@misc{Osmolovskii_Nikolai_Equivalence_2007,
 author={Osmolovskii, Nikolai and Maurer, Helmut},
 copyright={Creative Commons Attribution BY 4.0 license},
 journal={Raport Badawczy = Research Report},
 address={Warszawa},
 howpublished={online},
 year={2007},
 publisher={Instytut Badań Systemowych. Polska Akademia Nauk},
 publisher={Systems Research Institute. Polish Academy of Sciences},
 language={eng},
 abstract={In Part 1 of this paper (Osmolovskii and Maurer, 2005), we have summarized the main results on the equivalence of two quadratic forms from which second order necessary and sufficient conditions can be derived for optimal bang-bang control problems. Here, in Part 2, we give detailed proofs and elaborate explicit rela- tions between Lagrange multipliers and elements of the critical cones in both approaches. The main analysis concerns the derivation of formulas for the first and second order derivatives of trajectories with respect to variations of switching times, initial and final time and ini- tial point. This leads to explicit representations of the second order derivatives of the Lagrangian for the induced optimization problem. Based on a suitable transformation, we obtain the elements of the Hessian of the Lagrangian in a form which involves only first order variations of the nominal trajectory. Finally, a careful regrouping of all terms allows us to find the desired equivalence of the two quadratic forms.},
 type={Text},
 title={Equivalence of second order optimality conditions for bang–bang control problems. Part 2 : Proofs, variational derivatives and representations},
 URL={http://www.rcin.org.pl/Content/217459/PDF/RB-2007-85.pdf},
}