\documentclass{abstract}
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\title{The Minimum Principle for State-Constrained Optimal Control Problems with Time Delays}
\author{Helmut Maurer}
\institute{University of M\"unster, Institute of Computational and Applied Mathematics}
\email{maurer@math.uni-muenster.de}
% \coauthor{Coauthor 1,  Coauthor 2, \and Coauthor 3} % comment out if not needed
\coauthor{Laurenz G\"ollmann and  Richard Vinter}
\mymaketitle

\section{Abstract}
We study the following state-constrained optimal control problem with multiple time delays:
$$
\mbox{Minimize}\quad J(u,x)=g(x(T))
$$
over  measurable functions  $u: [0,T] \rightarrow \mathbb{R}^m$
and arcs $\, x \in W^{1,1}([0,T],\mathbb{R}^n)$
satisfying
% subject to the retarded differential equation, boundary conditions
% and mixed control-state inequality constraints
%%=============================================================
\begin{eqnarray*}
&& \dot x(t)=f(t,x(t-d_0),\ldots,x(t-d_k),u(t-d_0),\ldots,u(t-d_k))
\; \mbox{a.e.}\; t \in [0,T],
\\
&& x(t) = x_0(t) \quad \forall \; t \in [-d_k,0], \quad  \psi(x(T))=0,
\\
&& u(t) = u_0(t) \quad   \forall \; t \in [-d_k,0),
\quad u(t) \in U \subset \mathbb{R}^m \;\mbox{a.e.} \; t \in [0,T],
% \\
% && \psi(x(T))=0,
\\
&& S(t,x(t-d_0),\ldots,x(t-d_k)) \leq 0
\; \, \mbox{a.e.}\;\; t\in [0,T],
\end{eqnarray*}
the data for which comprise an interval $[0,T]$, constant time delays \\
$0=d_0 < d_1 < \ldots < d_k , \, (k \geq 1)$, Lipschitz functions  $ g:\mathbb{R}^{n}\rightarrow \mathbb{R},
\, f:[0,T]\times\mathbb{R}^{(k+1)\cdot n}\times\mathbb{R}^{(k+1)\cdot m}\rightarrow \mathbb{R}^n, \,
 \psi :\mathbb{R}^{n}\rightarrow \mathbb{R}^q \;(0\leq q\leq n),$ and
$ S:[0,T]\times\mathbb{R}^{(k+1)n}\rightarrow \mathbb{R} $  defining the state constraint. The  initial functions satisfy
$x_0(\cdot) \in L^{\infty}([0,T],\mathbb{R}^n)$ and $\, u_0(\cdot) \in L^{\infty}([0,T],\mathbb{R}^m)$.

 We assume that the delays {\it  commensurate} which means that there exists $\Delta > 0$ and integers $0=n_0 < n_1 < \ldots < n_k$ with $ d_j = \Delta \cdot n_j$ for $j=0,1,\ldots,k$.
The transformation technique proposed by Guinn \cite{Guinn} then permits the replacement of the delayed optimal control problem by a non-delayed control problem on the interval $[0,\Delta]$ to which we may apply
the Minimum Principle for non-delayed  problems with state constraints; cf. e.g. Maurer \cite{Maurer}, Vinter \cite{Vinter}. The backward transformation of the adjoint variables and multipliers
furnishes  a Minimum Principle for the delayed control problem; cf. also G\"ollmann, Maurer  \cite{Goellmann-Maurer}.

We briefly sketch numerical discretization and optimization methods which are similar to those in
\cite{Goellmann-Maurer}. The theory and numerical approach are illustrated by two examples  in chemical engineering, resp., in biomedicine that extend optimal control models in  Dadebo, Luus
\cite{Dadebo-Luus}, resp., Stengel et al. \cite{Stengel}.




\paragraph{Acknowledgments}
% Funding acknowledgments (if required)
H. Maurer and R. Vinter were supported by the European Union under the 7th Framework Programme
FP7--PEOPLE--2010--ITN, Grant agreement number 264735--SADCO.

\begin{thebibliography}{9}
%
\bibitem{Dadebo-Luus}
{\sc S. Dadebo and R. Luus},
{\em Optimal control of time-delay systems by dynamic  programming},
Optimal Control Applications \& Methods, 13 (1992), pp.~29--41.
%
\bibitem{Goellmann-Maurer}
{\sc L. G\"ollmann, H. Maurer}, {\em Theory and applications of optimal control problems with multiple time-delays}, Special Issue on Computational Methods for Optimization and Control, J. of Industrial and Management Optimization, 10 (2014),  pp.~413--441.
%
\bibitem{Guinn}
{\sc T. Guinn}, {\em Reduction of delayed optimal control problems to non-delayed problems},
J. Optimization Theory and Applications, 18 (1976), pp~371--377.
%
\bibitem{Maurer}
{\sc H. Maurer}: {\em On the Minimum Principle for optimal control problems with state constraints},
Rechenzentrum der Universit\"at M\"unster, Report 41, M\"unster, Germany, 1979.
%
\bibitem{Stengel}
{\sc R. F. Stengel, R. Ghigliazza, N. Kulkarni and O. Laplace},
{\em Optimal control of innate immune response},
Optimal Control Applications and Methods, 23 (2002), pp.~ 91--104.
%
\bibitem{Vinter}
{\sc R.~Vinter}, {\em Optimal Control}, Modern Birkh\"auser Classics, Reprint of the 2000 Edition, Birkh\"auser, Boston, 2010.

\end{thebibliography}

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