On Turnpike and Dissipativity Properties of Continuous-Time Optimal Control Problems\thanksref{footnoteinfo
% Abstract of not more than 200 words. This paper investigates the relations between three different properties, which are of importance in optimal control problems: dissipativity of the underlying dynamics with respect to a specific supply rate, optimal operation at steady state, and the turnpike property. We show in a continuous-time setting that if along optimal trajectories a strict dissipation inequality is satisfied, then this implies optimal operation at this steady state and the existence of a turnpike at the same steady state. Finally, we establish novel converse turnpike results, i.e., we show that the existence of a turnpike at a steady state implies optimal operation at this steady state and dissipativity with respect to this steady state. We draw upon a numerical example to illustrate our findings.
DissTurn_final.tex
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\begin{document}



\begin{frontmatter}
%\runtitle{Insert a suggested running title}  % Running title for regular 
                                              % papers but only if the title  
                                              % is over 5 words. Running title 
                                              % is not shown in output.

\title{On Turnpike and Dissipativity Properties of Continuous-Time Optimal Control Problems\thanksref{footnoteinfo}} % Title, preferably not more 
                                                % than 10 words.

\thanks[footnoteinfo]{A preliminary version of this paper was presented at  CDC 2014 \cite{epfl:faulwasser14e}.  
Corresponding author: Timm Faulwasser.
}

\author[KIT]{Timm Faulwasser}\ead{ timm.faulwasser@kit.edu},    % Add the 
\author[UCSB]{Milan Korda}\ead{milan.korda@engineering.ucsb.edu},  
\author[EPFL]{Colin N. Jones}\ead{colin.jones@epfl.ch},  
\author[EPFL]{Dominique Bonvin}\ead{dominique.bonvin@epfl.ch}               % e-mail address 

\address[KIT]{Institute for Applied Computer Science, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany}
\address[UCSB]{Department of Mechanical Engineering, University of California, Santa Barbara}
\address[EPFL]{Laboratoire d'Automatique, \'Ecole Polytechnique F\'ed\'erale de Lausanne, CH-1015 Lausanne, Switzerland}  % Please supply                                              


          
\begin{keyword}                           % Five to ten keywords,  
dissipativity, turnpike properties, converse turnpike results, optimal operation at steady state, optimal control, economic MPC             % chosen from the IFAC 
\end{keyword}                             % keyword list or with the 
                                          % help of the Automatica 
                                          % keyword wizard






\begin{abstract}                          % Abstract of not more than 200 words.
This paper investigates the relations between three different properties, which are of importance in optimal control problems: dissipativity of the underlying dynamics with respect to a specific supply rate, optimal operation at steady state, and the turnpike property. We show in a  continuous-time setting that if along optimal trajectories   a strict dissipation inequality is satisfied, then this implies optimal operation at this steady state and the existence of a turnpike at the same steady state. 
Finally, we establish novel converse turnpike results, i.e., we show that the existence of a turnpike at a steady state implies optimal operation at this steady state and dissipativity with respect to this steady state. 
We draw upon a numerical example to illustrate our findings.  
\end{abstract}
\end{frontmatter}


The notion of turnpike property of an optimal control problem (OCP)---introduced in \cite{Dorfman58} in the late 1950s---
is used to describe the phenomenon that in many finite-horizon OCPs the optimal solutions for different initial conditions approach a neighborhood of the best steady state, stay within this neighborhood for some time, and might leave this neighborhood towards the end of the optimization horizon. Turnpike phenomena have been observed in different types of OCPs: with/without terminal constraints  \cite{Carlson91, Clarke13a,  Trelat15a} and with/without discounted cost functionals \cite{Wuerth09, Zaslavski14a, Gurman04a}. 
Turnpikes have received widespread interest in the context of optimal control of economics \cite{McKenzie76, Carlson91}. The works by \cite{Anderson87a, Wilde72a, Rao99a, Sahlodin15} show how turnpike phenomena can be used to approximate solutions of OCPs with long horizons appearing in applications.\footnote{We remark that occasionally turnpike phenomena are denoted by varying names: \cite{Anderson87a, Wilde72a} refer to turnpikes as a \textit{dichotomy of optimal control problems}, while \cite{Rao99a} uses the phrase \textit{hypersensitive optimal control problems}.} Turnpikes also appear in OCPs arising in economic MPC formulations \cite{Wuerth09, epfl:faulwasser15g, epfl:faulwasser15a, kit:faulwasser17a, Gruene13a}.
%
%
%
%      \vspace*{-5mm} 
      Recently, \cite{Stieler14a,Gruene16a} discussed different aspects of turnpike phenomena in a discrete-time setting with constraints and in a continuous-time setting without constraints \cite{Trelat15a}.
Taking into account the large number of publications on turnpike phenomena, it is quite surprising that only very few works state a precise definition of turnpike properties, see \cite{Zaslavski14a, Stieler14a}. 
Often, turnpike results for specific OCPs are proven without a rigorous definition of the turnpike property itself \cite{McKenzie76, Carlson91, Clarke13a, Gurman04a}. 
 While such an approach simplifies the construction of many turnpike results, it hinders establishing \textit{converse turnpike theorems}. 
    \vspace*{-3mm} 
 
  The main goal of this paper is to analyze the relation between three different properties that arise in the context of finite-horizon continuous-time OCPs: system dissipativity with respect to a specific supply rate (which depends on the cost function of the OCP), optimal operation at steady state, and the existence of a turnpike at that steady state. 
    Recently, a related  discrete-time analysis has been presented under the assumptions of local controllability of the turnpike and turnpike-like behavior of nearly optimal solutions \cite{Gruene16a}.
        The present paper takes a different route by avoiding such assumptions in the continuous-time case. Its contributions are as follows: 
While the preliminary version of this paper \cite{epfl:faulwasser14e} discussed \textit{state} turnpikes, we extend these results and provide a framework for the definition of different turnpike properties of OCPs, i.e., we suggest to distinguish \textit{state, input-state}, and \textit{extremal turnpikes} of OCPs. 
%We show for time-invariant OCPs that a dissipativity condition combined with a finite-time reachability assumption is sufficient to guarantee state and input-state turnpike properties.   Additionally, we show that discrete-time results \cite{Mueller14a, Angeli12a} on the relation between dissipativity and optimal operation at steady state carry over to the continuous-time setting. 
%We generalize a turnpike result from  \cite{epfl:faulwasser14e}, i.e. we show that satisfaction of a disspation inequality along optimal trajectories combined with exponential reachability implies turnpike behavior.
Our main contribution are novel converse turnpike results that  require neither local controllability of the turnpike nor turnpike-like behavior of nearly optimal solutions as in \cite{Gruene16a}. In particular, we show that the existence of a turnpike implies optimal operation at steady state; we prove that exactness of turnpikes implies dissipativity, whereby exactness of a turnpike means that the optimal solutions are at the turnpike steady state for some parts of the optimization horizon; and we show that under mild local assumptions on the cost function of the OCP, the existence of a turnpike implies satisfaction of a strict dissipation inequality along optimal solutions. 

 %
 %  We draw upon a continuous chemical reactor to demonstrate that the dissipativity-based turnpike conditions can be verified via sum-of-squares techniques. 
%   \vspace*{-2mm} 
   %\vspace*{-2mm} 
   The remainder of this paper is structured as follows: Section \ref{sec:prelim} introduces a formal definition of turnpike and dissipativity properties as well as the definition of optimal operation at steady state. Section \ref{sec:D} discusses implications of dissipativity. Section \ref{sec:SS} investigates the relation between optimal operation at steady state and dissipativity, while Section \ref{sec:T} presents converse turnpike results. % Sufficient equivalence conditions are stated in Section \ref{sec:equi}. 
   To demonstrate how some of our conditions can be verified, we draw upon the numerical example of a chemical reactor in Section \ref{sec:example}.


\section{Preliminaries and Problem Statement} \label{sec:prelim}
We briefly recall the notions of optimal operation at steady state, dissipativity with respect to a steady state, and turnpike properties of OCPs. 

\subsection{Optimal Steady-State Operation}
We consider the nonlinear system given by
\begin{align}
\dot x &= f(x,u), \quad x(0) = x_0,  \label{eq:sys_def} 
\end{align}
where the states $x \in \Rnx$ and the inputs $u \in \Rnu$ are constrained to lie in the compact sets 
$\mcl{X} \subset \Rnx$ and $\mcl{U} \subset \Rnu$.  We assume that the vector field $f:\Rnx\times\Rnu \to \Rnx$ is Lipschitz on~$\mathcal{X}\times \mathcal{U}$. A solution to  \eqref{eq:sys_def}, starting at $x_0$ at time $0$, driven by the input $u:[0,\infty) \to \mcl{U}$, is denoted as $x(\cdot, x_0, u(\cdot))$.

   \vspace*{-2mm} Consider the maximal control-invariant set $\mcl{X}_0 \subseteq \mcl{X}$ given by
\begin{multline}
\mcl{X}_0 = \left\{x_0 \in \mcl{X}\,|\,\exists\, u(\cdot)\in\mcl{L}([0,\infty), \mcl{U}):\right. \\
\left.\forall t \geq 0 ~x(t,x_0, u(\cdot)) \in \mcl{X}\right\},
\end{multline}
where $\mcl{L}([0,\infty), \mcl{U})$ denotes the class of measurable functions on $[0,\infty)$ taking values in the compact set $\mcl{U}\subset \Rnu$.
This set is the largest subset of $\mcl{X}$ that can be made positively invariant via a control $u(\cdot)$. Here, we assume that $\mathcal{X}_0 \not = \emptyset$. Furthermore, consider a finite-horizon OCP that aims at minimizing
the objective functional
\begin{equation} \label{eq:J}
J_T(x_0, u(\cdot)) = \frac{1}{T}\int_{0}^{T} F(x(t), u(t))\,\mathrm{d}t,
\end{equation}
where $F:\mcl{X} \times \mcl{U}\to \R$ is the cost function, and $T$ is the optimization horizon.  
We assume that $F$ is Lipschitz on $\mcl{X} \times \mcl{U}$.
The OCP reads 
\begin{subequations} \label{eq:OCP}
\begin{align}
\underset{u(\cdot) \in  \mcl{L}([0,T], \mcl{U})}{\inf} & \,J_T(x_0, u(\cdot))\\
%\end{equation}
\hspace*{-1cm}\textrm{ subject to} \nonumber \\ % \eqref{eq:sys_def} and 
%\begin{align} \label{eq:OCP_con}
%\forall t \in  [0,T]: \quad 
\dot x(t) & = f(x(t), u(t)), \quad x(0) = x_0 \\
\forall t \in  [0,T]: \quad x(t) &\in \mcl{X}, \quad u(t) \in \mcl{U}.
\end{align}
\end{subequations}
 The pair $\left(x(\cdot, x_0, u(\cdot)), u(\cdot)\right)$ is called admissible if  $u(\cdot) \in  \mcl{L}([0,T], \mcl{U})$ and if there exists a corresponding absolutely continuous solution $x(\cdot, x_0, u(\cdot))$, which  satisfies $x(t, x_0, u(\cdot)) \in \mcl{X}$  for all $t \in [0,T]$. 
 An optimal solution to \eqref{eq:OCP} is denoted by $u^\star(\cdot)$ and the corresponding state trajectory is written as $x^\star(\cdot, x_0, u^\star(\cdot))$.\footnote{Here, we assume for simplicity that the optimal solution exists and is attained. We refer to \cite{Vinter10, Lee67} for conditions ensuring the existence of optimal solutions to OCP~\eqref{eq:OCP}.}

\underline{Notational remarks:}  
We denote the dependence of optimal solutions to \eqref{eq:OCP} on the initial condition $x_0$ and the horizon length $T$  by writing 
$\OCP{x_0}$. Whenever it is convenient, input-state pairs are written as $z= (x,u)^T$ and the combined input-state constraints are written as $\mcl{Z} = \mcl{X}\times\mcl{U}$.
Throughout this paper, we use the superscript $\bar\cdot$ to denote a variable at steady state. Hence, we have $f(\bar z) = f(\bar x, \bar u) = 0$.
The set of admissible steady-state pairs is denoted as 
%\begin{subequations}
\begin{equation*}
\bar{\mcl{Z}}:= \left\{\bar z \in\mcl{Z}~|~0 = f(\bar z)\right\}.
\end{equation*}
Admissible trajectory pairs of $\OCP{x_0}$ are abbreviated by $z(\cdot, x_0) = (x(\cdot, x_0, u(\cdot)), u(\cdot))^T$.
For any function $\varphi$ with domain $\mbb{R}^{n_x+n_u}$ we write
\[
\varphi\left(z^\star(t, x_0)\right) := \varphi\left(x^\star(t, x_0, u^\star(\cdot, x_0)), \,u^\star(t, x_0)\right).
\]
While $\OCP{x_0}$ aims at optimizing the transient performance of system \eqref{eq:sys_def}, one can as well ask for the best stationary operating conditions. 
These conditions are given by the following steady-state problem: 
\begin{align} \label{eq:OPT_ss}
\underset{\bar z\,\in\,\mathbb{R}^{n_x+n_u}}{\inf} \, F(\bar z)\qquad \textrm{subject to } \bar z  \in \bar{\mcl{Z}}% \\%\label{eq:cost}\\
%\textrm{subject to\quad} & \bar z  \in \bar{\mcl{Z}}, \nonumber 
\end{align}
where $F$ is the same as in \eqref{eq:J}. 
A globally optimal solution to this static optimization problem is denoted as $\bar z^\star$.
The set of optimal steady-state pairs is denoted by $\bar{\mcl{Z}}^\star$, i.e.,
\begin{equation}
\bar{\mcl{Z}}^\star =
 \left\{\bar z^\star \in \bar{\mcl{Z}}~|~\bar z^\star  \text{ is optimal in\,\eqref{eq:OPT_ss}}\right\}.
\end{equation}
Henceforth, we assume that $\bar{\mcl{Z}}^\star \not= \emptyset$.
The sets $\bar{\mcl{X}}$ and $\bar{\mcl{X}}^\star$, with $\bar{\mcl{X}}^\star \subseteq \bar{\mcl{X}} \subset \mcl{X}$, denote the projection of $\bar{\mcl{Z}} \subset \Rnx\times\Rnu$ onto the state space $\Rnx$ and the projection of $\bar{\mcl{Z}}^\star \subset \Rnx\times\Rnu$ onto $\Rnx$, respectively.

In the operation of dynamic processes, it is of major interest to know whether the best infinite-horizon performance can be achieved at the best steady state or via permanent transient operation. Optimal operation over an infinite horizon is defined similar to \cite{Gruene13a,Angeli12a} as follows.
\begin{defi}[Optimal operation at steady state] \label{def:S}
System~\eqref{eq:sys_def} is said to be optimally operated at steady state if there exists a $\bar z = (\bar x, \bar u)^T \in \bar{\mcl{Z}}$ such that, for any initial condition $x_0 \in \mcl{X}_0$ and any infinite-time admissible pair $z(\cdot, x_0)$,
%$(x(\cdot, x_0, u(\cdot)), \,u(\cdot))$, 
we have
%{\{-2mm}}
\begin{equation} \label{eq:optOptSS}
\underset{T\to\infty}{\liminf} \, J_T(x_0, u(\cdot)) \geq F(\bar z). 
~ \vspace{-4mm}
\end{equation}
  
\end{defi}
The following lemma follows trivially from the above. 
\begin{lemm}\label{lem:opt_ss_op}
If system~\eqref{eq:sys_def} is optimally operated at  $\bar z$, then $\bar z$ is an optimal solution to \eqref{eq:OPT_ss}.
 
\end{lemm}


\subsection{Turnpike Properties of OCPs}
%

%
%
Since there is no generally valid definition of turnpike properties of continuous-time OCPs, we propose a definition motivated by a turnpike result given in \cite{Carlson91}.  To this end, consider the placeholder variable 
%\begin{equation}
$
\xi \in \left\{x, z\right\}, 
$
%\end{equation}
which, depending on the context, denotes the state or the input-state pair of \eqref{eq:sys_def}. Accordingly, $\xi^\star(\cdot, x_0)$ is either an optimal state trajectory $x^\star(\cdot)$ or an optimal pair $z^\star(\cdot)$. Likewise, 
$\bar \xi$ is either a steady state $\bar x$ or a steady-state pair $\bar z$. 
Using this placeholder variable, we define the set 
\begin{equation} \label{eq:Theta}
\Theta_{\xi,T}(\varepsilon) = \left\{t \in [0,T]: \left\| \xi^\star(t, x_0, u^\star(\cdot)) - \bar \xi\right\| > \varepsilon\right\}.
\end{equation}

\begin{defi}[Turnpike property of $\OCP{x_0}$] \label{def:turnpike}
The solution pairs $z^\star(\cdot, x_0)$ of $\OCP{x_0}$ are said to have a \textnormal{$\{$state, input-state$\}$ turnpike} with respect to $\bar \xi  \in \left\{\bar x, \bar z\right\}$ 
if there exists a function $\nu_\xi:[0,\infty)\to [0,\infty]$ such that, for all $x_0 \in \mathcal{X}_0$ and all $T>0$, we have
\begin{equation}\label{eq:TP}
\mu\left[\Theta_{\xi,T}(\varepsilon)\right]  <\nu_\xi(\varepsilon)< \infty \quad \forall\: \varepsilon >0,
\end{equation}
where $\mu[\cdot]$ is the Lebesgue measure on the real line.\\
The pairs $z^\star(\cdot, x_0)$ of \eqref{eq:OCP} are said to have an \textnormal{exact $\{$state, input-state$\}$ turnpike} if Condition \eqref{eq:TP} also holds for $\varepsilon = 0$, i.e., 
\begin{equation} \label{eq:exactTP}
\mu\left[\Theta_{\xi,T}(0)\right]  <  \nu_\xi(0) < \infty. ~ %\vspace{-2mm}
\end{equation}
 \end{defi}
%
%\vspace*{1mm}
The turnpike property states that, for any initial condition $x_0$ and any horizon length $T>0$, the time that the optimal solutions of $\OCP{x_0}$ spend outside an $\varepsilon$-neighborhood of $\bar \xi$ is bounded by $\nu_\xi(\varepsilon)$, where $\nu_\xi(\varepsilon)$ is independent of the horizon length $T$. In other words, for sufficiently long horizons $T$, the optimal solutions have to enter any arbitrarily small $\varepsilon$-neighborhood of $\bar \xi$. In the case of a state turnpike, i.e. $\xi = x$, the solutions have to be close to the steady state $\bar x$. This situation is sketched in Fig. \ref{fig:turnpike}. In the case of an input-state turnpike, i.e. $\xi = z$, also the inputs have to approach any arbitrarily small $\varepsilon$-neighborhood of $\bar u$.
Note that, the steady state $\bar x$, respectively, the steady-state pair $\bar z$ approached by the optimal solutions is commonly referred to as \textit{the turnpike}.
According to Def. \ref{def:turnpike}, the turnpike has to be global in the sense that it is the same for all horizon lengths $T \geq 0$ and all $x_0 \in \mcl{X}_0$. Naturally, it also possible to have local turnpike properties. 
%, i.e., for different sets of initial conditions different steady states are approached. 
Here, for the sake of simplicity, we focus on global turnpike properties. 

%\vspace*{-3mm}
If the stronger condition \eqref{eq:exactTP} holds, then, for sufficiently long horizons $T$, the optimal solutions have to enter the turnpike \textit{exactly} for some part of the horizon, i.e., the optimal solutions have to be at steady state for some part of the horizon. Hence, this case is denoted as exact turnpike, for an analysis  see \cite{epfl:faulwasser15a, kit:faulwasser17a}.

%From an optimal control point of view, one can distinguish three arcs along a turnpike solution: an arc along which the turnpike $\bar \xi$ is approached, an arc along which the optimal solutions stay close to $\bar\xi$, and an arc leading away from the $\bar \xi$.  In essence, the turnpike property of an OCP can be understood as the existence of the second arc, and Def. \ref{def:turnpike} requires that the time length of this arc increases with increasing horizon length $T$. Note that the approaching and leaving arcs do not need to exist. In other words, an optimal solution, which does not leave the neighborhood of $\bar \xi$, is also called a turnpike solution. Furthermore, note that the measure-based concept of Def. \ref{def:turnpike} in principle includes cases where optimal solution enters and leaves any $\varepsilon$-neighborhood of $\bar z$ several times. 

\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{./turnpike2.pdf}
\caption{Sketch of a state turnpike property. 
%: for different initial conditions $x_1, x_2$ the optimal solutions approach a neighborhood of the same steady state $\bar x$. 
\label{fig:turnpike}}
\end{center}
\end{figure}

\vspace*{-2mm}
\begin{rema}[Extremal turnpikes]~\\
As shown in \cite{Trelat15a} for OCPs without input or path constraints, the adjoint variables also stay close to their optimal steady-state values whenever states and inputs stay close to the turnpike. It is straightforward to modify Def. \ref{def:turnpike} to include this third case. Without further elaboration, we suggest to denote this case as an \textit{extremal turnpike}.
% since it requires that the solutions of the extremal system of $\OCP{x_0}$ stay close to steady state for major parts of the time horizon.  
\end{rema}
%In the present paper, we distinguish between state and input-state turnpikes. The main reason is that input-state turnpikes help establish converse turnpike results. 
%
%\vspace*{-2mm}
\begin{rema}[Turnpikes and reachability]\label{rem:turnpike_imp_reach}~\\
Def.~\ref{def:turnpike} implies that the turnpike steady state $\bar x$ is asymptotically reachable from all $x_0 \in \mathcal{X}_0$.  
\end{rema}


\subsection{Dissipativity}
%It has been shown for discrete-time systems that dissipativity is closely related to the question of optimal operation at steady state \cite{Angeli12a}. 
Next, we briefly recall the definition of dissipativity with respect to a steady state \cite{Angeli12a}. We refer to \cite{Willems72a, Moylan14a} for further details on dissipativity. 
Let $w: \mcl{X}\times\mcl{U}\to \R$  be given by
\begin{equation} \label{eq:def_w}
w(x,u) := F(x,u) - F(\bar z),
\end{equation}
with $\bar z\in\bar{\mcl{Z}}$, and $F$ is the cost function in \eqref{eq:J} and  \eqref{eq:OPT_ss}. Let $\mcl{K}$ denote the set of functions of class $\mcl{K}$. 
%\vspace*{-2mm} 
\begin{defi}[Dissipativity wrt  a steady state]\label{def:diss}
System  \eqref{eq:sys_def} is said to be \textit{dissipative on $\mcl{Z}$ with respect to $\bar z  \in \bar{\mcl{Z}}$} if 
there exists a non-negative storage function\footnote{Note that the required properties of $S$ differ in different works: in \cite{epfl:faulwasser14e, Mueller14a} boundedness is assumed, while in \cite{Angeli12a} the storage $S$ can take real values instead of non-negative real values.} $S:\mcl{X} \to \R^+_0$ such that for all $x_0 \in \mcl{X}$, all $T\geq 0$ and all $u(\cdot) \in  \mcl{L}([0,T], \mcl{U})$ satisfying $x(t, x_0, u(\cdot)) \in \mcl{X}$ for all $t \in [0,T]$ we have
%\vspace*{-1mm}
\begin{subequations} \label{eq:DI}
\begin{equation}\label{eq:diss}
%\vspace*{-2mm} 
S(x_T) - S(x_0) \leq \int_{0}^{T} w(x(t), u(t))\,\mathrm{d}t,
%\frac{\partial S}{\partial x} f(x,u) \leq w(x,u)
\end{equation}
where  $x_T = x(T, x_0, u(\cdot))$. \\
If, in addition, for some $\alpha \in \mcl{K}$ and $\xi \in \left\{x, z\right\}$,
%\vspace*{-2mm}
\begin{multline} \label{eq:str_diss}
S(x_T) - S(x_0) \\ \leq \int_{0}^{T} -\alpha\left(\left\|\xi(t)-\bar \xi\right\|\right) + w(x(t), u(t))\,\mathrm{d}t
\end{multline}
\end{subequations}
then, 
\begin{enumerate}[(i)]
%\vspace*{-4mm}
\item for $\xi = x$, system \eqref{eq:sys_def} is said to be \textnormal{strictly dissipative on $\mcl{Z}$ with respect to the steady state $\bar x$};  
\item for $\xi = z$, system \eqref{eq:sys_def} is said to be \textnormal{strictly dissipative on $\mcl{Z}$ with respect to the steady-state pair $\bar z$};
\end{enumerate}
%\vspace*{-2mm}
\end{defi}
%In the next definition we restrict satisfaction of disspation inequalities to optimal pairs.
\begin{defi}[Dissipativity of $\OCP{x_0}$]~\\
If, for all  $x_0 \in \mcl{X}_0$, \eqref{eq:diss}---respectively \eqref{eq:str_diss}---is satisfied along any optimal pair of $\OCP{x_0}$,
 we say that  \emph{$\OCP{x_0}$ is dissipative---respectively strictly dissipative---with respect to $\bar \xi \in \{\bar x, \bar z\}$.} 
%\vspace*{-2mm}
\end{defi}
 Note that in the non-strict case, one can show that dissipativity of system \eqref{eq:sys_def} and dissipativity of $\OCP{x_0}$ are equivalent. 
Furthermore, strict dissipativity of \eqref{eq:sys_def} implies strict dissipativity of $\OCP{x_0}$ but not vice versa.\footnote{Note that in \cite{Stieler14a} \emph{dissipativitiy of a system with respect to a steady state} is denoted as \emph{dissipativity of an OCP}.} Due to these dissipativity notions, $w$ in \eqref{eq:def_w} is called a \textit{supply rate}. 
%If the inequality  \eqref{eq:DI} 
%In this paper, all technical results are based on the integral dissipation inequalities \eqref{eq:DI}. Only for the example presented in Section \ref{sec:example} will we rely on the stronger differential version
%\begin{equation}\label{eq:str_diss_diff}
%\frac{\partial S}{\partial x} f(x,u) \leq -\alpha\left(\left\|\xi-\bar \xi\right\|\right) + w(x,u). %, \quad \forall (x,u) \in \mcl{Z}.
%\end{equation}

%\vspace*{-2mm}
In order to verify (strict) dissipativity, one has two possibilities: compute a storage function, or rely on converse dissipativity results. Next, we recall such a result.
%
%
%
To this end, consider the OCP 
\vspace*{-2mm}
\begin{subequations} \label{eq:OCP_diss}
\begin{align} 
\sup_{u(\cdot) \in  \mcl{L}([0,T], \mcl{U}),\,T }&  -\int_0^T  w(x(t), u(t))\,\mathrm{d}t \label{eq:OCP_diss_Sa}\\
&\hspace*{-1cm}\textrm{subject to }   \nonumber \\
\dot x &= f(x,u), \quad x(0) = x_0, \\
\forall t \in  [0,T]: \quad x(t) &\in \mcl{X}, \quad u(t) \in \mcl{U},\\
T&\geq 0, \label{eq:OCP_diss_T} \vspace*{-2mm}
\end{align}
\end{subequations}
which allows a free end time $T$ in \eqref{eq:OCP_diss_T}.
If $F(\bar z) = 0$ holds, then \eqref{eq:OCP_diss} is the free end-time version of $\OCP{x_0}$.
Let $S^a(x_0)$ denote the optimal value function of  \eqref{eq:OCP_diss}, which is also called the available storage. Observe that $S^a(x_0)\geq 0$, since $T = 0$ is  allowed. The following result states a necessary and sufficient condition for dissipativity. 
\vspace*{-2mm}
\begin{thm}[Willems 1972 \cite{Willems72a}]\label{thm:dissCond} ~\\
 System \eqref{eq:sys_def} is dissipative on $\mcl{Z}= \mcl{X}\times\mcl{U}$ if and only if $S^a(x_0)$ is finite for all $x_0\in\mcl{X}$.
%Moreover, $0 \leq s^a(x_0) \leq S(x_0)$ for dissipative dynamical systems and $s^a$ itself is a possible storage function.
  Moreover, if the system is dissipative with respect to the supply rate w, then $0 \le S^a(x_0)\le S(x_0)$ for \emph{any} storage function $S$ and the function $S^a$ itself is a storage function.         
\end{thm}
The proof can be obtained by straightforward modification of the classical proof from \cite{Willems72a}. The next corollary is directly implied by $0 \le S^a(x_0)\le S(x_0)$.
\vspace*{-2mm}
%\begin{coro}
%\begin{itemize}
%\item System \eqref{eq:sys_def} is dissipative with bounded storage, if and only if, for all $x_0 \in \mcl{X}$, $s^a(x_0) \leq \hat S \leq \infty$.  
%\item System \eqref{eq:sys_def} is dissipative along optimal pairs, if and only if, 
%\end{itemize}
%\end{coro}
\begin{coro}
System \eqref{eq:sys_def} is dissipative with bounded non-negative storage function, if and only if, for all $x_0 \in \mcl{X}$, $S^a(x_0) \leq \hat S < \infty$.  
\end{coro}
\vspace*{-2mm}
%\begin{coro}
%System \eqref{eq:sys_def} is strictly dissipative with bounded storage, if and only if, there exists $\alpha\in\mcl{K}$, such that for all $x_0 \in \mcl{X}$, $S^a_{\xi, \alpha}(x_0) \leq \hat S \leq \infty$.  
%\end{coro}
 %Due to space limitations it is omitted here. %Furthermore, it is worth ot be remarked that Thm. \ref{thm:dissCond} can also be used to check dissipativity along 
\begin{rema}[Verifying strict dissipativity]\label{rema:str_diss}~\\
Note that in order to verifiy strict dissipativity on a compact set, one uses that Thm. \ref{thm:dissCond} applies to generic supply rates $w$, i.e. one swaps $w$ with $-\alpha + w$ in \eqref{eq:OCP_diss_Sa} and shows that, for at least one $\alpha\in\mcl{K}$, the available storage $S^a$ is finite.
%To show dissipativity of $\OCP{x_0}$, one swaps "$u(\cdot) \in  \mcl{L}([0,T], \mcl{U})$" in \eqref{eq:OCP_diss_Sa} with "$u(\cdot)$ is optimal in $\OCP{x_0}$". Again Thm. \ref{thm:dissCond} holds with straightforward modification to the proof in \cite{Willems72a}. 
\end{rema}
\vspace*{-2mm}
\begin{rema}[Strict dissipativity of $\OCP{x_0}$]\label{rema:dissOCP}~\\
To show strict dissipativity of $\OCP{x_0}$, one swaps $w$ with $-\alpha + w$ and "$u(\cdot) \in  \mcl{L}([0,T], \mcl{U})$" in \eqref{eq:OCP_diss_Sa} with "$u(\cdot)$ is optimal in~$\OCP{x_0}$". Again Thm. \ref{thm:dissCond} holds with straightforward modification to the proof in \cite{Willems72a}.
\end{rema}
%\textcolor{red}{Do we need a modified version of Thm 1?}
%This classical result leads to the following sufficient dissipativity condition. 
%\begin{propo}[Sufficient dissipativity cond.] \label{prop:suff_diss}
%Suppose that, for all $T >0$, all $x_0\in \mcl{X}$, and some $\hat s \in [0, \infty)$, the optimal pairs $z^\star(t, x_0)$ of $\OCP{x_0}$ satisfy
%\begin{equation} \label{eq:sufficient_condition}
%\left|\int_0^T w(z^\star(t, x_0))\,\mathrm{d}t\right| < \hat s,
%\end{equation}
%then system \eqref{eq:sys_def} is dissipative with respect to $\bar z$ on $\mcl{X}\times\mcl{U}$.
%\end{propo}
%The proof of this result is provided in the appendix.



\subsection{Problem Statement} \label{sec:results}
The main purpose of this paper is to establish links between the three following formal statements/assumptions:
\begin{stat} \label{stat:D}
For all $x_0 \in \mcl{X}_0$, $\OCP{x_0}$ is strictly dissipative with respect to $\bar \xi \in \{\bar x, \bar z\}$ and with bounded storage function.
\end{stat}
\vspace*{-3mm}
It is worth mentioning that if Stmt. \ref{stat:D} is true for $\bar \xi = \bar z$, then it also holds for $\bar \xi = \bar x$.  
\vspace*{-3mm}
\begin{stat} \label{stat:S}
System \eqref{eq:sys_def} is opt. operated at $\bar z^\star \in \bar{\mcl{Z}}^\star$.
 \end{stat}
\vspace*{-3mm}
\begin{stat} \label{stat:TP}
For all $x_0 \in \mcl{X}_0$, the optimal solutions of $\OCP{x_0}$ have a $\{$state, input-state$\}$ turnpike at $\bar \xi \in \{\bar x, \bar z\}$.
 \end{stat}
 \vspace*{-2mm}
%\newpage
Subsequently, we show that 
\begin{itemize}
\vspace*{-2mm}
\item Stmt. \ref{stat:D} $\Rightarrow$ Stmt. \ref{stat:TP} (Thm. \ref{thm:D_imp_TP});
\item Stmt. \ref{stat:D} $\Rightarrow$ Stmt. \ref{stat:S} (Thm. \ref{thm:D_imp_S});
\item Stmt. \ref{stat:TP} $\Rightarrow$ Stmt. \ref{stat:S} (Thm. \ref{thm:TP_imp_S});
\item Stmt. \ref{stat:S} $\Rightarrow$ Stmt. \ref{stat:D} (Thm. \ref{thm:S_imp_D});
\item Stmt. \ref{stat:TP} $\Rightarrow$ Stmt. \ref{stat:D} (Thm. \ref{thm:exTP_imp_D} and Thm. \ref{thm:TP_imp_D}).
\end{itemize}
%\vspace*{-2mm}
%whereby some of the relations invoke additional assumptions introduced later.
 An overview of the results to be shown is sketched in Fig. \ref{fig:implications}.
 \begin{figure}[t]
\begin{center}
\includegraphics[width=0.49\textwidth]{./D_T_O_v4.pdf}
\caption{Implications between dissipativity, optimal operation at steady state and turnpikes investigated in this paper. \label{fig:implications}}
\end{center}
\end{figure}
 %
%, we further show that
%\begin{itemize}
%\vspace*{-3mm}
%\item Stmt. \ref{stat:S} $\Rightarrow$ Stmt. \ref{stat:D} (Thm. \ref{thm:S_imp_D});
%\item Stmt. \ref{stat:TP} $\Rightarrow$ Stmt. \ref{stat:D} (Thms. \ref{thm:exTP_imp_D} \& \ref{thm:TP_imp_D}). 
%\end{itemize}
%\vspace*{2mm} 
%Finally, combining these results, we will find sufficient conditions establishing equivalence of Statements \ref{stat:D}--\ref{stat:TP}.
%These relations are sketched in Fig. \ref{fig:implications}. 
%%
%\begin{figure}[t]
%\begin{center}
%\includegraphics[width=0.49\textwidth]{./figures/D_T_O_mod_II.pdf}
%\caption{Implications between dissipativity, optimal operation at steady state and the existence of turnpike. \label{fig:implications}}
%\end{center}
%\end{figure}
%

For some of these relations we will invoke a reachability assumption. 
%\begin{ass} \label{ass:reach}
%There exists a time $T_{\bar{\mcl{X}}^\star} < \infty$ such that, for all $x_0\in \mcl{X}_0$, there exists an optimal steady state $\bar x^\star\in \bar{\mcl{X}}^\star$ reachable in time $T_{\bar{\mcl{X}}^\star}$, i.e.,
%\[\bigcup_{\bar x^\star \in \bar{\mcl{X}}^\star} \mathcal{C}(\bar x^\star,T_{\bar{\mcl{X}}^\star}) \supseteq \mcl{X}_0. 
%\vspace*{-4mm}
%\]
%  \end{ass}
%We remark that Ass. \ref{ass:reach} is only explicitly needed for the proof of Thm. \ref{thm:D_imp_TP}. It can easily be relaxed to exponential reachability.
%Due to space limitations, we do not investigate this in detail.
%\textcolor{red}{
\begin{ass}[Exponential reachability] \label{ass:reach}~\\
For all $x_0\in \mcl{X}_0$, there exists an optimal steady state pair $\bar z^\star\in \bar{\mcl{Z}}^\star$, an infinite-time admissible input $ u_\infty(\cdot, x_0) \in \mcl{L}([0,\infty), \mcl{U})$, and constants $c, \lambda \in \mathbb{R}^+$, independent of $x_0$, such that
\[
\left\| (x(t, x_0,  u_\infty(\cdot, x_0)), \,u_\infty(\cdot, x_0))^T - \bar z^\star\right\| \leq c e^{-\lambda t}. 
\vspace*{-4mm}
\]
 \end{ass}
% }
This assumption is satisfied if one can steer the state to any small neighborhood of $ \bar x^\star$ in finite time, $ \bar x^\star$ corresponds to a   steady-state pair $( \bar x^\star,  \bar u^\star) \in \inte Z$, and \eqref{eq:sys_def} has a stabilizable Jacobian linearization at this point.  



%\begin{ass} \label{ass:reach}
%For all $x_0 \in \mcl{X}_0$ the set of steady states is reachable within some finite time: $\exists\, T_\bar{\mcl{X}}%< \infty$ such that $\mcl{X}_0  \subset \mcl{R}(\bar{\mcl{X}}, T_\bar{\mcl{X}})$.
%\[\bigcup_{x^\star \in \bar{\mcl{X}}^\star} R(x^\star,T_{\bar{\mcl{X}}^\star}) \supset \mcl{X}_0 \]
%\end{ass}

\section{Implications of Dissipativity} \label{sec:D}
%Next, we  investigate the implications of Stmt. \ref{stat:D} (dissipativity).
%We begin with a technical statement. 
%\begin{lemm} \label{lem:diss_opt}~\\
%\vspace*{-7mm}
%\begin{enumerate}[(i)]
%\item If Stmt. \ref{stat:D} is true with $\bar\xi = \bar z$, then $\bar z = \bar z^\star$, i.e., $\bar z$ is the unique optimal solution to~\eqref{eq:OPT_ss}.
%\item  If Stmt. \ref{stat:D} is true with $\bar\xi = \bar x$, then $\bar{x} = \bar{x}^\star$, i.e., $\bar{x} $ is the unique solution to~\eqref{eq:OPT_ss} in the sense that there exists a not necessarily unique $\bar u$ such that $\bar z = (\bar x, \bar u)$ is optimal in~\eqref{eq:OPT_ss} and $\bar x$ is unique.  
%\end{enumerate}
%\vspace*{-5mm}
%\end{lemm}
%\begin{pf}
%We begin with part (i), i.e., $\xi = z$.  
%Evaluating the strict dissipativity inequality~\eqref{eq:str_diss} along any trivial pair with $z(t) \equiv \bar z^\star$   yields 
%$
%0 \leq F(\bar z^\star) - F(\bar z) -\alpha(\|\bar z^\star-\bar z\|) $.
%Since $\bar z$ is admissible in~(\ref{eq:OPT_ss}), we have 
%$
%F(\bar z^\star) - F(\bar z)   \le 0.
%$
%Combining the last two inequalities, and using the fact that $\alpha$ is of class $\mcl{K}$, yields $\bar z = \bar z^\star$. 
%To prove part (ii) note that $\xi = x$ implies the existence of $\bar u \in \mcl{U}$ such that $0 = f(\bar x, \bar u)$. The same arguments as above show that strictness of \eqref{eq:str_diss} implies $\bar x = \bar x^\star$. \QED
%\end{pf}
%%Note that in part (ii) of the lemma it is possible to have two steady-state pairs $(\bar x, \bar u_1), (\bar x, \bar u_2)$ with $F(\bar x, \bar u_1) = F(\bar x, \bar u_2), u_1\neq u_2$ and to be dissipative with respect to $\bar x$. 
%%This shows that, for $\xi = x$, strict dissipativity defines a unique value of the optimal steady state but not of the optimal steady-state pair. 
%\vspace*{-4mm}
%As far as uniqueness of optimal solutions of \eqref{eq:OPT_ss} is concerned, the class-$\mcl{K}$ function in \eqref{eq:str_diss} can be regarded as a substitute of a convexity  assumption on $F$.
%%
%Lem. \ref{lem:diss_opt} tells that it suffices to investigate the relations between Stmts. \ref{stat:D} and \ref{stat:TP} for $\bar \xi = \bar\xi^\star \in \{\bar x^\star, \bar z^\star \}$. 
%
\begin{thm}[Dissipativity $\Rightarrow$ turnpike]  \label{thm:D_imp_TP}~\\
Let Stmt.~\ref{stat:D}   hold for $\xi \in \{x, z\}$ and Ass.~\ref{ass:reach} holds for $\bar z^\star~\in~\bar{\mcl{Z}}$.
Then, Stmt. \ref{stat:TP} holds, i.e., $\OCP{x_0}$ has a turnpike at $\bar \xi^\star\in \{\bar x ^\star, \bar z^\star\}$.  
\end{thm}
\vspace*{-4mm}
\begin{pf}
We first consider $\xi =z$. Let $z^\star(\cdot,x_0)$ be an optimal solution to $\OCP{x_0}$. The integral dissipation inequality~(\ref{eq:str_diss}) gives
\begin{align*}
S(x^\star(T)) - S(x^\star(0))\le &  -\int_0^T \alpha(\| z^\star(t,x_0) - \bar z^\star \|)\,\mathrm{d}t \\ & + 
\int_0^T\hspace{-2mm} F(z^\star(t)) - F(\bar z^\star)\,\mathrm{d}t.
\end{align*}
%
By Ass.~\ref{ass:reach}, $\bar x^\star$ is exponentially reachable from every $x_0\in\mcl{X}_0$. Hence, the second integral is bounded from above by $K_F:=\frac{c}{\lambda}L_F < \infty$ independently of $T$, where $L_F$ is a Lipschitz constant of $F$ and $c,\lambda$ are from Ass.~\ref{ass:reach}. In addition, since $S$ is bounded, the lhs is bounded in absolute value, independently of $T$, by some $K_S < \infty$. Hence, we have
\begin{align*}
\int_0^T \alpha(\| z^\star(t,x_0) - \bar z^\star \|)\,\mathrm{d}t \le K_S +K_F < \infty.
\end{align*}
Using \eqref{eq:Theta} and that $\alpha \in \mcl{K}$, we obtain
\begin{equation*}
\int_0^T \alpha(\| z^\star(t,x_0) - \bar z^\star \|)\,\mathrm{d}t \ge %  \int_{\Theta_{z,T}(\varepsilon)}\hspace{-3.8mm} \alpha(\| z^\star(t,x_0) - \bar z^\star \|)\,\mathrm{d}t  \\ \ge
 \alpha(\varepsilon)\mu[\Theta_{z,T}(\varepsilon)].
\end{equation*}
It follows that
$
\mu[\Theta_{z,T}(\varepsilon)] \le (K_S+K_F) / \alpha(\varepsilon) =: \nu_z(\varepsilon),
$
where $\nu_z(\varepsilon)$ does not depend on $T$.
The case $\xi =x$ can be proved easily along the same lines. \hfill~ \QED
\end{pf}
%\vspace*{-4mm}
%
%
%This result is an extension of a classical result given in \cite[Theorem 4.4]{Carlson91}. Therein, a state turnpike property is proved under the assumption of strict convexity of $F$ and via consideration of a terminal constraint $x(T) =x_T$ for the OCP \eqref{eq:OCP}. Statements similar to Thm. \ref{thm:D_imp_TP} have been derived for the discrete-time case in \cite{Gruene13a,Stieler14a}.
%
%Dissipativity also implies opt. operation at steady state. 
\begin{thm}[Dissipativity $\Rightarrow$ opt. operation at $\bar z^\star$] \label{thm:D_imp_S}
If Stmt.~\ref{stat:D} holds at $\bar z^\star$, then Stmt.~\ref{stat:S} holds, i.e., system \eqref{eq:sys_def} is optimally operated at the steady state $\bar z^\star$. 
\end{thm}
%\vspace*{-4mm}
\begin{pf}
(By contradiction). Assume that there exists an infinite-time optimal pair $z^\star_\infty(\cdot, x_0)$ and a sequence $\{T_k\}_{k=1}^\infty$, with $T_{k+1}\ge T_k$ and $T_k\to\infty$, such that
\begin{equation}\label{eq:proofDimpSs^aux}
\lim_{k\to\infty} J_{T_k}(x_0, u^\star_\infty(\cdot)) \le  F(\bar z^\star) - \sigma
\end{equation}
for some $\sigma > 0$. Evaluating the dissipation inequality~\eqref{eq:diss} along $z^\star_\infty(\cdot, x_0)$ and dividing by $T_k$ gives
\[
\frac{1}{T_k}[S(x^\star_\infty(T_k)) - S(x^\star_\infty(0))] \le     J_{T_k}(x_0, u^\star_\infty(\cdot)) - F(\bar z^\star).
\]
Since the storage function $S$ is bounded, the lhs of the above inequality converges to zero for $T_k \to \infty$, 
whereas the rhs converges to $-\sigma< 0$, which is a contradiction. \QED
\end{pf}
\vspace*{-5mm}
In the proof of Thm.~\ref{thm:D_imp_S}, we never invoked the strict dissipativity term $\alpha(\|\xi - \bar \xi^\star\|)$. Hence, the following stronger statement holds.
\begin{thm}\label{thm:D_imp_SII}~\\%[Dissipativity $\Rightarrow$ opt. operation at $\bar z^\star$]  
For all $x_ 0 \in \mcl{X}_0$, let  $\OCP{x_0}$ be dissipative (not necessarily strictly) with bounded storage and with respect to the steady-state pair~$\bar z^\star$. Then, Statement~\ref{stat:S} holds, i.e., the system \eqref{eq:sys_def} is optimally operated at the steady state~$\bar z^\star$.  
\end{thm}
%We remark that Thms. \ref{thm:D_imp_TP}--\ref{thm:D_imp_SII} hold true if dissipativity of $\OCP_T{x_0}$ is replaced by dissipativity of \eqref{eq:sys_def}.

\section{Implications of Optimal Operation \newline
at Steady State} \label{sec:SS}
In order to discuss the implications of optimal operation at steady state, we will extend a discrete-time result given in \cite{Mueller14a} to the continuous-time setting.
To this end,
consider the set
\begin{subequations} \label{eq:sets}
\begin{multline} \label{eq:steerable_set}
\mcl{C}(\bar x^\star, T) := \left\{x_0 \in  \mcl{X} \,|\,\exists u(\cdot) \in \mcl{L}([0,T], \mcl{U}), \right.\\ 
\hspace*{.9cm}\, \forall t\in[0,T]:  x(t, x_0, u(\cdot)) \in \mcl{X}, \\\left.
\, x(T, x_0, u(\cdot)) = \bar x^\star\right\},
\end{multline}
which is the set of initial conditions $x_0 \in \mcl{X}$ that can be steered, in some finite time $T \in (0,\infty)$, by means of an admissible input, to $\bar x^\star \in \mcl{X}$.
Likewise, we define the set
\begin{multline} \label{eq:reachable_set}
\mcl{R}(\bar x^\star, T):= \left\{x_T\in  \mcl{X} \,|\,\exists u(\cdot) \in \mcl{L}([0,T], \mcl{U}), \right.\\ 
\hspace*{.9cm}\, \forall t\in[0,T]:  x(t, \bar x^\star, u(\cdot)) \in \mcl{X}, \\
\left.\, x(T, \bar x^\star, u(\cdot)) = x_T\right\}, 
\end{multline}
\end{subequations}
which containts all states that can be reached, in some finite time $T \in (0,\infty)$, by means of an admissible input, starting from $\bar x^\star$. 
%recall the steerable set $\mcl{C}(\bar x^\star, T)$ and the reachable set $\mcl{R}(\bar x^\star, T)$ defined in \eqref{eq:sets}.
Since, by construction, $\bar x^\star$ is contained in both sets, we have $\mcl{C}(\bar x^\star, T) \cap \mcl{R}(\bar x^\star, T) \neq \emptyset$.
Now, let 
\begin{multline} \label{eq:X_T}
\mcl{X}_T :=  \left\{x_0 \in  \mcl{X} \,|\,\forall t\in[0,T]: \,\exists u(\cdot) \in \mcl{L}([0,T], \mcl{U})  \right.\\ \left.
%z(t, x_0, u(\cdot)) \in \mcl{Z},
\hspace*{-.6cm}\, x(t, x_0, u(\cdot)) \in \mcl{C}(\bar x^\star, T) \cap \mcl{R}(\bar x^\star, T) \right\} 
\end{multline}
be the set of initial conditions $x_0$, for which (i) there exists a corresponding admissible pair $z(\cdot, x_0, u(\cdot))$, and (ii) the inclusion $x(t, x_0, u(\cdot))\in \mcl{C}(\bar x^\star, T) \cap \mcl{R}(\bar x^\star, T)$ holds for all $t\in[0,T]$. Subsequently, we use the shorthand notation $\mcl{Z}_T :=\mcl{X}_T\times\mcl{U} $.

\begin{thm}[Opt. operation at $\bar z^\star \Rightarrow$ dissipativity] \label{thm:S_imp_D}
If Stmt. \ref{stat:S} holds, then, for any $T \geq 0$, Stmt. \ref{stat:D} holds with $\mcl{Z}$ replaced by  $\mcl{Z}_T $, i.e.,  system \eqref{eq:sys_def} is dissipative on $\mcl{Z}_T $.
\end{thm}
 The proof follows along the same lines as the discrete-time version \cite[Thm. 4]{Mueller14a} and is thus omitted. 
\begin{coro} % \label{thm:S_imp_D}
If Stmt. \ref{stat:S} holds, then, for any $T \geq 0$, Stmt. \ref{stat:D} holds with $\mcl{X}_0 = \mcl{X}_T $, i.e.,  $\OCP{x_0}$ is dissipative.
\end{coro}

%\textcolor{red}{Does Muller14a give dissipativity of OCP?}
%\begin{pf}
%The proof is essentially a continuous-time version of \cite[Theorem 4]{Mueller14a}.
%First, observe that  
%\[
%\mcl{X}_T \subset  \mcl{C}(\bar x^\star, T) \cap \mcl{R}(\bar x^\star, T) \subset \mcl{X}_{2T}.
%\]
%The first relation follows from the definition of $\mcl{X}_T $. The second relation can be shown as follows:
%for each $x\in   \mcl{C}(\bar x^\star, T) \cap \mcl{R}(\bar x^\star, T)$, there exists an admissible input such that
%$x(T,  x, u(\cdot)) = \bar x^\star$ and $x(2T, x, u(\cdot)) = x$.
%%$x(T, \bar x^\star, u(\cdot)) = x$ and $x(2T, \bar x^\star, u(\cdot)) = \bar x^\star$. 
%In other words, at any $x\in   \mcl{C}(\bar x^\star, T) \cap \mcl{R}(\bar x^\star, T)$, we can construct an admissible periodic orbit passing through $\bar x^\star$.
%
%Second, choose any $T\geq 0$ and assume, for simplicity and without loss of generality, that $F(\bar z^\star) = 0$.
%For the sake of contradiction, assume that Stmt. \ref{stat:S} holds and that the system is not dissipative on $\mcl{Z}_T $. 
%Replace $\mcl{X}$ by $\mcl{X}_T$  in \eqref{eq:OCP_diss} and in Thm. \ref{thm:dissCond}. 
%Then, according to Thm. \ref{thm:dissCond}, it must hold that
%\[
%\forall \rho> 0 ~ \exists \tilde x \in \mcl{X}_T: \quad s^a(\tilde x) \geq \rho,
%\]
%where $s^a(\tilde x)$ is the supremum in \eqref{eq:OCP_diss}.
%This implies that, for each $\rho \geq 0$, there exist  (i) $\tilde x \in \mcl{X}_T$, (ii) an admissible pair $z_\rho(\cdot, \tilde x, u_\rho(\cdot))$ satisfying for all $t\in[0,\infty):\, z_\rho(t, \tilde x, u_\rho(\cdot))\in \mcl{Z}_T $, and (iii) a time instant $T_\rho \in [0, \infty)$ such that
%\begin{equation}\label{eq:contradiction_DissI}
%\int_0^{T_\rho} F(z_\rho(t,\tilde x) )\,\mathrm{d}t \leq -\rho.
%\end{equation}
%Choose any $\rho > \delta + 2T\hat F$ with $\delta > 0$ and $\hat w = \sup_{z \in \mcl{Z}}\,F(z)$. At time $T_\rho$, the construction of $\mcl{C}(\bar x^\star, T)$ and $\mcl{R}(\bar x^\star, T)$ implies that 
%\[
%x_\rho = x_\rho(T_\rho, \tilde x, u_\rho(\cdot)) \in \mcl{C}(\bar x^\star, T) \cap \mcl{R}(\bar x^\star, T).
%\]
%Therefore, there exists a periodic admissible pair 
%\[
%z_\pi(t, x_\rho, u_\pi(\cdot)) \in \mcl{Z}, \quad \forall t\in[0,2T],
%\] 
%with $x_\pi(0, x_\rho) = x_\pi(2T, x_\rho) = x_\rho$ and $x_\pi(T, x_\rho) = \bar x^\star$.
%
%Now, let $\hat T = T_\rho+2T$. Consider the following input
%\begin{equation}
%\hat u\left(k\hat T+t\right) :=\left\{
%\begin{matrix}
%u_\rho(t),& \textrm{for} ~ t \in [0, T_\rho), k \in \mbb{N} \\
%u_\pi(t),& \textrm{for} ~ t \in [T_\rho, \hat T), k \in \mbb{N}
%\end{matrix}
%\right..
%\end{equation}
%Using the initial condition $\tilde x$, this input generates the periodic admissible pair $\hat z(\cdot, \tilde x, \hat u(\cdot))$.
%Evaluating the integral over the supply rate for this admissible pair yields
%\begin{multline}
%\int_0^{(k+1)\hat T} F(\hat z(k\hat T+t,\tilde x) )\,\mathrm{d}t  \\
%= (k+1)\left( \int_0^{T_\rho} F(z_\rho(t,\tilde x) )\,\mathrm{d}t + \int_{T_\rho}^{\hat T} F(z_\pi(t,\tilde x) )\,\mathrm{d}t\right).
%\end{multline}
%Using \eqref{eq:contradiction_DissI}, this can be rewritten as
%\begin{equation} \label{eq:contradiction_DissII}
%\int_0^{(k+1)\hat T} \hspace*{-.25cm} F(\hat z(k\hat T+t,\tilde x) )\,\mathrm{d}t  
%\leq (k+1) (-\rho+   2T\hat w ) \leq -\delta,
%\end{equation}
%where the last inequality follows from $\rho > \delta + 2T\hat F$ and $k\in\mbb{N}$.
%
%Evaluating performance of the admissible pair  $\hat z(\cdot, \tilde x, \hat u(\cdot))$  for $\hat T\to\infty$ yields
%\begin{multline*}
%%\liminf_{T\to\infty}\frac{1}{T}\int_0^{T} F(\hat z(t,\tilde x) )\,\mathrm{d}t  \\
%\liminf_{k\to\infty} \frac{1}{(k+1)\hat T} \int_0^{(k+1)\hat T} \hspace*{-.25cm}F(\hat z(k\hat T+t,\tilde x) )\,\mathrm{d}t \\
%\stackrel{\eqref{eq:contradiction_DissII}}{\leq} \frac{-\rho+   2T\hat w  }{\hat T} \leq -\frac{\delta}{\hat T}.
%\end{multline*}
%Recall that we assumed w.l.o.g. $F(\bar z^\star) = 0 $. Furthermore, it follows from the last inequality that 
%\[
%\liminf_{T\to\infty} J_T(\tilde x, \hat u(\cdot)) \leq -\delta(\hat T)^{-1} < 0,
%\]
%which contradicts \eqref{eq:optOptSS}, cf. Def. \ref{def:S}.
%Hence, we arrive at a contradiction, i.e., \eqref{eq:sys_def} has to be dissipative on $\mcl{Z}_T $ with supply rate \eqref{eq:def_w}.
%\QED
%\end{pf}
%Theorems \ref{thm:D_imp_S} and \ref{thm:S_imp_D} yield the following corollary.
%\begin{coro}[Opt.\,operation\,at\,$\bar z^\star\Leftrightarrow$\,dissipativity] \label{cor:S_imp_D}
%Let $\mcl{Z}_T = \mcl{Z}$ for some $T\geq 0$. Then, 
%%Stmt. \ref{stat:D}, with $\mcl{Z}$ replaced by  $\mcl{Z}_T$, is equivalent to Stmt. \ref{stat:S}, i.e.,  
%system \eqref{eq:sys_def} is dissipative on $\mcl{Z}_T$ with respect to $\bar z^\star$ if and only if it is optimally operated at $\bar z^\star$.  
%\end{coro}


\section{Converse Turnpike Results} \label{sec:T}
After discussing the implications of dissipativity and optimal operation at steady state, we now turn to converse turnpike results.

\begin{thm}[Turnpike $\Rightarrow$  opt. operation at $\bar z$] \label{thm:TP_imp_S}~\\
Let Stmt. \ref{stat:TP} hold for $\bar \xi \in \{\bar x, \bar z\}$. If either $\bar \xi = \bar z$ or  $\bar\xi = \bar x$ and
\begin{equation} \label{eq:cond_ThmConvTP}
\frac{\partial F}{\partial u} = 0, \quad \forall  (x,u) \in \mcl{Z},	
\end{equation}
 then Stmt. \ref{stat:S} holds, i.e., \eqref{eq:sys_def} is optimally operated at ~$\bar{z}$. 
 \end{thm}
\vspace*{-5mm}
\begin{pf}
We first consider the case  $\bar \xi = \bar z$. 
%Since we assume that $F(x,u)$ is independent of $u$, we simply write $F(x)$ throughout the proof. 
Fix $x_0\in \mathcal{X}_0$ and, for contradiction, assume that there exists an infinite-time admissible pair $z_\infty(\cdot, x_0)$ and a sequence $\{T_k\}_{k=1}^\infty$ with $T_{k+1}\ge T_k$ and $T_k\to\infty$ such that
\begin{equation}\label{eq:proof_aux}
\lim_{k\to\infty} J_{T_k}(x_0,u_\infty(\cdot)) \le  F(\bar z) - \sigma
\end{equation}
for some $\sigma > 0$. 
Next, observe that the turnpike property and Lipschitz continuity of $F(x,u)$ imply that there exists a function $\gamma_z$, independent of $T$, such that
\[ 
\mu[\Omega_{z,T}(\varepsilon)] < \gamma_z(\varepsilon),
\] 
with
$
\Omega_{z,T}(\varepsilon) := \{t\in [0,T] \mid F(z_T^\star(t, x_0)) - F(\bar z)| > \varepsilon \}$.
%and where $z_T^\star(\cdot)$ denotes an optimal pair of $\OCP{x_0}$. 
Set $m :=\min_{z\in\mathcal{Z}}F(z)$ and $\bar \Omega := [0,T_k]\setminus \Omega_{z,T_k}(\varepsilon) $. Then, for arbitrary $\varepsilon > 0$, we have
\begin{align*}
\int_0^{T_k}F(z_{T_k}^\star(t))\,\mathrm{d}t &= \int_{\Omega_{z,T}(\varepsilon)} \hspace{-0.5cm}  F(z_{T_k}^\star(t))\,\mathrm{d}t + \int_{\bar \Omega}F(z_{T_k}^\star(t))\,\mathrm{d}t  \\
 \hspace{-2.4cm} \ge m\mu[\Omega_{z,T}(\varepsilon)] &+  \int_{\bar \Omega} \hspace{-0.2cm}F(\bar{z})-\varepsilon\,dt \\
%\end{align*}
%\begin{align*}
 %\hspace{-2.4cm} 
= m\mu[\Omega_{z,T}(\varepsilon)] &+  \int_0^{T_k} \hspace{-0.2cm}F(\bar{z})-\varepsilon\, dt - \int_{\Omega_{z,T}(\varepsilon)} \hspace{-0.4cm} F(\bar{z})-\varepsilon\,\mathrm{d}t \\
   %\hspace{-2.4cm}  
= m\mu[\Omega_{z,T}(\varepsilon)] &+ T_k(F(\bar{z})-\varepsilon) - \mu[\Omega_{z,T}(\varepsilon)](F(\bar{z})-\varepsilon).
\end{align*}
Since $\mu[\Omega_{z,T}(\varepsilon] < \gamma_z(\varepsilon)$, independently of $T_k$, dividing by $T_k$ and letting $k\to\infty$ gives
\[
\lim_{k\to\infty} \frac{1}{T_k}\int_0^{T_k}F(z_{T_k}^\star(t))\,\mathrm{d}t  \ge F(\bar{z}) - \varepsilon .
\]
Selecting $\varepsilon < \sigma$ leads to a contradiction to~(\ref{eq:proof_aux}), since the pair $z_\infty(\cdot, x_0)$ truncated 
to  $[0,T_k]$ is admissible for $\OCP{x_0}$ for any $T_k \ge 0$.

%\vspace*{-2mm} 
Recall that, for $\bar \xi = \bar x$,  \eqref{eq:cond_ThmConvTP} implies that $F$ does not depend on $u$. 
Hence, swapping $z$ with $x$ in the derivations above, proves the assertion for  $\bar \xi = \bar x$. \QED
\end{pf}
\vspace*{-5mm}
Thm.~\ref{thm:TP_imp_S} and Lem.~\ref{lem:opt_ss_op} lead to the following corollary. %says that a turnpike can occur only at an optimal steady state.
\begin{coro}[Turnpikes are opt. steady states] \label{coro2} 
Let Stmt. \ref{stat:TP} hold for $\bar \xi \in \{\bar x, \bar z\}$ and assume that  the conditions of Theorem~\ref{thm:TP_imp_S} hold. Then, for $\bar \xi = \bar z$,
the turnpike $\bar z$ is optimal in~(\ref{eq:OPT_ss}), i.e., $\bar z = \bar z^\star$; and, for $\bar \xi = \bar x$, there exists a $\bar u\in\mcl{U}$ such that $(\bar{x},\bar u)$  is optimal in~(\ref{eq:OPT_ss}).  
\end{coro}
%\begin{pf}
%Follows from Theorem~\ref{thm:TP_imp_S} and Lemma~\ref{lem:opt_ss_op}.\QED
%\end{pf}
%\begin{rema}
\vspace*{-2mm} Corollary \ref{coro2} is a consequence of the turnpike definition used in this paper, which implies asymptotic reachability of $\bar{x}$ from all $x_0\in \mcl{X}_0$ (see Remark~\ref{rem:turnpike_imp_reach}). If, instead, a local definition of turnpike is used, analogous local results can be established.  
%\end{rema}
%
%\vspace*{-2mm}

At this point, one may wonder whether or not Stmts. \ref{stat:D}--\ref{stat:TP} are equivalent. 
In the view of Thm. \ref{thm:D_imp_S} \& \ref{thm:S_imp_D} this boils down to the question of
how large the gap between strict and non-strict dissipativity, i.e., between \eqref{eq:diss} and \eqref{eq:str_diss}, is.
One intuitive way to close this gap is assuming that $F$ has a unique minimizer $\bar{z}^\star$ over $\mathcal{Z}$, and this unique minimizer is a steady state, i.e., $\bar{z}^\star \in \bar{\mathcal{Z}}$. This, however, would imply to restrict the class of cost functions significantly. 
Another approach, pursued in \cite{Gruene16a} in a discrete-time setting, requires local controllability close to $\bar z$ and turnpike-like behavior of nearly optimal solutions. 
Next, we follow a different route and present two distinct set of conditions guaranteeing that the existence of a turnpike implies dissipativity. %of \eqref{eq:diss}.
\begin{thm}[Exact turnpike $\Rightarrow$ dissipativity]\label{thm:exTP_imp_D}~\\
Let Stmt. \ref{stat:TP}  hold with $\mcl{X}_0 = \mcl{X}$, and let the turnpike be exact and of the input-state type ($\xi = z$). 
Then,
\vspace*{-3mm}
\begin{enumerate}[(i)]
\item  system \eqref{eq:sys_def} is dissipative with respect to $\bar z^\star$ on $\mcl{Z}$ and with bounded storage function, and
\item %for any $\alpha \in \mcl{K}$, there exists a storage function $S$ depending on $\alpha$, s.t.  
$\OCP{x_0}$ is strictly dissipative with respect to $\bar z^\star$.  
\end{enumerate}
%\vspace*{-5mm}
\end{thm}
\vspace*{-5mm}
\begin{pf}
%We begin with assertion (a).
\underline{Part (i):} Since, for all $T \geq 0$, the optimal pairs $z^\star(\cdot, x_0)$ show turnpike behavior, we have 
\begin{multline*}
|s^a(x_0)|  := \left|\int_0^T w(z^\star(t, x_0))\,\mathrm{d}t\right| \\
\leq \int_{\Theta_{z, T}(\varepsilon)}\hspace*{-5mm} |w(z^\star(t, x_0)) |\,\mathrm{d}t  
+ \int_{[0,T]\setminus \Theta_{z, T}(\varepsilon) } \hspace*{-8mm} |w(z^\star(t, x_0)) |\,\mathrm{d}t. 
\end{multline*}
Due to Lipschitz continuity of $F$, for all $T \geq 0$, all $x_0 \in \mcl{X}$, and all $\varepsilon >0$, the second integral on the rhs is bounded from above by $T L_F\varepsilon$, where $L_F$ is a Lipschitz constant of $F$.
Hence, the last inequality leads to
\begin{equation*}
%\left|\int_0^T w(z^\star(t, x_0))\,\mathrm{d}t \right|  
|s^a(x_0)|
\leq \mu[\Theta_{z, T}(\varepsilon)] \hat w + T L_F\varepsilon, %(T - \mu[\Theta_{\xi, T}(\varepsilon)]) L_F\varepsilon, 
\end{equation*}
where $\hat w := \sup_{z \in\mcl{Z}} |w(z)| < \infty$.
The turnpike is exact. We may set $\varepsilon  = 0$ and obtain
$
%\left|\int_0^T w(z^\star(t, x_0))\,\mathrm{d}t \right|
|s^a(x_0)|
\leq \nu_z(0) \hat w < \infty$.
Hence, for all $T\geq 0$ and all $x_0 \in \mcl{X}_0$, the supremum in \eqref{eq:OCP_diss} is finite.  
%Note that the preceding developments hold for $\bar \xi = \bar x$  and $\bar\xi = \bar z$.  
Using Thm. \ref{thm:dissCond}, we conclude that \eqref{eq:sys_def} is dissipative  with bounded storage function
on $\mcl{Z}$. 

\vspace*{-2mm}
\underline{Part (ii):} Choose any $\alpha \in \mcl{K}$, and split the time horizon into $\Theta_{z, T}(\varepsilon)$ and $[0,T]\setminus\Theta_{z, T}(\varepsilon)$. Similar to part (i), this leads to
%\begin{multline*}
%\left|\int_0^T -\alpha(\|z^\star(t, x_0) - \bar z^\star\|) + w(z^\star(t, x_0))\,\mathrm{d}t\right| \leq \\
% \int_{\Theta_{\xi, T}(\varepsilon)} \alpha(\|z^\star(t, x_0) - \bar z^\star\|) + |w(z^\star(t, x_0)) |\,\mathrm{d}t  \\
%+ \int_{[0,T]\setminus \Theta_{\xi, T}(\varepsilon) } \alpha(\|z^\star(t, x_0) - \bar z^\star\|) + |w(z^\star(t, x_0)) |dt. 
%\end{multline*}
%This can be rewritten as
\begin{align*}
|s^a(x_0)| &:= \left|\int_0^T \hspace*{-2mm} -\alpha(\|z^\star(t, x_0) - \bar z^\star\|) + w(z^\star(t, x_0))\,\mathrm{d}t \right|  \\
&\leq \mu[\Theta_{z, T}(\varepsilon)] \left(\hat\alpha+ \hat w\right) + T (\alpha(\varepsilon)+  L_F\varepsilon), 
\end{align*}
where $\hat\alpha := \sup_{z \in \mcl{Z}}\, \alpha(\|z - \bar z^\star\|) < \infty$.
Setting $\varepsilon  = 0$ allows concluding that, for all $T \geq 0$ and all $x_0 \in \mcl{X}_0$,
%\begin{multline*}
%\left|\int_0^T  -\alpha(\|z^\star(t, x_0) - \bar z^\star\|) + w(z^\star(t, x_0))\,\mathrm{d}t \right|\\
%
%\[
$| s^a(x_0)| < \infty$.
%\leq \nu_\xi(0) \left(\hat\alpha+\hat w\right) < \infty. $
%\]
%\end{multline*}
Note that Thm. \ref{thm:dissCond} also holds if one restricts the class of considered input signals in \eqref{eq:OCP_diss} (Rem. \ref{rema:dissOCP}).  
Hence, we conclude that along optimal pairs of $\OCP{x_0}$ the strict inequality \eqref{eq:str_diss} holds for the chosen $\alpha \in \mathcal{K}$. 
\QED
\end{pf}
\vspace*{-5mm}
Let $\mcl{B}_\rho(\bar z)$ denote an open ball of radius $\rho$ centered at $\bar z$.
\vspace*{-2mm}
\begin{ass}[$\bar z^\star$  is local minimizer of $F$ on $\mcl{Z}$] \label{ass:loc_bnd}
 There exists a constant $\rho >0$ and $\alpha_\rho \in \mcl{K}$ such that
 \begin{equation} \label{eq:loc_bnd}
 \alpha_\rho(\|z - \bar{z} \|) \leq F(z) - F(\bar z), \quad \forall z \in \mcl{B}_\rho(\bar z) \cap \mcl{Z}. 
% \vspace*{-4mm} 
 \end{equation} 
\end{ass}
%Note that Ass. \ref{ass:loc_bnd} requires $\bar z^\star$ to be a local minimizer of $F$ on $\mcl{Z}$. 
%The above assumption is satisfied if either of the following conditions holds:
%\vspace*{-3mm}
%\begin{itemize}
%\item $F$ is lower bounded on the set of admissible steady-state pairs, i.e., there exists $\alpha_\rho\in\mcl{K}$, such that, for all $\bar z  \in \bar{\mcl{Z}}$,  
%$
%\alpha_\rho(\|z - \bar{z} \|) \leq F(z) - F(\bar z)
%$
%  and $\bar z \in \interior\left\{z = \mbb{R}^{n_x +n_u} \,|\, f(\bar z) =0\right\}$.\footnote{To see that this implies Ass. \ref{ass:loc_bnd}, note that there exists an open ball of radius $\delta$ such that $\mcl{B}_\delta(\bar z) \subset  \interior\left\{z = \mbb{R}^{n_x +n_u} \,|\, f(\bar z) =0\right\}$. Satisfaction of Ass. \ref{ass:loc_bnd} follows directly, as the construction of $\mcl{B}_\delta(\bar z)$ and $\bar{\mcl{Z}}$ implies 
%$ \mcl{B}_\delta(\bar z) \cup \bar{\mcl{Z}} = \mcl{B}_\delta(\bar z) \cup {\mcl{Z}}$.}
%\item There exists $\rho >0$, such that $F$ is strongly convex on  $ \mcl{B}_\rho(\bar z)$.
%\end{itemize}
%
%Next, we present a converse result using Ass. \ref{ass:loc_bnd}.
% 
\vspace*{-4mm}
\begin{thm}[Turnpike $\Rightarrow$ strict dissipativity] \label{thm:TP_imp_D}
Suppose that %Ass. \ref{ass:reach}, 
 Ass. \ref{ass:loc_bnd} and Stmt. \ref{stat:TP} hold,  and let the turnpike be of the input-state type ($\xi = z$). \\
Then,  there exists a bounded storage function $S$ depending on $\alpha_\rho$, s.t.  $\OCP{x_0}$ is strictly dissipative with respect to $\bar z^\star$. 
%the strict dissipation inequality \eqref{eq:str_diss} holds along  optimal pairs $z^\star(\cdot, x_0)$ of $\OCP{x_0}$.  
\vspace*{-4mm} 
\end{thm}
\vspace*{-2mm}
\begin{pf}
Consider $\alpha_\rho \in \mcl{K}$ from Ass.  \ref{ass:loc_bnd}
and the integral
\begin{equation} \label{eq:strDI_opt}
s^a(x_0)= \int_0^T  \alpha_\rho(\|z^\star(t, x_0) - \bar z^\star\|) - w(z^\star(t, x_0))\,\mathrm{d}t
\end{equation}
evaluated along optimal pairs $z^\star(\cdot, x_0)$ of $\OCP{x_0}$. 
We split the time horizon into $\Theta_{z, T}(\varepsilon)$ and $[0,T]\setminus\Theta_{z, T}(\varepsilon)$. 
Evaluating the rhs of \eqref{eq:strDI_opt} on $\Theta_{z, T}(\varepsilon)$ yields
\begin{subequations} \label{eq:bnd1}
\begin{multline}
\int_{\Theta_{z, T}(\varepsilon)} \hspace{-2mm} \alpha_\rho(\|z^\star(t, x_0) - \bar z^\star\|) - w(z^\star(t, x_0))\,\mathrm{d}t \\
 \leq \mu[\Theta_{z, T}(\varepsilon)](\hat\alpha+\hat w) 
\end{multline}
where $\hat\alpha_\rho:= \sup_{z \in \mcl{Z}}\, \alpha_ \rho(\|z - \bar z^\star\|) < \infty$, and $\hat w$ is as defined in proof of Thm. \ref{thm:exTP_imp_D}. The turnpike property of $\OCP{x_0}$ implies, for all $T>0$ and all $x_0 \in \mcl{X}_0$, that
\begin{equation}
\mu[\Theta_{z, T}(\varepsilon)](\hat\alpha+\hat w) \leq \nu_z(\varepsilon)(\hat\alpha+\hat w)  < \infty.
\end{equation}
\end{subequations}
%cf. \eqref{eq:TP}.
To evaluate the rhs of \eqref{eq:strDI_opt} on  $[0,T]\setminus\Theta_{z, T}(\varepsilon)$, recall that the turnpike of $\OCP{x_0}$
implies that for $T$ sufficiently large, we have
%\[
$
\|z^\star(t, x_0)-\bar z\|\leq \rho,\quad \forall t \in [0,T]\setminus\Theta_{z, T}(\varepsilon),
$
%\]
where $\rho$ is from Ass. \ref{ass:loc_bnd}. 
%In other words, pick a horizon long enough to guarantee that $z^\star(t, x_0)$ enters $\mcl{B}_\rho(\bar z) \cap \mcl{Z}$. 
Using \eqref{eq:loc_bnd} we obtain
\begin{equation} \label{eq:bnd2}
\int_{[0,T]\setminus\Theta_{z, T}(\varepsilon)} \hspace{-3mm}  \alpha_\rho(\|z^\star(t, x_0) - \bar z^\star\|) - w(z^\star(t, x_0))\,\mathrm{d}t \leq 0.
\end{equation}
Combining \eqref{eq:bnd1} and \eqref{eq:bnd2} yields
$
 s^a(x_0) <  \nu_z(\varepsilon)(\hat\alpha+\hat w) < \infty.
$
Recalling Rem. \ref{rema:dissOCP}, we conclude that along optimal pairs  \eqref{eq:str_diss} holds for $\alpha_\rho$. \QED
\end{pf}

%\section{Sufficient Equivalence Conditions} \label{sec:equi}
%At this point, one may wonder whether one can show the equivalence of Statements \ref{stat:D}--\ref{stat:TP}. 
%In the view of Corollary \ref{cor:SS_imp_Dis}, this boils down to the question of
% how large the gap between strict and non-strict dissipativity, i.e., between \eqref{eq:diss} and \eqref{eq:str_diss}, is. To this end, we investigate under what conditions dissipativity implies strict dissipativity.
% 
%\begin{ass} \label{ass:uniqueMin}
%The cost function $F$ has a unique minimizer $\bar{z}^\star$ over $\mathcal{Z}$, and this unique minimizer is a steady state, i.e., $\bar{z}^\star \in \bar{\mathcal{Z}}$.
% 
%\end{ass}
%
%Under this assumption, dissipativity and strict dissipativity are equivalent.
%\begin{lemm}[Strict and non-strict dissipativity] \label{lem:strDissEq} ~\\
% If Assumption~\ref{ass:uniqueMin} holds and the system \eqref{eq:sys_def} is dissipative with respect to the supply rate $w(z) = F(z) - F(\bar{z}^\star)$, then the system is strictly dissipative with respect to the same supply rate.  
%\end{lemm}
%\begin{pf}
%First, we observe that there exists a class-$\mathcal{K}$ function $\alpha$ such that 
%\begin{equation}\label{classKLowerBound}
%w(z) \ge \alpha(\|z - \bar{z}^\star \|) \quad \forall\, z \in \mathcal{Z}.
%\end{equation}
%This follows from \cite[Lemma~18]{Kellett14} or \cite[Chap.~2]{Hahn67} using Assumption~\ref{ass:uniqueMin} and the compactness of $\mathcal{Z}$. 
%
%Next, using the dissipativity premise and by adding and subtracting $w$ on the right side of \eqref{eq:diss}, we obtain
%\[
%S(x_T) - S(x_0)  \leq \int_{0}^{T} 2 w(x(t),u(t)) - w(x(t),u(t))\,\mathrm{d}t.
%\]
%Combining the last inequality and~(\ref{classKLowerBound}) gives
%\begin{multline*}
%\textstyle{\frac{1}{2}}\left(S(x_T) - S(x_0)\right) \\
% \leq \displaystyle \int_{0}^{T} -  \textstyle{\frac{1}{2}}\alpha(\|z(t) - \bar{z}^\star \|) + w(x(t),u(t))\,\mathrm{d}t,
%\end{multline*}
%which is a strict dissipation inequality with the supply rate $w$, the storage function $S/2$, and the class-$\mathcal{K}$ function $\alpha/2$. This proves that the system is strictly dissipative with respect to the supply rate $w(x,u)$ for the case $\xi = z$ (see Definition~\ref{def:diss}). Since $\|x -\bar x^\star\| \leq \|z -\bar z^\star\|$, the case $\xi = x$ follows directly.\QED
%\end{pf}
%
%Now, we are ready to state conditions ensuring the equivalence of Statements \ref{stat:D}--\ref{stat:TP}.
%
%\begin{thm}[Equivalence of Statements \ref{stat:D}--\ref{stat:TP}] ~\\
%If
%\begin{enumerate}[(i)]
%\vspace*{-3mm}
%\item $F$ satisfies Assumption~\ref{ass:uniqueMin},
%\item $\mcl{Z}= \mcl{Z_T}$, with $\mcl{Z}_T = \mcl{X}_T \times\mcl{U}$, $\mcl{X}_T$ from  \eqref{eq:X_T}, and  $\mcl{X}_0 = \mcl{X}_T$, 
%\end{enumerate}
%\vspace*{-3mm}
%then, for $\xi = z$, Statements \ref{stat:D}--\ref{stat:TP} are equivalent.  
%\end{thm}
%\begin{pf}
%Due to part (i) and (ii), we know from Corollary \ref{cor:SS_imp_Dis} and Lemma \ref{lem:strDissEq} that Stmt. \ref{stat:D} $\Leftrightarrow$ Statement  \ref{stat:S}. 
%The condition  $\mcl{Z}= \mcl{Z_T}$  in part (ii) implies that  Ass. \ref{ass:reach} holds. Hence, we infer that Stmt. \ref{stat:D} $\Rightarrow$ Statement  \ref{stat:TP} (Thm. \ref{thm:D_imp_TP}).
%Finally, Thm. \ref{thm:TP_imp_S} shows that Stmt. \ref{stat:TP} $\Rightarrow$ Statement  \ref{stat:S}. Hence, we have Stmt. \ref{stat:D} $\Leftrightarrow$ Statement  \ref{stat:S} $\Leftrightarrow$ Stmt. \ref{stat:TP}. \QED
%\end{pf}
%
\section{Example: Chemical Reactor} \label{sec:example}

We consider the example of a continuously stirred chemical  reactor.
% in which three endothermal chemical reactions $A \stackrel{k_1}{\to} B  \stackrel{k_2}{\to} C$ and $2A  \stackrel{k_3}{\to} D$ take place. 
A model of the reactor, including the concentration of species $A$ and $B$, $c_A, c_B$ in mol/l and the reactor temperature $ \vartheta$ in $^\circ$C as state variables, reads
\begin{subequations} \label{eq:sys_vanVuss}
\begin{align}
\dot c_A & = -r_A(c_A, \vartheta) + (c_{in}-c_A)u_1 \\
\dot c_B & =\phantom{-} r_B(c_A, c_B,  \vartheta)  -c_Bu_1 \\
\dot  \vartheta  & = \phantom{-}h(c_A, c_B,  \vartheta) + \alpha(u_2- \vartheta) + ( \vartheta_{in}- \vartheta)u_1, 
\end{align}
\end{subequations}
where $r_A 	   =k_1c_A +k_3c_A^2, \quad r_B  = k_1c_A-k_2c_B$,\quad $h = -\delta( k_1 c_A \Delta H_{AB} + k_2c_B \Delta H_{BC}+k_3c_A^2 \Delta H_{AD})$
%and 
%\vspace*{-2mm}
%\begin{align*} 
%r_A 	   &=\phantom{-}k_1c_A +k_3c_A^2, \quad r_B  = k_1c_A-k_2c_B \\
%%r_A(\cdot) & =\phantom{-} k_1(T)(c_A-c_B) \\
%h &= -\delta\Big( k_1 c_A \Delta H_{AB} + k_2c_B \Delta H_{BC}+k_3c_A^2 \Delta H_{AD}\Big) %\vspace*{-5mm}%\\                & \phantom{=}   + k_2(T)c_A^2
%%k_i &= \phantom{-}k_{i0}e^{\frac{-E_i}{ \vartheta+ \vartheta_0}}, \quad i = 1,2,3. \label{eq:arr}  
%\end{align*}
 and $k_i = \phantom{-}k_{i0}e^{\frac{-E_i}{ \vartheta+ \vartheta_0}}, \quad i = 1,2,3$.
 The system parameters can be found in \cite{Rothfuss96b}. 
 %The inputs $u_1, u_2$ are the normalized flow rate of $A$ through the
%reactor in 1/h and the temperature in the cooling jacket in $^\circ$C. 
The states and inputs are subject to the constraints $
c_A \in [0, 6] \frac{mol}{l}, c_B \in [0, 4]\frac{mol}{l},  \vartheta \in [70, 150]^\circ C$ and $u_1 \in [3, 35]\frac{1}{h}, u_2 \in [0, 200]^\circ C$.
%\begin{equation}  \label{eq:con_vanVuss}
% \begin{array}{lll}
%c_A \in [0, 6] \frac{mol}{l} & c_B \in [0, 4]\frac{mol}{l} &  \vartheta \in [70, 150]^\circ C \\
%u_1 \in [3, 35]\frac{1}{h}  & u_2 \in [0, 200]^\circ C . &
% \end{array}
%\end{equation}
We consider the problem of maximizing the production rate of $c_B$; thus $F$ in \eqref{eq:J} and \eqref{eq:OPT_ss} is
%\begin{equation} 
$F(c_B, u_1) = -\beta c_Bu_1, \beta >0.$
%\end{equation}
%
%
%
%\begin{figure*}[t]
%\begin{center}
%\includegraphics[width=0.9\textwidth]{./figures/Turnpike_cropped.eps}
%\caption{Simulation results for the Van de Vusse reactor \eqref{eq:sys_vanVuss}.\label{fig:example}}
%\end{center}
%\end{figure*}
%
%\subsection*{Numerical Verification of Dissipativity} % \label{sec:ex2}
To numerically verify dissipativity, we approximate the exponential term $k_i(\theta)$ by its fourth-order Taylor expansion at $\vartheta = 110^\circ$C.
% This yields a reasonable approximation of  $k_i(\theta)$ over the box  $[70, 200]$. 
This way the system dynamics become polynomial and a polynomial storage function can be sought using sum-of-squares (SOS) programming. 
Indeed, by choosing a quadratic function $\alpha := \bar{\alpha} \|x - \bar{x}^\star \|^2$, where $\bar{\alpha} > 0 $ is an optimization variable, all data in \eqref{eq:str_diss} become polynomial. We solve the following problem
\begin{subequations} \label{eq:OPT_sos}
\begin{align}
%\hspace{0.18\textwidth}  
&\underset{\bar{\alpha} \in [0,1], S(\cdot)\in \mathbb{R}_d[x]}{\max} \, \bar{\alpha}   \\
\textrm{subject to } &\forall (x,u)\in \mcl{X}\times \mcl{U}\nonumber \\ 
 w(x,u) &- \bar{\alpha}\|x - \bar x^\star \|^2 -\frac{\partial S}{\partial x}f(x,u) \geq 0, \label{eq:sos}
\end{align}
\end{subequations}
where $\mathbb{R}_d[x]$ denotes the vector space of polynomials of fixed degree $d$. The non-negativity constraint~(\ref{eq:sos}) is replaced by a sufficient SOS constraint; here we use the standard Putinar condition~\cite{Putinar93} imposed for each of the four vertices of $\mcl{U}$ separately (this is possible since $u$ enters the dynamics affinely and $\mcl{U}$ is convex). Instead of the integral inequality~\eqref{eq:str_diss} we consider its differential counter part in \eqref{eq:sos}.
This leads to a semidefinite programming problem, which is solved using SeDuMi~\cite{sedumi}. Details of sum-of-squares programming are omitted for brevity; see \cite{lasserreBook,Ebenbauer09a}. The strict dissipativity constant $\bar{\alpha}$ is constrained to $[0,1]$ for numerical reasons.
The optimal steady state $\bar x^\star = [2.1756, 1.1049, 128.53]^T$, $\bar u^\star = [ 35, 142.76]^T$ is computed using an SQP-method; its global optimality is verified using Gloptipoly~\cite{gloptipoly3}.
Seeking a polynomial storage function of degree five via \eqref{eq:OPT_sos} is feasible with $\bar{\alpha} = 1$, thus verifying strict dissipativity with respect to $\bar x^\star$ on $\mcl{X}\times\mcl{U}$.\\
%\subsection*{Numerical Verification of Turnpike Properties}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.495\textwidth]{./sim_results.png}  \\
%\includegraphics[width=0.41\textwidth]{./figures/p2.png}
\caption{Simulation results for  \eqref{eq:sys_vanVuss}: 
blue -- $T = 0.5$; dash-dot red -- $ T = 0.25$; green -- optimal steady state; black and dash-dot black -- constraints. \label{fig:example}}
\end{center}
\end{figure}
To illustrate turnpike behavior, we solve \eqref{eq:OCP} as given above
%\begin{equation} \label{eq:OCP_example}
%\begin{array}{c}
%\underset{u(\cdot)}{\minimize}  \,  \displaystyle \frac{1}{T}\int_0^T  -\beta c_B(t)u_1(t)\,\mathrm{d}t \\
%\textrm{subject to \eqref{eq:sys_vanVuss} and \eqref{eq:con_vanVuss}}. 
%\end{array}
%\end{equation}
%To foster comparability with the numerically obtained storage function, the Arrhenius terms \eqref{eq:arr} are again approximated by the fourth-order Taylor series developed at $\vartheta = 110^\circ$C. 
%The OCP is solved
 for two initial conditions with a piecewise-constant input parametrization with \cite{Houska11a}. 
For the initial condition $(c_A(0), c_B(0), \vartheta(0))  = (1.5, 1.2, 140)$, we consider an optimization horizon of $T = 0.5$; for $(c_A(0), c_B(0), \vartheta(0))  = (5.1, 0, 100)$, we use $T =0.25$. 
The plots in the upper part of Fig.~\ref{fig:example} show the state trajectories $c_A, c_B$ and $\vartheta$ and their optimal steady-state values.  Clearly, the optimal solutions exhibit the turnpike property. 
The lower part of Fig. \ref{fig:example} depicts the inputs $u_1, u_2$ (last two plots). Note that the input $u_1$ is always at its upper limit. The third plot from the bottom illustrates the strict dissipation inequality~\eqref{eq:str_diss}. To this end, we define  the residual
%\vspace*{-2mm}
\[
\Delta(t) = S(x(\tau))\Big|_{0}^{t} %\left.
-\int_0^t  \hspace*{-2mm} w(x(\tau),u(\tau)) -\bar\alpha\|x(\tau)-\bar x^\star \|\,\mathrm{d}\tau %\right|_{ \begin{array}{l l}   \tiny x(t) =&x(t, x_0, u^\star(\cdot))\\  \scriptsize{ u(t) =}& u^\star(t)\end{array}}
%\delta(t) := \left.\left(\frac{\partial S}{\partial x} f(x,u)+ \alpha(\|x-\bar x^\star \|) - w(x,u)\right)\right|_{ \begin{smallmatrix} { x =x(t)}\\ { u = u(t)}\end{smallmatrix}},
%\vspace*{-2mm}
\]
where $\bar\alpha\|x-\bar x^\star \|$, $w(x,u)$ and the storage $S$ are the ones computed in \eqref{eq:OPT_sos}.
%Note that $\Delta(t) \leq 0$ implies that \eqref{eq:str_diss} is satisfied along optimal solutions. 

%In Fig. \ref{fig:example} (lower part, left side) $\delta(t)$ is plotted along the optimal solutions to \eqref{eq:OCP_example}.  We  see that the dissipation inequality \eqref{eq:str_diss_diff} is satisfied for both cases. 





\section{Summary and Conclusions}
Before concluding this paper, it is worth summing up the main points and highlighting subtle differences between the presented results. 
Thms. \ref{thm:D_imp_TP}--\ref{thm:D_imp_SII}  rely on strict dissipativity of $\OCP{x_0}$, i.e. they require (strict) dissipativity to hold along all optimal solutions. Taking this into account, it is evident that equivalence of strict dissipativity of $\OCP{x_0}$ (Stmt. \ref{stat:D}) and a turnpike property of $\OCP{x_0}$ (Stmt. \ref{stat:TP}) is guaranteed under mild conditions, cf. 
 Ass. \ref{ass:reach} \& \ref{ass:loc_bnd} in Thm. \ref{thm:TP_imp_D}.
 It seems to be the generality of optimal operation at steady state (Stmt. \ref{stat:S}) which in search for equivalence conditions requires stronger assumptions such as local controllability of the turnpike steady state and turnpike-like behavior of near optimal solutions, cf.  \cite{Gruene16a}. 
 
%We also remark that  for OCPs with polynomial problem data one can verify the strict dissipativity condition \eqref{eq:str_diss} in its differential form by means of sum-of-squares programming, see \cite{epfl:faulwasser14e} for an example.
%In contrast, Corollary \ref{cor:S_imp_D} reveals that equivalence of  Stmts. \ref{stat:D}--\ref{stat:S} requires much stronger assumptions, cf. also \cite{Gruene16a, Mueller14a}. 


For continuous-time optimal control problems, this paper has investigated the relationship between dissipativity properties, optimal operation at steady state, and turnpike properties. We extended discrete-time results to show that
dissipativity of $\OCP{x_0}$ implies (a) optimal operation at steady state and (b) a turnpike property of optimal solutions. 
On top, we derived novel converse turnpike results showing that under mild assumptions a tunrpike in $\OCP{x_0}$ implies dissipativity. Finally, we commented on sufficient conditions guaranteeing the equivalence of the properties. 



%Finally, we demonstrated by means of a numerical example that, for polynomial systems, dissipativity, and thus also turnpikes of optimal control problems, can be verified via sum-of-squares programming.  

	
\bibliographystyle{plain}        % Include this if you use bibtex 
%\bibliography{C:/Users/Timm/Documents/DATA/Literatur/literature_latin1}           % and a bib file to produce the 
%                                 % bibliography (preferred). The
%                                 % correct style is generated by
%                                 % Elsevier at the time of printing.

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\newblock McGraw-Hill, New York, 1958.

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   % Sections and subsections are supported  
                                        % in the appendices.
\end{document}
DissTurn_final.bbl
\begin{thebibliography}{10}

\bibitem{Anderson87a}
B.D.O. Anderson and P.V. Kokotovic.
\newblock Optimal control problems over large time intervals.
\newblock {\em Automatica}, 23(3):355--363, 1987.

\bibitem{Angeli12a}
D.~Angeli, R.~Amrit, and J.B. Rawlings.
\newblock On average performance and stability of economic model predictive
  control.
\newblock {\em IEEE Trans. Automat. Contr.}, 57(7):1615--1626, 2012.

\bibitem{Carlson91}
D.A. Carlson, A.~Haurie, and A.~Leizarowitz.
\newblock {\em Infinite Horizon Optimal Control: Deterministic and Stochastic
  Systems}.
\newblock Springer Verlag, 1991.

\bibitem{Clarke13a}
F.~Clarke.
\newblock {\em Functional Analysis, Calculus of Variations and Optimal
  Control}.
\newblock Number 264. Springer, 2013.

\bibitem{Stieler14a}
T.~Damm, L.~Gr{\"u}ne, M.~Stieler, and K.~Worthmann.
\newblock An exponential turnpike theorem for dissipative optimal control
  problems.
\newblock {\em SIAM Journal on Control and Optimization}, 52(3):1935--1957,
  2014.

\bibitem{Dorfman58}
R.~Dorfman, P.A. Samuelson, and R.M. Solow.
\newblock {\em Linear Programming and Economic Analysis}.
\newblock McGraw-Hill, New York, 1958.

\bibitem{Ebenbauer09a}
C.~Ebenbauer, T.~Raff, and F.~Allg{\"o}wer.
\newblock Dissipation inequalities in systems theory: An introduction and
  recent results.
\newblock In {\em R. Jeltsch and G. Wanner (ed.), Invited Lectures of the
  International Congress on Industrial and Applied Mathematics 2007}, pages
  23--42, 2009.

\bibitem{epfl:faulwasser15a}
T.~Faulwasser and D.~Bonvin.
\newblock On the design of economic {NMPC} based on an exact turnpike property.
\newblock {\em IFAC-PapersOnLine}, 48(8):525 -- 530, 2015.

\bibitem{epfl:faulwasser15g}
T.~Faulwasser and D.~Bonvin.
\newblock On the design of economic {NMPC} based on approximate turnpike
  properties.
\newblock In {\em 54th IEEE Conference on Decision and Control}, pages 4964 --
  4970, Osaka, Japan, December 15-18 2015.

\bibitem{kit:faulwasser17a}
T.~Faulwasser and D.~Bonvin.
\newblock Exact turnpike properties and economic {NMPC}.
\newblock {\em European Journal of Control}, 2017.
\newblock In press.

\bibitem{epfl:faulwasser14e}
T.~Faulwasser, M.~Korda, C.N. Jones, and D.~Bonvin.
\newblock Turnpike and dissipativity properties in dynamic real-time
  optimization and economic {MPC}.
\newblock In {\em Proc. of the 53rd IEEE Conference on Decision and Control},
  pages 2734--2739, Los Angeles, California, USA, 2014.

\bibitem{Gruene13a}
L.~Gr{\"u}ne.
\newblock Economic receding horizon control without terminal constraints.
\newblock {\em Automatica}, 49(3):725--734, 2013.

\bibitem{Gruene16a}
L.~Gr{\"u}ne and M.A. M{\"u}ller.
\newblock On the relation between strict dissipativity and turnpike properties.
\newblock {\em Sys. Contr. Lett.}, 90:45 -- 53, 2016.

\bibitem{Gurman04a}
V.I. Gurman and M.Y. Ukhin.
\newblock Turnpike solutions in optimization of regional development
  strategies.
\newblock {\em Automation and Remote Control}, 65(4):603--611, 2004.

\bibitem{gloptipoly3}
D.~Henrion, J.B. Lasserre, and J.~L{\"o}fberg.
\newblock Gloptipoly 3: moments, optimization and semidefinite programming.
\newblock {\em Optimization Methods and Software}, 24:761--779, 2009.

\bibitem{Houska11a}
B.~Houska, H.J. Ferreau, and M.~Diehl.
\newblock {ACADO} toolkit -- an open-source framework for automatic control and
  dynamic optimization.
\newblock {\em Optimal Control Applications and Methods}, 32(3):298--312, 2011.

\bibitem{lasserreBook}
J.B. Lasserre.
\newblock {\em Moments, Positive Polynomials and Their Applications,}.
\newblock Imperial College Press, 2009.

\bibitem{Lee67}
E.B. Lee and L.~Markus.
\newblock {\em Foundations of Optimal Control Theory}.
\newblock The SIAM series in applied mathematics. John Wiley \& Sons New York,
  London, Sydney, 1967.

\bibitem{McKenzie76}
L.W. McKenzie.
\newblock Turnpike theory.
\newblock {\em Econometrica: Journal of the Econometric Society},
  44(5):841--865, 1976.

\bibitem{Moylan14a}
P.~Moylan.
\newblock {\em Dissipative Systems and Stability}.
\newblock http://www.pmoylan.org, 2014.

\bibitem{Mueller14a}
M.A. M{\"u}ller, D.~Angeli, and F.~Allg{\"o}wer.
\newblock On necessity and robustness of dissipativity in economic model
  predictive control.
\newblock {\em IEEE Trans. Automat. Contr.}, 60(6):1671--1676, 2015.

\bibitem{sedumi}
I.~P{\'o}lik, T.~Terlaky, and Y.~Zinchenko.
\newblock {SeDuMi}: a package for conic optimization.
\newblock In {\em IMA Workshop on Optimization and Control}, 2007.

\bibitem{Putinar93}
M.~Putinar.
\newblock Positive polynomials on compact semi-algebraic sets.
\newblock {\em Indiana University Mathematics Journal,}, 42:969--984, 1993.

\bibitem{Rao99a}
A.V. Rao and K.D. Mease.
\newblock Dichotomic basis approach to solving hyper-sensitive optimal control
  problems.
\newblock {\em Automatica}, 35(4):633--642, 1999.

\bibitem{Rothfuss96b}
R.~Rothfu{\ss}, J.~Rudolph, and M.~Zeitz.
\newblock Flatness based control of a nonlinear chemical reactor model.
\newblock {\em Automatica}, 32:1433--1439, 1996.

\bibitem{Sahlodin15}
A.M. Sahlodin and P.I. Barton.
\newblock Optimal campaign continuous manufacturing.
\newblock {\em Industrial \& Engineering Chemistry Research},
  54(45):11344--11359, 2015.

\bibitem{Trelat15a}
E.~Tr{\'e}lat and E.~Zuazua.
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