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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 $$\widehat\xi$$.

##### MSC:
 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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