\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
Eighth Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
{\em Electronic Journal of Differential Equations},
Conf. 19 (2010), pp. 161--175.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{8mm}}
\begin{document} \setcounter{page}{161}
\title[\hfilneg EJDE-2010/Conf/19/\hfil Optimal control]
{Optimal control of a waste water cleaning plant}
\author[E. V. Grigorieva, E. N. Khailov\hfil EJDE/Conf/19 \hfilneg]
{Ellina V. Grigorieva, Evgenii N. Khailov} % in alphabetical order
\address{Ellina V. Grigorieva \newline
Department of Mathematics and Computer Science\\
Texas Woman's University \\
Denton, TX 76204, USA}
\email{EGrigorieva@mail.twu.edu}
\address{Evgenii N. Khailov \newline
Department of Computer Mathematics and Cybernetics\\
Moscow State Lomonosov University \\
Moscow, 119991, Russia}
\email{Khailov@cs.msu.su}
\thanks{Published September 25, 2010.}
\subjclass[2000]{49J15, 49N90, 93C10, 93C95}
\keywords{Optimal control problem; nonlinear model;
\hfill\break\indent waste water cleaning process}
\begin{abstract}
In this work, a model of a waste water treatment plant is
investigated. The model is described by a nonlinear system of two
differential equations with one bounded control. An optimal control
problem of minimizing concentration of the polluted water at the
terminal time $T$ is stated and solved analytically with the use of
the Pontryagin Maximum Principle. Dependence of the optimal solution
on the initial conditions is established. Computer simulations of a
model of an industrial waste water treatment plant show the advantage
of using our optimal strategy. Possible applications are discussed.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction}
While water is the most abundant life sustaining substance on the
planet, clean, fresh water is in many localities often the most scarce.
The supply of fresh water over the land masses is limited by chaotic
weather effects. Meanwhile, human populations and the success of our
civilizations rely on stable and sustainable supplies of clean, fresh
water. As population densities increase, the maintenance of supplies
of potable water tend to become dependent on the efficiencies of fresh
water recovery methods.
The activated sludge process (ASP) is a biochemical process for treating
sewage and industrial waste-water that uses air (or oxygen) and
microorganisms to biologically oxidize organic pollutants, producing
a waste sludge (or floc) containing the oxidized material. The optimal
operation of the waste water processes with biological treatment is
challenging because of the strong effluent requirements, the complexity
of theses processes as an object of control and the need to reduce the
operation cost. The USA has strict requirements on the effluent quality
of the ASP. Similar strict requirements were adopted during the last
decade in Europe and in South Africa \cite{Lit16}.
In general, an activated sludge process has an aeration tank where air
(or oxygen) is injected and thoroughly mixed into the waste-water and
a settling tank (usually referred to as a "clarifier" or "settler").
Flocculation-agglomeration is a process where a solute comes out of
solution in the form of floc or flakes. Part of the waste sludge is
recycled to the aeration tank where the remaining waste sludge is
removed for further treatment and ultimate disposal. A diagram of the
process is shown in Figure \ref{fig1}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig1} %Sludge.eps
\end{center}
\caption{Diagram of flocculation-agglomeration process}\label{fig1}
\end{figure}
During the last decades various control strategies for the ASP have
been developed. Simple strategies are limited to the maintenance of
some desired values of easily determinable process parameters like
food-microorganism ratio, sludge recycle flowrate or oxygen
concentration in the aeration basin \cite{Lit7}. In more complex
models, the behavior of the sludge process also depends on several
working conditions e.g. air compressor power to regulate the mean
oxygen concentration \cite{Lit15}. Establishing optimal working
conditions and control strategies is frequently accomplished with the
aid of mathematical models
\cite{Lit2,Lit3,Lit8,Lit11,Lit16}.
Relevant work in the investigation and comparison of control strategies
was done by
\cite{Lit4,Lit5,Lit9,Lit10,Lit13,Lit14}.
Obviously, the solution depends on the model. What unites all these
papers is that either the considered models are so complex that they
cannot be solved analytically or that the controls are not bounded and
therefore the realism of the model is questionable. The model proposed
in \cite{Lit2} is simple enough that it can be investigated analytically.
On the other hand, it properly corresponds to the main steps of the ASP
and water cleaning control process. In \cite{Lit2} an optimal control
problem of the minimization of the waste concentration in the ASP was
stated and the Pontryagin Maximum Principle \cite{Lit12} was offered
for its solution. However, the complete analysis of the corresponding
boundary value problem for the Maximum Principle was not conducted.
The author simply offered a numerical solution to the problem at
different piecewise constant controls.
This work deals with the complete analysis of the model proposed by
\cite{Lit2}, but with a different objective function. In Section 2,
we discuss the model. In Section 3, we establish the properties of the
state variables. In Section 4, we state optimal control problem of
minimizing water pollution concentrations at the terminal time $ T $
and find optimal solutions. In Section 5, we investigate how optimal
solutions depend on the initial conditions. A numerical simulation of
the ASP at different parameters of the model is conducted in Section 6.
Finally, Section 7 presents our conclusions.
\section{The model}
Let us consider the model of an activated sludge process. A simplified
diagram can be shown in the Figure \ref{fig2}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig2} % System.eps
\caption{Simplified diagram of an activated sludge process}\label{fig2}
\end{center}
\end{figure}
Here $ u(t) $ is the inflow rate of the recirculated biomass (gal/min),
$ b $ is the inflow rate of substrate - polluted water (gal/min),
$ a_{2} $ the concentration of substrate (lb/gal), $ a_{1} $
concentration of bacteria (lb/gal).
This process can be described by the following system of differential
equations
\begin{equation}\label{num2.1}
\begin{gathered}
\dot{x}(t) = u(t)a_{1} + \mu_{0}\frac{x(t)s(t)}{k + s(t)} -
(b + u(t))x(t), \\
\dot{s}(t) = b a_{2} - \frac{\mu_{0}}{Y} \frac{x(t)s(t)}{k + s(t)} -
(b + u(t))s(t), \quad t \in [0,T], \\
x(0) = x_{0} > 0, \; s(0) = s_{0} > 0.
\end{gathered}
\end{equation}
We consider function $ u(t) $ as a control function and the set
$ D(T) $ is the set of all Lebegue measurable functions $ u(t) $ such
that $ 0 < u_{1} \le u(t) \le u_{2} $ for almost all $ t \in [0,T] $.
The recycle sludge rate $ u(t) $ is not allowed to take values below
a certain lower limit $ u_{1} $ in order to prevent the biomass from
being swept out of the aeration tank. An upper limit $ u_{2} $ for
$ u(t) $ is given by the limited power of the recycle pump.
Here $ x(t) $ is the concentration of biomass, $ s(t) $ is the
concentration of polluted water, $ Y $ is the substrate utilization
- yield coefficient, $ \mu_{0} $ is the maximal specific rate of
bacteria growth, $ k $ the saturation coefficient.
For the model \eqref{num2.1} assume $ k \gg s $. This case is
realistic since one normally tries to keep the substrate-to-biomass
ratio comparatively low. Denoting $ \mu = \frac{\mu_{0}}{k} $ the
simplified system can be written as
\begin{equation}\label{num2.2}
\begin{gathered}
\dot{x}(t) = u(t)a_{1} + \mu x(t)s(t) - (b + u(t))x(t), \\
\dot{s}(t) = b a_{2} - \frac{\mu}{Y} x(t)s(t) - (b + u(t))s(t), \; t \in [0,T], \\
x(0) = x_{0} > 0, \; s(0) = s_{0} > 0. \\
\end{gathered}
\end{equation}
Numerical modeling conducted in \cite{Lit2} shows that the second term
in the first equation of \eqref{num2.2} can be ignored. Finally we
obtain the following system
\begin{equation}\label{num2.3}
\begin{gathered}
\dot x(t) = u(t) a_{1} - (b + u(t))x(t), \quad t \in [0,T], \\
\dot s(t) = b a_2 - \frac{\mu}{Y} x(t)s(t) - (b + u(t))s(t), \\
x(0) = x_{0} > 0, \; s(0) = s_{0} > 0.
\end{gathered}
\end{equation}
This model and its investigation will be considered further.
\section{Properties of the state variables}
We have the following statement, which can be easily proven using
direct integration of the system \eqref{num2.3}.
\begin{lemma}\label{Lemma1}
Let $ u(\cdot)\in D(T) $ be some control function. Then there exist
corresponding to this control, $ u(t) $, solutions $ x(t) $, $ s(t) $
to system \eqref{num2.3}, which on the closed interval $ [0,T] $
satisfy the inequalities:
$$
x(t) > 0, \quad s(t) > 0.
$$
\end{lemma}
Analyzing system of equations \eqref{num2.3} with the use of
Lemma \ref{Lemma1} we can conclude that if at some moment of time
$ t \in [0,T] $ we have that $ x(t) = a_{1} $, then
$$
\dot x(t) = - ba_{1} < 0.
$$
By analogy if at some moment $ t $ we have $ s(t) = a_{2} $,
then we obtain relationship
$$
\dot s(t) = - \frac{\mu}{Y}a_{2}x(t) - u(t)a_{2} < 0.
$$
The validity of these relationships leads to the following statement.
\begin{lemma}\label{Lemma2}
Let $ u(\cdot) \in D(T) $ be some control function. Suppose that at
some moments of time $ \tau_{1}, \tau_{2} \in [0,T) $ the following
relationships hold
$$
x(\tau_{1}) \le a_{1}, \quad s(\tau_{2}) \le a_{2},
$$
then we have $ x(t) < a_{1} $ for any $ t \in (\tau_{1},T] $ and
$ s(t) < a_{2} $ for any $ t \in (\tau_{2},T] $.
\end{lemma}
From results of the Lemma \ref{Lemma2} it follows the statement.
\begin{lemma}\label{Lemma3}
Let $ u(\cdot) \in D(T) $ be some control function. Suppose that at
some moments of time $ \eta_{1}, \eta_{2} \in (0,T) $ the following
relationships hold
$$
x(\eta_{1}) > a_{1}, \quad s(\eta_{2}) > a_{2},
$$
then we have $ x(t) > a_{1} $ for any $ t \in [0,\eta_{1}) $ and
$ s(t) > a_{2} $ for any $ t \in [0,\eta_{2}) $.
\end{lemma}
Moreover, we have the statement.
\begin{lemma}\label{Lemma4}
Let $ u(\cdot) \in D(T) $ be some control function. Suppose that at
some moments of time $ \theta_{1}, \theta_{2} \in (0,T) $ the
following relationships hold
$$
x(\theta_{1}) \ge a_{1}, \quad s(\theta_{2}) \ge a_{2},
$$
then we have inequalities:
$$
\dot x(\theta_{1}) < 0, \quad \dot s(\theta_{2}) < 0
$$
respectively.
\end{lemma}
The validity of the Lemma \ref{Lemma4} follows from the equations
\eqref{num2.3}.
\section{Optimal control problem of minimizing pollution at terminal
time~$ T $}
Let $ s(t) $ be the pollution concentration at moment $ t $. Then an
integrated relative increase of the amount of pollution by time $ t $
can be written as
$$
\int_{0}^{t} \frac{\dot s(t)}{s(t)}dt = \ln{\frac{s(t)}{s_0}}, \;
t \in [0,T].
$$
For system \eqref{num2.3} we will consider an optimal control problem
of minimizing of the integrated relative increase of the pollution by
time $ T $, which is equivalent to
\begin{equation}\label{num4.1}
J(u) = s(T) \to \min_{u(\cdot) \in D(T)}.
\end{equation}
The existence of the optimal control $ u_{*}(t) $ and corresponding
to it optimal solutions $ x_{*}(t) $, $ s_{*}(t) $ for the optimal
control problem \eqref{num2.3},\eqref{num4.1} follows from \cite{Lit6}.
In order to solve problem \eqref{num2.3},\eqref{num4.1} we will apply
the Pontryagin Maximum Principle (\cite{Lit12}). For the optimal control
$ u_{*}(t) $ and corresponding optimal trajectories $ x_{*}(t) $,
$ s_{*}(t) $ there exist nontrivial solutions $ \psi_{*}(t) $,
$ \varphi_{*}(t) $ of the adjoint system
\begin{equation}\label{num4.2}
\begin{gathered}
\dot{\psi}_{*}(t) = (b + u_{*}(t))\psi_{*}(t) +
\frac{\mu}{Y}s_{*}(t)\varphi_{*}(t), \\
\dot{\varphi}_{*}(t) = \left(\frac{\mu}{Y} x_{*}(t) +
(b + u_{*}(t))\right)\varphi_{*}(t), \\
\psi_{*}(T) = 0, \quad \varphi_{*}(T) = -1,
\end{gathered}
\end{equation}
for which the control $ u_{*}(t) $ is given by
\begin{equation}\label{num4.3}
u_{*}(t) = \begin{cases}
u_{2} & \text{if } L(t) > 0, \\
\forall u \in [u_{1},u_{2}] & \text{if } L(t) = 0, \\
u_{1} & \text{if } L(t) < 0,
\end{cases}
\end{equation}
where
$$
L(t) = (a_{1} - x_{*}(t))\psi_{*}(t) - s_{*}(t)\varphi_{*}(t), \;
t \in [0,T]
$$
is the switching function. As it follows from \eqref{num4.3}, the
function $ L(t) $ determines the type of the optimal control
$ u_{*}(t) $.
Systems of equations \eqref{num2.3},\eqref{num4.2} and relationships
\eqref{num4.3} form the two point boundary value problem for the
Maximum Principle. Let us study this problem in depth.
Suppose control $ u(t) $, trajectories $ x(t) $, $ s(t) $ and functions
$ \psi(t) $, $ \varphi(t) $ satisfy this boundary value problem. Then
such a function $ u(t) $ is called extremal control, trajectories
$ x(t) $, $ s(t) $ are extremal trajectories and functions $ \psi(t) $,
$ \varphi(t) $ are called corresponding to them solutions of the adjoint
system \eqref{num4.2}.
For the functions $ \psi(t) $, $ \varphi(t) $ the following statement
is true.
\begin{lemma}\label{Lemma5}
Let $ u(t) $ be the extremal control, $ x(t) $ and $ s(t) $ are extremal
trajectories and $ \psi(t) $, $ \varphi(t) $ are the corresponding to
them solutions of the adjoint system \eqref{num4.2}. Then the following
inequalities hold
$$
\psi(t) > 0, \quad \varphi(t) < 0
$$
for all $ t \in [0,T) $.
\end{lemma}
The proof of this statement is based on integration of the system
\eqref{num4.2} with the use of Lemma \ref{Lemma1}.
Let $ L(t) $ be the switching function corresponding to the extremal
control $ u(t) $, extremal trajectories $ x(t) $, $ s(t) $ and
solutions $ \psi(t) $, $ \varphi(t) $ of the adjoint system
\eqref{num4.2}. The following statements is true.
\begin{lemma}\label{Lemma6}
There exists such time $ \theta \in [0,T) $ that for the extremal
control $ u(t) $ the equality $ u(t) = u_{2} $ is valid for all
$ t \in (\theta,T] $.
\end{lemma}
\begin{proof}
For the switching function $ L(t) $ from terminal conditions of the
system \eqref{num4.2} we have that
$$
L(T) = (a_{1} - x(T))\psi(T) - s(T)\varphi(T) = s(T).
$$
From Lemma \ref{Lemma1} we obtain the inequality $ L(T) > 0 $. Since
$ L(t) $ is continuous function, there exists value
$ \theta \in [0,T) $ such that $ L(t) > 0 $ for all
$ t \in (\theta,T] $. Furthermore, from \eqref{num4.3} we have
$ u(t) = u_{2} $ for $ t \in (\theta,T] $.
\end{proof}
\begin{lemma}\label{Lemma7}
If $ x_{0} \le a_{1} $, then for the extremal control $ u(t) $ the
relationship $ u(t) = u_{2} $ holds for all $ t \in [0,T] $.
\end{lemma}
\begin{proof}
From results of Lemma \ref{Lemma1}, Lemma \ref{Lemma2}, Lemma \ref{Lemma5}
for the switching function $ L(t) $ the inequality $ L(t) > 0 $ is true
for all $ t \in [0,T] $. Therefore, the desired result follows from
\eqref{num4.3}.
\end{proof}
\begin{lemma}\label{Lemma8}
The switching function $ L(t) $ has at most one zero in the interval
$ (0,T) $.
\end{lemma}
\begin{proof}
Note that the derivative of the switching function $ L(t) $ is
\begin{equation}\label{num4.4}
\dot{L}(t) = b a_{1}\psi(t) - b a_{2}\varphi(t) +
\frac{\mu}{Y}(a_{1} - x(t))s(t)\varphi(t), \; t \in [0,T].
\end{equation}
Let $ t_{0} \in (0,T) $ such that $ L(t_{0}) = 0 $. It means that
$$
a_{1} - x(t_{0}) = \frac{s(t_{0})\varphi(t_{0})}{\psi(t_{0})}.
$$
Substituting this expression into the formula \eqref{num4.4} we obtain
$$
\dot{L}(t_{0}) = b a_{1}\psi(t_{0}) - b a_{2}\varphi(t_{0}) +
\frac{\mu}{Y}\frac{s^{2}(t_{0})\varphi^{2}(t_{0})}{\psi(t_{0})}.
$$
It follows from the results of Lemma \ref{Lemma1}, Lemma \ref{Lemma5}
that $ \dot{L}(t_{0}) > 0 $. Since $ \dot{L}(t) $ is continuous, then
switching function $ L(t) $ has on the interval $ [0,T] $ the form
$$
L(t)
\begin{cases}
< 0, & \text{if } 0 \le t < t_{0}, \\
= 0, & \text{if } t = t_{0}, \\
> 0, & \text{if } t_{0} < t \le T.
\end{cases}
$$
This completes the proof.
\end{proof}
From Lemma \ref{Lemma6}, Lemma \ref{Lemma7}, Lemma \ref{Lemma8} and
relationship \eqref{num4.3} we obtain the following statement.
\begin{lemma}\label{Lemma9}
If $ L(0) \ge 0 $, then the extremal control $ u(t) $ is constant
function of the type
$$
u(t) = u_{2}, \; t \in [0,T].
$$
Alternatively, if $ L(0) < 0 $, then the extremal control $ u(t) $ is
a piecewise constant function of the type
$$
u(t) =
\begin{cases}
u_{1}, & \text{if } 0 \le t \le \theta, \\
u_{2}, & \text{if } \theta < t \le T,
\end{cases}
$$
where $ \theta \in (0,T) $ is the moment of switching, defined from
$ L(\theta) = 0 $.
\end{lemma}
From the properties of the switching function $ L(t) $ and
Lemma \ref{Lemma9} we established that the two point boundary value
problem for the Maximum Principle
\eqref{num2.3},\eqref{num4.2},\eqref{num4.3} has a unique solution
$ u(t) $, $ x(t) $, $ s(t) $, $ \psi(t) $, $ \varphi(t) $,
$ t \in [0, T] $, which as it follows from \cite{Lit1} is the optimal
solution for problem \eqref{num2.3},\eqref{num4.1}.
Optimal control $ u_{*}(t) $ has one of the two forms:
\begin{equation}\label{num4.5}
u_{*}(t) = u_{2}, \; t \in [0,T],
\end{equation}
and
\begin{equation}\label{num4.6}
u_{*}(t) =
\begin{cases}
u_{1}, & \text{if } 0 \le t \le \theta_{*}, \\
u_{2}, & \text{if } \theta_{*} < t \le T,
\end{cases}
\end{equation}
where $ \theta_{*} \in (0,T) $ is the moment of switching.
\section{Dependence of the optimal control on initial conditions}
In this section, we will find the initial conditions that correspond
to optimal controls of types \eqref{num4.5} and \eqref{num4.6}.
Therefore, consider the system \eqref{num2.3} with initial conditions
$ (x_{0},s_{0}) $.
Let $ u_{*}(t) $, $ x_{*}(t) $, $ s_{*}(t) $ be optimal solution to
the problem \eqref{num2.3},\eqref{num4.1}, and $ \psi_{*}(t) $,
$ \varphi_{*}(t) $ corresponding to them solutions of the adjoint
system \eqref{num4.2}.
First, we will consider the case that the optimal control $ u_{*}(t) $,
$ t \in [0,T] $ is given by formula \eqref{num4.5}. The inequality
$ L(0) \ge 0 $ can be rewritten as
$$
(a_{1} - x_{0})\psi_{*}(0) - s_{0}\varphi_{*}(0) \ge 0,
$$
or
\begin{equation}\label{num5.1}
s_{0} \ge (a_{1} - x_{0})\frac{\psi_{*}(0)}{\varphi_{*}(0)}.
\end{equation}
Next, we introduce a function
$ q(t) = \frac{\psi_{*}(t)}{\varphi_{*}(t)} $, which satisfies the
system
\begin{equation}\label{num5.2}
\begin{gathered}
\dot q(t) = -\frac{\mu}{Y}x_{*}(t)q(t) + \frac{\mu}{Y}s_{*}(t), \quad
t \in [0,T], \\
q(T) = 0.
\end{gathered}
\end{equation}
The solution of the Cauchy problem \eqref{num5.2} is written as
\begin{equation}\label{num5.3}
q(t) = -\frac{\mu}{Y} \int_{t}^{T}
e^{\frac{\mu}{Y}\int_{t}^{r} x_{*}(\xi)d\xi}s_{*}(r)dr.
\end{equation}
Then the inequality \eqref{num5.1} can be rewritten as
\begin{equation}\label{num5.4}
s_{0} \ge (a_{1} - x_{0})q(0).
\end{equation}
From the expression \eqref{num5.3} we obtain the formula
\begin{equation}\label{num5.5}
q(0) = -\frac{\mu}{Y} \int_{0}^{T}
e^{\frac{\mu}{Y}\int_{0}^{r} x_{*}(\xi)d\xi}s_{*}(r)dr.
\end{equation}
To show that the value $ q(0) $ depends on the initial conditions
$ (x_{0},s_{0}) $, integrate the system \eqref{num2.3} with control
$ u_{*}(t) $, $ t \in [0,T] $ given by \eqref{num4.5}. Integration
yields the formulas:
\begin{gather}
x_{*}(t) = x_{0}e^{-(b + u_{2})t} + \frac{a_{1}u_{2}}{b + u_{2}}
\left(1 - e^{-(b + u_{2})t}\right), \quad t \in [0,T], \label{num5.6}\\
\begin{aligned}
s_{*}(t) &= s_{0}e^{-(b + u_{2})t} \cdot
e^{-\frac{\mu}{Y}\int_{0}^{t} x_{*}(\xi)d\xi} \\
&\quad + a_{2}b \int_{0}^{t} e^{-(b + u_{2})(t - r)} \cdot
e^{-\frac{\mu}{Y}\int_{r}^{t} x_{*}(\xi)d\xi}dr.
\end{aligned} \label{num5.7}
\end{gather}
Substituting expressions \eqref{num5.6} and \eqref{num5.7} into
\eqref{num5.5}, we obtain
$$
q(0) = -\sigma s_{0} - g(x_{0}),
$$
where the value
$$
\sigma = \frac{\mu}{(b + u_{2})Y}\left(1 - e^{-(b + u_{2})T}\right)
$$
is positive, as well as the function
\begin{equation}\label{num5.8}
g(x_{0}) = \frac{\mu a_{2}b}{Y}
\int_{0}^{T}\Big(
\int_{0}^{r}e^{-(b + u_{2})(r - \eta)} \cdot
e^{\frac{\mu}{Y}\int_{0}^{\eta}x_{*}(\xi)d\xi}d\eta\Big)dr
\end{equation}
is also positive. Function \eqref{num5.8} depends on $ x_{0} $ by
formula \eqref{num5.6}.
Then \eqref{num5.4} can be rewritten as
\begin{equation}\label{num5.9}
s_{0}(1 + \sigma(a_{1} - x_{0})) \ge -(a_{1} - x_{0})g(x_{0}).
\end{equation}
If $ a_{1} - x_{0} \ge 0 $, then \eqref{num5.9} holds.
This means that for any initial conditions $ (x_{0},s_{0}) $ for which
$ a_{1} - x_{0} \ge 0 $, the optimal control $ u_{*}(t) $ has type
\eqref{num4.5} in agreement with Lemma \ref{Lemma7}.
If $ a_{1} - x_{0} < 0 $, then from \eqref{num5.9} it follows that
$ 1 + \sigma(a_{1} - x_{0}) > 0 $. Then inequality \eqref{num5.9}
becomes
\begin{equation}\label{num5.10}
s_{0} \ge -\frac{(a_{1} - x_{0})g(x_{0})}{1 + \sigma(a_{1} - x_{0})}.
\end{equation}
Now, consider a function
$$
s_{0} = f(x_{0}) = -\frac{(a_{1} - x_{0})g(x_{0})}{1 + \sigma(a_{1} - x_{0})}
$$
on the interval $ x_{0} \in [a_{1},a_{1} + \frac{1}{\sigma}) $.
Let us examine the properties of the function $ s_{0} = f(x_{0}) $.
Analyzing formulas \eqref{num5.6},\eqref{num5.8} we obtain
relationships:
$$
f(a_{1}) = 0, \quad
\lim_{x_{0} \to a_{1} + \frac{1}{\sigma}} f(x_{0}) = + \infty.
$$
Using \eqref{num5.8} we will find derivatives of the function
$ g(x_{0}) $. We have the expressions:
\begin{gather*}
\dot g(x_{0}) = \big(\frac{\mu}{Y}\big)^{2} a_{2}b
\int_{0}^{T}\Big( \int_{0}^{r}e^{-(b + u_{2})(r - \eta)} \cdot
e^{\frac{\mu}{Y}\int_{0}^{\eta}x_{*}(\xi)d\xi} \cdot
\Big(\int_{0}^{\eta}e^{-(b + u_{2})\xi}d\xi\Big)d\eta \Big)dr,
\\
\ddot g(x_{0}) = \big(\frac{\mu}{Y}\big)^{3} a_{2}b
\int_{0}^{T}\Big( \int_{0}^{r}e^{-(b + u_{2})(r - \eta)} \cdot
e^{\frac{\mu}{Y}\int_{0}^{\eta}x_{*}(\xi)d\xi} \cdot
\Big(\int_{0}^{\eta}e^{-(b + u_{2})\xi}d\xi\Big)^{2}d\eta
\Big)dr,
\end{gather*}
from which the inequalities immediately follow:
\begin{equation}\label{num5.11}
\dot g(x_{0}) > 0, \quad \ddot g(x_{0}) > 0.
\end{equation}
The corresponding derivatives of the function $ f(x_{0}) $ are:
\begin{gather*}
\dot f(x_{0}) = \frac{g(x_{0})}{(1 + \sigma(a_{1} - x_{0}))^{2}} -
\frac{(a_{1} - x_{0})\dot g(x_{0})}{1 + \sigma(a_{1} - x_{0})},
\\
\ddot f(x_{0}) = \frac{2\sigma g(x_{0})}{(1 + \sigma(a_{1} - x_{0}))^{3}} +
\frac{2\dot g(x_{0})}{(1 + \sigma(a_{1} - x_{0}))^{2}} -
\frac{(a_{1} - x_{0})\ddot g(x_{0})}{1 + \sigma(a_{1} - x_{0})}.
\end{gather*}
Using \eqref{num5.11} we see that on the interval
$ x_{0} \in (a_{1},a_{1} + \frac{1}{\sigma}) $ the following
inequalities are valid:
$$
\dot f(x_{0}) > 0, \quad \ddot f(x_{0}) > 0.
$$
Therefore, function $ s_{0} = f(x_{0}) $ is increasing and concave up.
The graph of this function is shown in Figure \ref{fig3}.
It follows from \eqref{num5.10} that for all initial values
$ (x_{0},s_{0}) $ for which
$$
a_{1} - x_{0} < 0, \quad 1 + \sigma(a_{1} - x_{0}) > 0, \quad
s_{0} \ge f(x_{0})
$$
the optimal control $ u_{*}(t) $ has type \eqref{num4.5}.
Now, consider the case that the optimal control $ u_{*}(t) $,
$ t \in [0,T] $ has type \eqref{num4.6}. The inequality $ L(0) < 0 $
implies the existence of switching $ \theta_{*} \in (0,T) $, for which
\begin{equation}\label{num5.12}
L(\theta_{*}) = 0.
\end{equation}
Equality \eqref{num5.12} can be rewritten as
$$
(a_{1} - x_{*}(\theta_{*}))\psi_{*}(\theta_{*}) -
s_{*}(\theta_{*})\varphi_{*}(\theta_{*}) = 0,
$$
or
\begin{equation}\label{num5.13}
s_{*}(\theta_{*}) = (a_{1} - x_{*}(\theta_{*}))q(\theta_{*}),
\end{equation}
where the function $ q(t) $ is defined by the Cauchy problem
\eqref{num5.2}. From \eqref{num5.3} we obtain the formula
\begin{equation}\label{num5.14}
q(\theta_{*}) = -\frac{\mu}{Y} \int_{\theta_{*}}^{T}
e^{\frac{\mu}{Y}\int_{\theta_{*}}^{r} x_{*}(\xi)d\xi}s_{*}(r)dr.
\end{equation}
As in the previous case, we find how the value $ q(\theta_{*}) $
depends on the initial conditions $ (x_{0},s_{0}) $. For this
we will integrate
the system \eqref{num2.3} with control $ u_{*}(t) $, $ t \in [0,T] $
given by \eqref{num4.6}. We have formulas:
\begin{equation}\label{num5.15}
x_{*}(t) =
\begin{cases}
x_{0}e^{-(b + u_{1})t} + \frac{a_{1}u_{1}}{b + u_{1}}
\left(1 - e^{-(b + u_{1})t}\right), \quad
\text{if } 0 \le t \le \theta_{*}, \\[4pt]
\Big(x_{0}e^{-(b + u_{1})\theta_{*}}
+ \frac{a_{1}u_{1}}{b + u_{1}}
\left(1 - e^{-(b + u_{1})\theta_{*}}\right)\Big)
e^{-(b + u_{2})(t - \theta_{*})} \\
+ \frac{a_{1}u_{2}}{b + u_{2}}
\left(1 - e^{-(b + u_{2})(t - \theta_{*})}\right), \quad
\text{if } \theta_{*} < t \le T,
\end{cases}
\end{equation}
and
\begin{equation}\label{num5.16}
s_{*}(t) =
\begin{cases}
s_{0}e^{-(b + u_{1})t}
e^{-\frac{\mu}{Y}\int_{0}^{t} x_{*}(\xi)d\xi} \\
+ a_{2}b \int_{0}^{t} e^{-(b + u_{1})(t - r)}
e^{-\frac{\mu}{Y}\int_{r}^{t}x_{*}(\xi)d\xi}dr, \quad
\text{if } 0 \le t \le \theta_{*}, \\[4pt]
\Big(s_{0}e^{-(b + u_{1})\theta_{*}}
e^{-\frac{\mu}{Y}\int_{0}^{\theta_{*}} x_{*}(\xi)d\xi} \\
+ a_{2}b \int_{0}^{\theta_{*}} e^{-(b + u_{1})(\theta_{*} - r)}
e^{-\frac{\mu}{Y}\int_{r}^{\theta_{*}}x_{*}(\xi)d\xi}dr\Big) \\
\times e^{-(b + u_{2})(t - \theta_{*})} \cdot
e^{-\frac{\mu}{Y}\int_{\theta_{*}}^{t} x_{*}(\xi)d\xi} \\
+ a_{2}b \int_{\theta_{*}}^{t} e^{-(b + u_{2})(t - r)}
e^{-\frac{\mu}{Y}\int_{r}^{t}x_{*}(\xi)d\xi}dr, \quad
\text{if } \theta_{*} < t \le T.
\end{cases}
\end{equation}
Substituting expressions \eqref{num5.15} and \eqref{num5.16} into
\eqref{num5.14}, we obtain
$$
q(\theta_{*}) = - \nu_{\theta_{*}}
\left(s_{0}\alpha_{\theta_{*}}(x_{0}) + \beta_{\theta_{*}}(x_{0})\right) -
h_{\theta_{*}}(x_{0}).
$$
Here the value
\begin{equation}\label{num5.17}
\nu_{\theta_{*}} =
\frac{\mu}{(b + u_{2})Y}\left(1 - e^{-(b + u_{2})(T - \theta_{*})}\right)
\end{equation}
is positive, and functions:
\begin{equation} \label{num5.18}
\begin{gathered}
\alpha_{\theta_{*}}(x_{0}) = e^{-(b + u_{1})\theta_{*}} \cdot
e^{-\frac{\mu}{Y}\int_{0}^{\theta_{*}} x_{*}(\xi)d\xi},
\\
\beta_{\theta_{*}}(x_{0}) =
a_{2}b \int_{0}^{\theta_{*}} e^{-(b + u_{1})(\theta_{*} - r)} \cdot
e^{-\frac{\mu}{Y}\int_{r}^{\theta_{*}}x_{*}(\xi)d\xi}dr,
\\
h_{\theta_{*}}(x_{0}) = \frac{\mu a_{2}b}{Y}
\int_{\theta_{*}}^{T}\Big(
\int_{\theta_{*}}^{r}e^{-(b + u_{2})(r - \eta)} \cdot
e^{\frac{\mu}{Y}\int_{\theta_{*}}^{\eta}x_{*}(\xi)d\xi}d\eta
\Big)dr
\end{gathered}
\end{equation}
are also positive. Functions \eqref{num5.18} depend on $ x_{0} $ by
formula \eqref{num5.15}. Moreover, it is easy to see that at
$ \theta_{*} = 0 $ the following relationships hold
\begin{equation}\label{num5.19}
\nu_{\theta_{*}} = \sigma, \quad
\alpha_{\theta_{*}}(x_{0}) = 1, \quad
\beta_{\theta_{*}}(x_{0}) = 0, \quad
h_{\theta_{*}}(x_{0}) = g(x_{0}).
\end{equation}
Then equality \eqref{num5.13} can be rewritten as
\begin{equation}\label{num5.20}
\begin{aligned}
&\alpha_{\theta_{*}}(x_{0})s_{0}
\left(1 + \nu_{\theta_{*}}(a_{1} - x_{*}(\theta_{*}))\right) \\
&= - \beta_{\theta_{*}}(x_{0})
\left(1 + \nu_{\theta_{*}}(a_{1} - x_{*}(\theta_{*}))\right)
- (a_{1} - x_{*}(\theta_{*}))h_{\theta_{*}}(x_{0}).
\end{aligned}
\end{equation}
Also from the formula \eqref{num5.13} and
Lemma \ref{Lemma1}, Lemma \ref{Lemma5} it follows that
$ x_{*}(\theta_{*}) > a_{1} $. Then from Lemma \ref{Lemma3} we find
that $ x_{0} > a_{1} $. It means that if the optimal control
$ u_{*}(t) $ has type \eqref{num4.6}, then for initial conditions
$ (x_{0},s_{0}) $
the inequality $ 1 + \sigma(a_{1} - x_{0}) \le 0 $ may be satisfied.
Let us show that if for initial conditions $ (x_{0},s_{0}) $ the
inequalities:
\begin{equation}\label{num5.21}
a_{1} - x_{0} < 0, \quad 1 + \sigma(a_{1} - x_{0}) > 0
\end{equation}
hold, then the point $ (x_{0},s_{0}) $ is below the graph of the
function $ s_{0} = f(x_{0}) $ (see Figure \ref{fig3}).
First, let us establish the positivity of the left hand side of the
equality \eqref{num5.20}. It is sufficient to show the validity of
the inequality
\begin{equation}\label{num5.22}
1 + \nu_{\theta_{*}}(a_{1} - x_{*}(\theta_{*})) > 0.
\end{equation}
Consider the auxiliary function
$$
\rho(\theta_{*}) = x_{*}(\theta_{*}) - a_{1}
- \frac{1}{\nu_{\theta_{*}}}
$$
for all $ \theta_{*} \in [0,T) $. As a consequence of
the first formula
of \eqref{num5.19} we have at $ \theta_{*} = 0 $ the relationship
\begin{equation}\label{num5.23}
\rho(0) = x_{0} - a_{1} - \frac{1}{\sigma} < 0.
\end{equation}
From \eqref{num5.15} and \eqref{num5.17} we find expressions:
\begin{equation}\label{num5.24}
\frac{dx_{*}}{d\theta_{*}}(\theta_{*}) =
- (b + u_{1})\Big(x_{0} - \frac{a_{1}u_{1}}{b + u_{1}}\Big)
e^{-(b + u_{1})\theta_{*}} < 0,
\end{equation}
and
$$
\frac{d\nu_{\theta_{*}}}{d\theta_{*}} =
- \frac{\mu}{Y}e^{-(b + u_{2})(T - \theta_{*})} < 0.
$$
Note that the derivative of the function $ \rho(\theta_{*}) $ is
$$
\dot\rho(\theta_{*}) = \frac{dx_{*}}{d\theta_{*}}(\theta_{*}) +
\frac{1}{\nu_{\theta_{*}}^{2}}\cdot\frac{d\nu_{\theta_{*}}}{d\theta_{*}}.
$$
By \eqref{num5.24} it is seen that the function $ \rho(\theta_{*}) $
is decreasing for all $ \theta_{*} \in (0,T) $. From \eqref{num5.23}
the negativity of the function $ \rho(\theta_{*}) $ for
$ \theta_{*} \in [0,T) $ follows. This fact implies the validity of
the inequality \eqref{num5.22}.
Then \eqref{num5.20} can be rewritten as
$$
s_{0} = F_{\theta_{*}}(x_{0}) =
- \frac{(a_{1} - x_{*}(\theta_{*}))h_{\theta_{*}}(x_{0})}
{\alpha_{\theta_{*}}(x_{0})
\left(1 + \nu_{\theta_{*}}(a_{1} - x_{*}(\theta_{*}))\right)} -
\frac{\beta_{\theta_{*}}(x_{0})}{\alpha_{\theta_{*}}(x_{0})}.
$$
Here the right hand side of this equality defines the function
$ F_{\theta_{*}}(x_{0}) $. From \eqref{num5.19} for $ \theta_{*} = 0 $
it is clear that
$$
F_{\theta_{*}}(x_{0}) = f(x_{0}).
$$
Therefore, the initial conditions $ (x_{0},s_{0}) $ for which
corresponding optimal control $ u_{*}(t) $ has type \eqref{num4.6}
belong to the graph of the function $ s_{0} = F_{\theta_{*}}(x_{0}) $.
The fact that we need to establish can be restate as follows.
Let us show that for the same values
$ x_{0} \in (a_{1},a_{1} + \frac{1}{\sigma}) $ and
$ \theta_{{*}} \in (0,T) $ the graph of the function
$ s_{0} = F_{\theta_{*}}(x_{0}) $ is below the graph of the function
$ s_{0} = f(x_{0}) $.
To prove this fact, we consider the equality \eqref{num5.12},
or alternatively, \eqref{num5.20} as the implicit equation
\begin{equation}\label{num5.25}
L(\theta_{*},x_{0},s_{0}(\theta_{*})) = 0.
\end{equation}
At $ \theta_{*} = 0 $ the points $ (x_{0},s_{0}) $ of the graph of the
function $ s_{0} = f(x_{0}) $ satisfy this equation.
Now, let us differentiate the equation \eqref{num5.25} by
$ \theta_{*} \in (0,T) $. We obtain the expression
\begin{equation}\label{num5.26}
\frac{\partial L}{\partial t}(\theta_{*},x_{0},s_{0}(\theta_{*})) +
\frac{\partial L}{\partial s_{0}}(\theta_{*},x_{0},s_{0}(\theta_{*}))
\cdot\frac{ds_{0}}{d\theta_{*}}(\theta_{*}) = 0.
\end{equation}
From Lemma \ref{Lemma8} and relationships
\eqref{num5.20},\eqref{num5.22} it follows that the corresponding
partial derivatives of the function $ L(\theta_{*},x_{0},s_{0}) $ are
positive. Then from \eqref{num5.26} it follows that
$$
\frac{ds_{0}}{d\theta_{*}}(\theta_{*}) < 0, \quad \theta_{*} \in (0,T).
$$
Therefore, for the same values
$ x_{0} \in (a_{1},a_{1} + \frac{1}{\sigma}) $ the value $ s_{0} $ of
the graph of the function $ s_{0} = F_{\theta_{*}}(x_{0}) $ is less
than the value $ s_{0} $ of the graph of the function
$ s_{0} = f(x_{0}) $.
Hence, when the optimal control $ u_{*}(t) $ has the type
\eqref{num4.6} and the inequalities \eqref{num5.21} hold, the initial
conditions $ (x_{0},s_{0}) $ satisfy the inequality
$$
s_{0} < f(x_{0}).
$$
Thus, the desired result is established.
Finally, let us introduce the sets:
\begin{gather*}
S = \{ (x_{0},s_{0}) \in \mathbb{R}^{2} : x_{0} > 0, \; s_{0} > 0\},\\
P = \{(x_{0},s_{0}) \in S : x_{0} \le a_{1}\}
\cup \{ (x_{0},s_{0}) \in S : a_{1} < x_{0} < a_{1}
+ \frac{1}{\sigma}, \; s_{0} \ge f(x_{0})\},\\
\begin{aligned}
Q &= \{ (x_{0},s_{0}) \in S : a_{1} < x_{0} < a_{1} + \frac{1}{\sigma},
\; s_{0} < f(x_{0}) \} \\
&\quad \cup \{(x_{0},s_{0}) \in S : x_{0} \ge a_{1}
+ \frac{1}{\sigma}\}.
\end{aligned}
\end{gather*}
It is clear that $ S = P \cup Q $. Sets $ P $ and $ Q $ are shown
in Figure \ref{fig3}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig3} % Picture1.eps
\caption{Graph of the function $ s_{0} = f(x_{0}) $ and sets
$ P $, $ Q $}\label{fig3}
\end{center}
\end{figure}
The preceding arguments show us the following statement.
\begin{theorem}\label{Theorem1}
The following cases are valid:
\begin{itemize}
\item[(a)] if the optimal control $ u_{*}(t) $, $ t \in [0,T] $
has the type \eqref{num4.5}, then corresponding initial conditions
$ (x_{0},s_{0}) $ satisfy the inclusion $ (x_{0},s_{0}) \in P $,
\item[(b)] if the optimal control $ u_{*}(t) $, $ t \in [0,T] $
has the type \eqref{num4.6}, then corresponding initial conditions
$ (x_{0},s_{0}) $ satisfy the inclusion $ (x_{0},s_{0}) \in Q $.
\end{itemize}
\end{theorem}
The converse of this statement is also true.
\begin{theorem}\label{Theorem2}
The following cases are valid:
\begin{itemize}
\item[(a)] if initial conditions $ (x_{0},s_{0}) $ satisfy the inclusion
$ (x_{0},s_{0}) \in P $, then corresponding optimal control
$ u_{*}(t) $, $ t \in [0,T] $ has the type \eqref{num4.5},
\item[(b)] if initial conditions $ (x_{0},s_{0}) $ satisfy the inclusion
$ (x_{0},s_{0}) \in Q $, then corresponding optimal control
$ u_{*}(t) $, $ t \in [0,T] $ has the type \eqref{num4.6}.
\end{itemize}
\end{theorem}
\begin{proof}
We will first prove the case (a). Let the initial conditions
$ (x_{0},s_{0}) $ satisfy the inclusion $ (x_{0},s_{0}) \in P $.
Then from preceding arguments it follows that the optimal control
$ u_{*}(t) $, $ t \in [0,T] $ has the type \eqref{num4.5} or type
\eqref{num4.6}. Type \eqref{num4.6} is impossible since from
Theorem \ref{Theorem1} we obtain the contradictory inclusion
$ (x_{0},s_{0}) \in Q $. Therefore, the optimal control $ u_{*}(t) $,
$ t \in [0,T] $ has type \eqref{num4.5}.
Case (b) is proved by analogous arguments.
\end{proof}
The Theorem \ref{Theorem2} allows us to select the optimal control
policy based on initial concentrations $ (x_{0},s_{0}) $ of biomass
and substrate.
\section{Computer modeling}
Our theoretical results obtained in the previous section allow to
select optimal successful strategy of ASP based on the knowledge of
the parameters of the model \eqref{num2.3} and initial conditions
$ (x_{0},s_{0}) $. In \cite{Lit2} parameter-estimation and verification
of the model measurement values from a waste water plant were obtained
for every hour of an operating period of one week.
Values of $ s(t) $ will be determined by total organic carbon content
in the influent and $ x(t) $ by the concentration of the suspended
solid in the aeration tank.
Let us show our results for the following model parameters:
$$
\begin{array}{ccc}
u_{1} = 0.1 \; \text{lb/min}, & u_{2} = 1.0 \; \text{lb/min},
& T = 10 \; \text{hours}, \\
a_{1} = 0.7 \; \text{lb/gal}, & a_{2} = 0.9 \; \text{lb/gal},
& Y = 3.0, \\
x_{0} = 1.5 \; \text{lb/gal}, & s_{0} = 2.0 \; \text{lb/gal},
& \mu = 0.1.
\end{array}
$$
The following relationships are valid:
$$
a_{1} - x_{0} = -0.8 < 0, \quad
1 + \sigma(a_{1} -x _{0}) = 0.976 > 0.
$$
Then the optimal control $ u_{*}(t) $ has the type \eqref{num4.6}
with one moment of switching at $ \theta_{*} = 2 $ hours,
which was obtained numerically. So, if we select such optimal policy
$ u_{1} \to u_{2} $, then the concentration of the polluted
water $ s(t) $ will be minimized at moment $ T $, final operation time
(see Figure \ref{fig4}).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig4} %Picture2.eps
\caption{Optimal concentration of the polluted water $ s_{*}(t) $}\label{fig4}
\end{center}
\end{figure}
\section{Conclusions}
Activated sludge process involves complex and subtle relationships
among a relatively large number of variables. The model investigated
in our paper is not intended to be the best ASP model. However, it is
nonlinear and it has a bounded control, which makes it quite interesting
from the mathematical point of view.
We found the type of optimal control by means of the so-called switching
function. This allowed us to reduce a complex two point boundary value
problem for the Maximum Principle to one of finite dimensional
optimization.
Our mathematical investigation of the activated sludge process can be
summarized by components:
(1) Complete investigation of a model \eqref{num2.3} of the activated
sludge process with one bounded control.
(2) Development of an optimal control strategy for the recycle flow
rate analytically.
(3) Investigation of how the selected optimal control strategy depends
on the initial conditions.
(4) Computer simulation of the controlled activated sludge process for
different model parameters.
Based on our theory, we find that the optimal analytical solution may
decrease waste water plant operation cost. Thus, if $ (x_{0},s_{0}) $
measured at moment $ t = 0 $ satisfies inclusion $ P $, then the
optimal control function $ u_{*}(t) $ the rate of the recycle sludge,
first must take the lower value, $ u_{1} $ until time $ \theta_{*} $
and then switch to the upper lever $ u_{2} $.
Finally, it should be noted that the ideas presented in this study can
be applied to other control systems with similar properties.
\subsection*{Acknowledgements}
The authors thank Dr. Paul Deignan for his
helpful suggestions and style recommendations.
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\end{document}