\documentclass{abstract}
\begin{document}

\title{Sufficient Conditions for Strong Local Optimality with Applications to Biomedical Problems}
\author{Urszula Ledzewicz}
\institute{Southern Illinois University, USA}
\email{uledzew@siue.edu}
\coauthor{Heinz Sch\"attler}

\mymaketitle

\section{Abstract}

We consider optimal control problems over a fixed interval for multi-input
bilinear dynamical systems with both $L_1$ and $L_2$-type objectives
in the presence of control constraints. Problems of this type arise as mathematical models for cancer
chemotherapy over an a priori specified fixed therapy horizon. The extremals resulting
from an application of the Pontryagin maximum principle are analyzed. Conditions are given that allow to embed extremals
into a field of broken extremals leading to easily verifiable sufficient conditions for strong local optimality.
In the case of an $L_1$-type objective, flows of extremal bang-bang trajectories arise for which a simple
algorithmic procedure to verify local optimality will be formulated.
In the case of an $L_2$-type objective, sufficient conditions for
strong local optimality that are based on the existence of a bounded
solution to a matrix Riccati differential equation will be formulated.
The theory is illustrated with a $3$-compartment model for multi-drug cancer chemotherapy with cytotoxic
and cytostatic agents.


\paragraph{Acknowledgments}
This material is based upon work supported by the National Science Foundation
under collaborative research Grants Nos.\ DMS 1311729/1311733.

\newpage

\begin{thebibliography}{9}

\bibitem{Muizenberg1} \textsc{U.~Ledzewicz, H.~Sch\"{a}ttler, M.~Reisi Gahrooi and S.~Mahmoudian Dehkordi},
\emph{On the MTD paradigm and optimal control for multi-drug cancer chemotherapy},
Mathematical Biosciences and Engineering (MBE), \textbf{10}, (2013), pp.\ 803--819, doi:10.3934/mbe.2013.10.803.

\bibitem{het} \textsc{U.~Ledzewicz and H.~Sch\"{a}ttler},
\emph{On optimal chemotherapy for heterogeneous tumors},
J.\ of Biological Systems, (2014), to appear.

\bibitem{GOC} \textsc{H.~Sch\"{a}ttler and U.~Ledzewicz},
\emph{Geometric Optimal Control - Theory, Methods and Examples},
Springer Verlag, 2012.

\end{thebibliography}

\end{document}
