manifolds. The hypotheses made concern the positivity of the second variation of the finite-dimensional sub-problem obtained by perturbation of the switching times only and the injectivity of the reference trajectory \(\widehat\xi\).

49K15 Optimality conditions for problems involving ordinary differential equations
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
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