<figure><img src="image_1.png" /></figure>

HAL
open science

# Analytical construction of the free boundary for nonlinear impulse control problems and its application to the optimal design of aerator systems

Masasuke Shima

► To cite this version:

Masasuke Shima. Analytical construction of the free boundary for nonlinear impulse control problems and its application to the optimal design of aerator systems. [Research Report] IRIA-RR-194, IRIA. 1976, pp.76. hal-04716363

HAL Id: hal-04716363

https://inria.hal.science/hal-04716363v1

Submitted on 1 Oct 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
AR194

IRIA
laboria

Institut de Recherche
d'Informatique
et d'Automatique

Domaine de Voluceau
Rocquencourt
B.P. 5 78150 - Le Chesnay
France
Tél.: 954 90 20

laboratoire de recherche
en informatique
et automatique

<figure><img src="image_3.png" /></figure>

ANALYTICAL CONSTRUCTION
OF THE FREE BOUNDARY
FOR NONLINEAR IMPULSE
CONTROL PROBLEMS
AND ITS APPLICATION
TO THE OPTIMAL DESIGN
OF AERATOR SYSTEMS

Masasuke SHIMA

Rapport de Recherche N° 194

Septembre 1976
ANALYTICAL CONSTRUCTION OF THE FREE BOUNDARY FOR
NONLINEAR IMPULSE CONTROL PROBLEMS AND ITS APPLICATION TO THE
OPTIMAL DESIGN OF AERATOR SYSTEM

Masasuke SHIMA

Résumé :

On présente ici une méthode de construction analytique de la frontière libre pour le problème du contrôle impulsionnel, pour un système gouverné par une équation différentielle non-linéaire et déterministe et pour un coût non-linéaire à minimiser.

Sous des hypothèses facilement vérifiées en pratique, cette méthode permet de calculer analyti-
quement la frontière libre et de déterminer le contrôle impulsionnel optimal. En outre, dans le
cas où le calcul analytique de la frontière libre est difficile, la méthode donne un algorithme de
calcul numérique qui permet d'approcher la frontière libre avec la précision désirable et d'obtenir
la convergence vers la solution optimale.

La méthode est étendue au cas linéaire multidimensionnel avec coût quadratique et elle est ap-
pliquée au problème de la disposition optimale d'aérateurs le long d'une rivière polluée. Un exem-
ple numérique est donné pour expliquer le fonctionnement de la méthode et pour la comparer
avec l'application directe de la méthode des différences finies appliquée à l'Inéquation Quasi
Variationnelle associée au problème.

Résumé :

A procedure for analytical construction of Free Boundary is presented for nonlinear impulse control problems, where the system is governed by an ordinary differential equation with non-linear and deterministic structure and the performance index to be minimized is also nonlinear.

Under several conditions easily verified in practice, the procedure enables us to calculate the
Free Boundary analytically and determine the optimal impulse control. In addition, the proce-
dure yields an algorithm of numerical calculation by which the Free Boundary can be approxima-
ted with desired accuracy and the convergence to the optimal solution is assured.

The procedure is extended to the «linear quadratic» problem of higher dimension, which is ap-
plied to the optimal design problem of aerator system to be constructed along the long polluted
river. A numerical example is given to illustrate the procedure and compare it with the finite
difference scheme applied directly to the Quasi Variational Inequality of the problem.
# I. INTRODUCTION

In this paper, the optimal impulse control problem is studied for the deterministic nonlinear control system. By means of the impulse control, the state of the system can be changed instantaneously and discontinuously. Management system of stocks and the effect of aeration in the polluted water are the typical examples of the impulse control system. In the impulse control problem, we look for the point of compromise between the cost of impulse control and the better performance of the system.

The method to solve this problem is given by Bensoussan & Lions¹).
Applying the principle of optimality of the Dynamic Programming, they derived the Q.V.I. (Quasi Variational Inequality) to be satisfied by the optimal cost function. The solution of Q.V.I. is obtained indirectly by the appropriate approximation methods. If there exist stochastic disturbances to the system, Q.V.I. becomes a parabolic type and is easier to obtain the good approximation because of the existence of the diffusion term. But, if the system is deterministic, Q.V.I. is a hyperbolic type and is more difficult to treat from the viewpoint of the numerical calculation.

The procedure stated in this paper enables us to calculate the free boundary constructively for the hyperbolic Q.V.I., with which we can obtain the optimal timings and the optimal magnitudes of a series of impulse control. Even for the case where the numerical approximation is necessary, we can pay attention only to the accuracy of the numerical calculation, without the consideration for the convergence to the optimal solution. This advantage is illustrated in the numerical example studied in chapter IV.

The construction of this paper is as follows. In chapter II, the general theory is developed for the 1-dimensional nonlinear impulse control system. In chapter III, the theory is extended to the linear higher order control system with the quadratic performance index. In chapter IV, the optimal design problem of the aerator system along the long polluted river is solved as the application of the impulse control theory. The aerator can be
reinterpreted as the dam or the treatment equipment of the industrial
sewerage. Chapter V is devoted to the concluding remarks.

## II. OPTIMAL IMPULSE CONTROL OF THE NONLINEAR 1-DIMENSIONAL CONTROL SYSTEM

### II.1 Statement of the Problem

The state of the system is denoted by a scalar x. t is the time.
In the closed domain $\mathcal{D}$ ;

$$ \mathcal{D} = \{(t, x) : 0 \le t \le T, 0 \le x \le (C(t), C(t)) > 0\} \quad (II.1.1) $$

the behaviour of the system is described by the ordinary differential equation:

$$ \dot{x} = f(t, x) + v + w(t, x), \quad x(0) = x_0, \quad (0, x_0) \in \mathcal{D} \quad (II.1.2) $$

$T > 0$ is the operating time of the system. $f(t,x)$ is continuously differentiable in x and piecewise continuously differentiable in t. The system (II.1.1) with $v=0$ and $w(t,x)=0$ :

$$ \dot{x} = f(t, x) \quad (II.1.3) $$

is called "free system". The existence and the uniqueness of its solution

$$ X(t; t', x'), \quad X(t'; t', x') = x', \quad (t', x') \in \mathcal{D} \quad (II.1.4) $$

passing through $(t', x') \in \mathcal{D}$ is assured. Furthermore, it is assumed
that if $(t', x') \in \mathcal{D}$ ,

$$ (t, X(t; t', x')) \in \mathcal{D}, \quad 0 \le t \le T \quad (II.1.5) $$

$w(t,x)$ is the deterministic disturbance acting on the system impulsively
at times $0 < r_1 < r_2 < \dots < r_m < T$ and is expressed as

$$ w(t, x) = \sum_{i=1}^{m} b(t, x, \eta_i) \delta(t - \eta_i) \quad (II.1.6) $$

$b(t,x,\eta)$ is continuous in $t,x$ and $\eta$. $\eta$ expresses, for example, the
concentrations of DO and BOD of the industrial sewerage flowing into the
river. $\delta(\cdot)$ is the Dirac's delta function. It is assumed that the state
of the system does not go out of $\mathcal{D}$ due to the disturbance $w(t,x)$ .
u is the impulse control exerted on the system at times:

$$0 \le T_1 < T_2 < \dots < T_N < T \qquad (II.1.7)$$

with the magnitudes:

$$\xi_1, \xi_2, \dots, \xi_N \qquad (II.1.8)$$

so as to minimize the performance index:

$$J = \sum_{i=1}^{N} g(T_i, \bar{X}(T_i; 0, x_0), \xi_i) + \int_{0}^{T} h(t, \bar{X}(t; 0, x_0)) dt + Q(T, X(T; 0, x_0))^* \qquad (II.1.9)$$

There is a restriction on the magnitude $\xi$ of the impulse control which can be exerted at $(t,x) \in \mathcal{D}$. That is

$$0 \le \xi \le x. \qquad (II.1.10)$$

By this impulse control, the state of the system is transfered from $x$ to $x-\xi$ with the expense of $g(t,x,\xi)$.

In order to have the meaningful formulation as the impulse control problem, we assume the following conditions induced from the optimal design problem of the aerator systems (Chapter IV):

$$(1) \quad f(t,0) > 0, \quad 0 \le t \le T, \qquad (II.1.11)$$

in order to satisfy the condition (II.1.5) and exclude the obvious case $f(t,0)=0$. In other words, the state $x=0$ is the most desirable state, but we cannot keep the state at $x=0$.

(2) $g(t,x,\xi)$ is continuously differentiable in all arguments and monotone nondecreasing in $x$ and $\xi$, satisfying the inequalities:

$$g(t,x,\xi) \ge \alpha > 0, \quad (t,x) \in \mathcal{D}, \quad 0 \le \xi \le x \qquad (II.1.12)$$

and

$$g(t,x,\xi) + g(t,x-\xi, \xi') \ge g(t,x,\xi+\xi'), \quad (t,x) \in \mathcal{D}, \quad 0 \le \xi, \xi', \xi+\xi' \le x. \qquad (II.1.13)$$

The fixed cost for the construction of the aerator can be regarded as $\alpha$.

(3) $h(t,x)$ is continuously differentiable in $t$ and $x$ and monotone increasing in $x$, satisfying the condition:

$$h(t,0) = 0, \quad 0 \le t \le T, \qquad (II.1.14)$$

and $h(t,x) > 0, (t,x) \in \mathcal{D}, x \ne 0$

$$\lim_{x \to \infty} h(t,x) = \infty, \quad 0 \le t \le T. \qquad (II.1.15)$$

$\bar{X}(t;0,x_0)$ is the solution of (II.1.2) corresponding to the initial condition $x(0)=x$ and the impulse control (II.1.7) and (II.1.8).
(4) $Q(t,x)$ is continuously differentiable in $t$ and $x$ and monotone increasing in $x$, satisfying the condition:

$$ 
\left. 
\begin{array}{l}
Q(t,0) = 0 \quad \text{and} \quad Q(t,x) > 0 \quad \text{for } x \neq 0 \quad \text{and} \quad (t,x) \in \mathcal{D}, \\
\lim_{x \to \infty} Q(t,x) = \infty, \quad 0 \le t \le T.
\end{array}
\right\} \quad (\text{II.1.16})
$$

## II.2 Q.V.I.

### II.2.1 Q.V.I.

We assume that there is no disturbance $w(t,x)$. Then, the problem is reduced to finding out the solution of Q.V.I.:

$$ \frac{\partial u}{\partial t} + f(t,x) \frac{\partial u}{\partial x} + h(t,x) \geq 0, \quad (\text{II.2.1}) $$

$$ u(t,x) \le \min_{0 \le \xi \le x} \{g(t,x,\xi) + u(t,x-\xi)\}, \quad (\text{II.2.2}) $$

$$ \left\{ \frac{\partial u}{\partial t} + f(t, x) \frac{\partial u}{\partial x} + h(t, x) \right\} \left[ u(t, x) - \min_{0 \le \xi \le x} \{ g(t, x, \xi) + u(t, x-\xi) \} \right] = 0 \quad (\text{II.2.3}) $$

to be satisfied for $(t,x) \in \mathcal{D}$ with the boundary condition:

$$ u(T, x) = Q(T, x), \quad (T, x) \in \mathcal{D}. \quad (\text{II.2.4}) $$

Once the solution $u(t,x)$ of Q.V.I. is obtained, we can solve the problem by means of the Free Boundary defined as follows¹):

$$ u(t,x) = \min_{0 \le \xi \le x} \{g(t,x,\xi) + u(t,x-\xi)\}, \quad 0 \le t \le T. \quad (\text{II.2.5}) $$

The solution $u(t,x)$ is usually approximated by the numerical calculation. But, it is known that the hyperbolic Q.V.I. is ill-conditioned from the viewpoint of the numerical calculation. Later, in the example of Chapter IV, it is shown that the degree of convergence to the optimal solution is very sensitive to the differential scheme used, which often gives the non-optimal solution. In the following, it is shown that the Free Boundary and the optimal impulse control can be determined exactly by solving Q.V.I. directly.

### II.2.2 Solution $u(t,x)$ in the Neighborhood of $(T,0)$

Suppose that the impulse control is not exerted on the whole interval $[0,T]$. Then, the value of the performance index calculated along the trajectory $X(t;t',x')$ of (II.1.3) is given by

$$ \varphi(t', x') \equiv \int_{t'}^{T} h(s, X(s; t', x')) ds + Q(T, X(T; t', x')). \quad (\text{II.2.6}) $$
It is clear that $\varphi(t,x)$ is continuously differentiable in $t$ and $x$ and satisfies the partial differential equation of the first order:

$$ \left. \begin{array}{l} \frac{\partial \varphi}{\partial t} + f(t, x) \frac{\partial \varphi}{\partial x} + h(t, x) = 0, \quad (t, x) \in \mathcal{D} \\[1em] \varphi(T, x) = Q(T, x). \end{array} \right\} \qquad (II.2.7) $$

Using (II.1.14) and (II.2.6), it is shown that $\varphi(t,x)$ is monotone increasing in $x$ and monotone decreasing in $t$ with

$$ \varphi(T, 0) = 0. \qquad (II.2.8) $$

If the explicit expression of $\varphi(t,x)$ is available, the problem can be solved analytically. Otherwise, it is necessary to integrate (II.2.6) numerically. It depends on the integrability of (II.2.7).

The minimal cost to exert the impulse control is $\alpha$. Since $\varphi(t,x)$ is continuous and satisfies (II.2.6), there exists a positive number $\delta$ such that if $(t,x) \in U_\delta$;

$$ U_\delta \equiv \{ (t, x) : (t-T)^2 + x^2 < \delta^2, (t, x) \in \mathcal{D} \}, \qquad (II.2.9) $$

then we have

$$ \varphi(t, x) < \alpha. \qquad (II.2.10) $$

Moreover, $\varphi(t,x)$ is nonnegative by the definition (II.2.6). Then, we have

$$ \min_{0 \le \xi \le x} [g(t, x, \xi) + \varphi(t, x - \xi)] \ge \alpha, \quad (t, x) \in \mathcal{D}. \qquad (II.2.11) $$

(II.2.10) and (II.2.11) yield the inequality:

$$ \varphi(t, x) < \min_{0 \le \xi \le x} [g(t, x, \xi) + \varphi(t, x - \xi)], \quad (t, x) \in U_{\delta}. \qquad (II.2.12) $$

In view of (II.2.7) and (II.2.12), it is known that $\varphi(t,x)$ satisfies Q.V.I. (II.2.1), (II.2.2), (II.2.3) and (II.2.4) at least in $U_\delta$.

Therefore, under the assumption of the existence and the uniqueness of the solution of Q.V.I., we obtain

$$ u(t, x) \equiv \varphi(t, x), \quad (t, x) \in U_{\delta} \qquad (II.2.13) $$

## II.3 Curve $\mathcal{P}$

By (II.2.13), we can solve Q.V.I. at least in $U_\delta$. Then, to what extent is (II.2.13) valid? How can we obtain $u(t,x)$ where (II.2.13) is not valid?
In the first place, let us study the limit of the validity of (II.2.13) by examining the equality (II.2.5) defining Free Boundary. according to the continuous differentiability of $g(t,x,\xi)$ and $\varphi(t,x)$, there always exists $\xi$ which gives the minimum of the expression:

$$ \min_{0 \le x \le \xi} \left\{ g(t, x, \xi) + \varphi(t, x - \xi) \right\} \qquad (II.3.1) $$

We assume the uniqueness of $\xi$ and denote it by $\xi(t,x)$.

If $0 < \xi(t,x) < x$, we have

$$ \frac{\partial g(t, x, \xi(t, x))}{\partial \xi} - \frac{\partial \varphi(t, x - \xi(t, x))}{\partial x} = 0 \qquad (II.3.2) $$

In view of the existence of $U_g$, it is clear that there exists the region where we need the impulse control at most once in order to control the system optimally. In such region, (II.3.1) is the minimal cost when we decide to exert the impulse control at $(t,x)$, including the cost of the impulse control and the cost of operating the system freely on the interval $[t,T]$.

Therefore, if the inequality:

$$ \varphi(t, x) < g(t, x, \xi(t, x)) + \varphi(t, x - \xi(t, x)) \qquad (II.3.3) $$

is satisfied, it is better not to give the impulse control at $(t,x)$.

This inequality is satisfied in $U_g$.

If the inequality:

$$ \varphi(t, x) > g(t, x, \xi(t, x)) + \varphi(t, x - \xi(t, x)) \qquad (II.3.4) $$

is satisfied, then it is better to exert the impulse control.

Thus, we define the curve $\vec{\Gamma}$ by $x(t)$ satisfying the expression:

$$ \mathcal{D}(t, x) \equiv \varphi(t, x) - \{g(t, x, \xi(t, x)) + \varphi(t, x - \xi(t, x))\} = 0 \qquad (II.3.5) $$

on $[0,T]$, where the existence and the uniqueness of $x(t)$ is assumed.

The curve $\vec{\Gamma}$ is defined by the curve $x(t) - \xi(t, x(t))$. The regions $\vec{\Gamma}^+$ and $\vec{\Gamma}^-$ are defined as follows.

$$ \vec{\Gamma}^{+} \equiv \{ (t, x) : \mathcal{D}(t, x) > 0, (t, x) \in \mathcal{D} \} \qquad (II.3.6) $$

$$ \vec{\Gamma}^{-} \equiv \{ (t, x) : \mathcal{D}(t, x) < 0, (t, x) \in \mathcal{D} \} \qquad (II.3.7) $$
Since $\varphi(t,x)$ is monotone increasing in $x$ and $\xi(t,x)$ is assumed to be defined uniquely, it is safely assumed that the configuration of $\Gamma^+$, $\Gamma$ and $\Gamma^-$ is shown by Fig.II-1.

If $(t',x') \in \Gamma^-$ and $(t,X(t;t',x')) \in \Gamma^-$ for $t' \le t \le T$, it is optimal to operate the system along the trajectory $X(t;t',x')$ of the Free System. Thus, there is no need of exerting the impulse control and (II.2.13) is valid. If there exists a trajectory in contact with $\Gamma$ as shown in Fig. II-1, the domain of validity of (II.2.13) is extended from $U_\delta$ to the shaded region of Fig.II-1.

However, what is the best policy for the case where the trajectory passing through $(t',x') \in \Gamma^-$ crosses $\Gamma$ and goes into $\Gamma^+$. In view of (II.3.4), we know that it is better to exert the impulse control in $\Gamma^+$. Then, the impulse control is to be exerted as soon as the trajectory reaches $\Gamma^?$. If this policy is optimal, $\Gamma$ is the Free Boundary of this problem. But, as is shown in the following, the policy to let the trajectory jump on $\Gamma$ is not optimal.

## II.4 Comparison of $J_t$ with $J_{t+\delta t}$ leads to the Curve $\Lambda$.

### II.4.1 Derivation of the Curve $\Lambda$

Let us exert the impulse control at $(t,x)$ with the magnitude $\xi(t,x)$ and denote the value of the performance index corresponding to this policy by $J_t$:

$$J_t = g(t, x, \xi(t, x)) + \varphi(t, x - \xi(t, x)). \quad (II.4.1)$$

In the same way, we operate the system along the trajectory $X(.;t,x)$ from $(t,x)$ to $(t+\delta t, X(t+\delta t; t, x))$ ($0<\delta t \ll 1$) and exert the impulse control with the magnitude $\xi(t+\delta t, X(t+\delta t; t, x))$. The cost $J_{t+\delta t}$ is given by

$$J_{t+\delta t} = g(t+\delta t, X(t+\delta t; t, x), \xi(t+\delta t; t, x)) + \int_t^{t+\delta t} h(s, X(s; t, x)) ds. \quad (II.4.2)$$

To calculate $J_{t+\delta t} - J_t$, the following estimations are useful.
$$ \delta x \equiv X(t+\delta t; t, x) - x = f(t, x)\delta t + o(\delta t), \qquad (II.4.3) $$

$$ \delta \xi \equiv \xi(t+\delta t, X(t+\delta t; t, x)) - \xi(t, x) = a(t, x)\delta t + o(\delta t). \qquad (II.4.4) $$

(II.4.4) is assured by the continuous differentiability of $g(t, x, \xi)$ and $\varphi(t, x)$ and the uniqueness of $\xi(t, x)$. $o(\delta t)$ is the higher order terms in $\delta t$. Using (II.4.3) and (II.4.4), we obtain

$$ J_{t+\delta t} - J_t = \underline{\underline{\Phi}}(t, x) \delta t + o(\delta t) \qquad (II.4.5) $$

$$ \text{where } \underline{\underline{\Phi}}(t,x) \equiv \frac{D}{Dt}g(t,x,\xi(t,x)) + \frac{D}{Dt}\varphi(t,x-\xi(t,x)) + h(t,x). \qquad (II.4.6) $$
$\frac{D}{Dt}$ means the derivative along the curve $X(.; t, x)$ or the curve

$X(.; t, x) - \xi(., X(.; t, x)):$

$$ \frac{D}{Dt}g(t,x,\xi(t,x)) = \frac{\partial \underline{\underline{\Phi}}(t,x,\xi(t,x))}{\partial t} + f(t,x) \frac{\partial \underline{\underline{\Phi}}(t,x,\xi(t,x))}{\partial x} + a(t,x) \frac{\partial \underline{\underline{\Phi}}(t,x,\xi(t,x))}{\partial \xi}, \qquad (II.4.6)' $$

$$ \frac{D}{Dt}\varphi(t,x-\xi(t,x)) = \frac{\partial \underline{\underline{\varphi}}(t,x-\xi(t,x))}{\partial t} + \{f(t,x)-a(t,x)\} \frac{\partial \underline{\underline{\varphi}}(t,x-\xi(t,x))}{\partial x}. $$

If $0 < \xi(t, x) < x$ and (II.3.2) is satisfied, we have

$$ \underline{\underline{\Phi}}(t,x) = \frac{\partial \underline{\underline{\Phi}}(t,x,\xi(t,x))}{\partial x} f(t,x) + \frac{\partial \underline{\underline{\Phi}}(t,x,\xi(t,x))}{\partial t} + \frac{\partial \underline{\underline{\varphi}}(t,x-\xi(t,x))}{\partial t} + \frac{\partial \underline{\underline{\varphi}}(t,x-\xi(t,x))}{\partial x} + h(t,x). \qquad (II.4.7) $$

If $\xi(t, x) = x$ and $\xi(t+\delta t, X(t+\delta t; t, x)) = X(t+\delta t; t, x)$, we have $\delta \xi = \delta x$ and

$$ \underline{\underline{\Phi}}(t,x) = \frac{\partial \underline{\underline{\Phi}}(t,x,\xi(t,x))}{\partial x} f(t,x) + \frac{\partial \underline{\underline{\Phi}}(t,x,\xi(t,x))}{\partial t} + \frac{\partial \underline{\underline{\varphi}}(t,0)}{\partial t} + \frac{\partial \underline{\underline{\varphi}}(t,0,\xi(t,x))}{\partial \xi} f(t,x) + h(t,x). \qquad (II.4.8) $$

Hence, there is scarecely no need of the explicit form of $a(t, x)$ in (II.4.4). In the particular case where $g(t, x, \xi) = \text{constant}$ and $\xi(t, x) = x$, (II.4.6) is reduced to

$$ \underline{\underline{\Phi}}(t,x) = \frac{\partial \underline{\underline{\varphi}}(t,0)}{\partial t} + h(t,x). \qquad (II.4.9) $$

In this case, the impulse control has the effect to pull back the state to the t-axis and its cost is always constant.

Now, we define the curve $\Lambda$ by $x(t)$ satisfying the equality:

$$ \underline{\underline{\Phi}}(t,x) = 0. \qquad (II.4.10) $$
The existence and uniqueness of $x(t)$ is assumed on $[0,T]$. The regions $\Lambda^+$ and $\Lambda^-$ are defined as follows.

$$ \Lambda^+ = \{ (t, x) : (t, x) \in \partial \Omega, \underline{\Psi}(t, x) > 0 \}, \quad (II.4.11) $$

$$ \Lambda^- = \{ (t, x) : (t, x) \in \partial \Omega, \underline{\Psi}(t, x) < 0 \}. \quad (II.4.12) $$

In the case of (II.4.9), for example, the assumption of uniqueness is assured by (II.1.14), (II.1.15) and the fact that $\frac{\partial \varphi(t, 0)}{\partial t} < 0$.

## II.4.2 Configuration of the Regions, Curves and Trajectories

From the definitions of the regions $\Pi^+$, $\Pi^-$, $\Lambda^+$ and $\Lambda^-$, we can conclude that it is not optimal to exert the impulse control at $(t,x) \in \Lambda^-$ even if $(t,x) \in \Pi^+$. If $(t,x) \in \Pi^-$, it is better not to exert the impulse control even for the case $(t,x) \in \Lambda^+$. If the trajectory traverses the region $\Pi^+ \cap \Lambda^+$, it is better to exert the impulse control as soon as possible. Therefore, if the trajectory crosses $\Lambda$ at $(t,x)$ from $\Lambda^-$ to $\Lambda^+$ and $(t,x) \in \Pi^+$, the impulse control is to be exerted at $(t,x)$.

Thus, the optimal policy is determined by the configuration of the curves $\Pi$ and $\Lambda$, the regions $\Pi^+$, $\Pi^-$, $\Lambda^+$, $\Lambda^-$ and the directions of the trajectories on $\Lambda$ and $\Pi$. Therefore, in order to obtain the Free Boundary and the optimal impulse control, it is sufficient to examine this configuration. The following theorem is useful.

**THEOREM 1.** Suppose that $(t,x) \in \Pi$. $(t,x) \in \Lambda^+$ if and only if the trajectory $X(t'; t, x)$ $(t' \ge t)$ goes into $\Pi^-$. $(t,x) \in \Lambda^-$ if and only if $X(t'; t, x)$ goes into $\Pi^+$. $\Pi$ and $\Lambda$ crosses at $(t,x)$ if and only if $X(t'; t, x)$ is in contact with $\Pi$ at $(t,x)$.

**Proof.** Suppose that $(t,x) \in \Pi$ and $X(t'; t, x)$ goes into $\Pi^-$. From the definition of $\Pi$, we have

$$ \varphi(t, x) = g(t, x, \xi(t, x)) + \varphi(t, x - \xi(t, x)). \quad (II.4.13) $$

At $(t+\delta t, X(t+\delta t))$, the definition of $\Pi^-$ yields

$$ \begin{aligned} \varphi(t+\delta t, X(t+\delta t)) < & g(t+\delta t, X(t+\delta t), \xi(t+\delta t, X(t+\delta t))) \\ & + \varphi(t+\delta t, X(t+\delta t)-\xi(t+\delta t, X(t+\delta t))). \end{aligned} \quad (II.4.14) $$
$X(t+\delta t)$ is the abbreviated form of $X(t+\delta t; t, x)$ ($0 < \delta t \ll 1$).

Subtracting (II.4.13) from (II.4.14) side by side and using (II.2.7) and (II.4.3), we have

$$ 
\begin{aligned}
\text{Left hand side} &= \varphi(t+\delta t, X(t+\delta t)) - \varphi(t, x) \\
&= \left\{ \frac{\partial \varphi}{\partial t} + \frac{\partial \varphi}{\partial x} f(t, x) \right\} \delta t + o(\delta t) \\
&= -\mathcal{H}(t, x) \delta t + o(\delta t).
\end{aligned} 
\quad (II.4.15) $$

On the other hand, by means of (II.4.1) and (II.4.2), we have

$$ 
\begin{aligned}
\text{Right hand side} &= g(t+\delta t, X(t+\delta t), \xi(t+\delta t, X(t+\delta t))) - g(t, x, \xi(t, x)) \\
&\quad + \varphi(t+\delta t, X(t+\delta t) - \xi(t+\delta t, X(t+\delta t))) - \varphi(t, x - \xi(t, x)) \\
&= J_{\delta t} + \int_{t}^{t+\delta t} \mathcal{H}(t', x(t'; t, x)) dt'.
\end{aligned} 
\quad (II.4.16) \\
\quad (II.4.17) $$

In view of (II.4.5), it is known that

$$ \text{Right hand side} = \{ \underline{\Phi}(t, x) - \mathcal{H}(t, x) \} \delta t + o(\delta t). \quad (II.4.18) $$

Therefore, we obtain the inequality:

$$ -\mathcal{H}(t, x) \delta t < \{ \underline{\Phi}(t, x) - \mathcal{H}(t, x) \} \delta t + o(\delta t) \quad (II.4.19) $$

or

$$ 0 < \underline{\Phi}(t, x) \delta t + o(\delta t). \quad (II.4.20) $$

Since $0 < t \ll 1$, (II.4.20) is equivalent to the inequality:

$$ \underline{\Phi}(t, x) > 0 \quad (II.4.21) $$

which means $(t,x) \in \Gamma$ and $(t,x) \in \Lambda^+$. The proof of the converse direction can be done by tracing back the calculation. For the case where $X(t'; t,x)$ goes into $\Gamma^+$, the proof can be done in the same way and is omitted here.

If $X(t'; t,x)$ is in contact with $\Gamma$ at $(t,x)$, we obtain the equality:

$$ -\mathcal{H}(t, x) = \underline{\Phi}(t, x) - \mathcal{H}(t, x) \quad (II.4.22) $$

in place of (II.4.19) by considering the first order term in $\delta t$. Thus, we have

$$ \underline{\Phi}(t, x) = 0 \quad (II.4.23) $$

which shows that $(t,x) \in \Gamma$ and $(t,x) \in \Lambda$. The proof of the converse is easy and omitted.

Q.E.D.
## II.5 Determination of the Optimal Policy (1)

### II.5.1 Configuration satisfying Theorem 1

Let us explain the meanings of Theorem 1 and determine the optimal policy by means of the simplest configuration shown in Fig.2 and satisfying the Theorem 1. This configuration appears in the optimal design problem of the aerator system studied in chapter IV.

According to the Theorem 1, the trajectory $X(.; \tau_1, x_1)$ is in contact with $\Gamma$ at $P_1$ where $\Gamma$ crosses $\Lambda$. In the configuration of Fig.II-2, this trajectory $X(t; \tau_1, x_1)$ for $t \neq \tau_1$ always remains in the region $\Gamma^-$. This conclusion is derived as follows. On the interval $(\tau_1, T], \Gamma$ is in $\Lambda^+$. Then, it is shown from the Theorem 1 that all the trajectories which cross $\Gamma$ necessarily go into $\Gamma^+$ from $\Gamma^-$. If the trajectory $X(t; \tau_1, x_1)$ is in $\Gamma^-$ on the interval $(\tau_1, T],$ there exists at least a trajectory crossing $\Gamma$ into $\Gamma^+$ on the interval $(\tau_1, T]$ because of the continuity of the trajectories of (II.1.3) with respect to the initial condition, which contradicts the Theorem 1.

Therefore, in the closed region $P_1R_1\text{CTR}_2P_1$ of Fig.II-2, every trajectory remains in $\Gamma^-$. And there is no need of the impulse control. The solution $u(t,x)$ of Q.V.I. coincides with $\varphi(t,x)$ in this region, which follows from the definition of $\Gamma^-$ and the reasoning in II.3. Hence, we have

$$u(t,x) = \varphi(t,x). \qquad (II.5.1)$$

In the closed region $P_1R_2R_3P_1$, it is easily seen that the trajectories never go out of $\Gamma^-$. Therefore, (II.5.1) is valid also in the closed region $P_1R_2R_3P_1$. Since the region $P_1R_4R_1P_1$ except the boundary $R_1R_4$ and $R_4P_1$ is in $\Lambda^-$, it is better to wait the impulse control until the trajectory reaches $\Lambda$, even if the trajectory traverses the region $\Gamma^+$. Therefore, the part $P_1R_4$ of $\Lambda$ becomes the Free Boundary of this problem and it is optimal to let the trajectory jump from $(t,x) \in \Lambda$ to the point $(t,x-\xi(t,x)) \in \tilde{\Lambda}$ by the impulse control.
In view of the condition (II.1.15), it is proved that $\tilde{\Lambda}$ is in $\Gamma^-$.
In addition, it is assumed that $\tilde{\Lambda}$ is inside the closed region $P_1 R_4 R_1 P_1$ of Fig.II-2. Then, for the configuration of Fig.II-2, we can conclude that it is sufficient to exert the impulse control only once for the trajectory passing the region $P_1 R_4 R_1 P_1$.

### II.5.2 Optimal Cost $u(t,x)$ in the Region $P_1 R_4 R_1 P_1$

Suppose that $(t,x)$ belongs to the region $P_1 R_4 R_1 P_1$. As is shown in Fig.II-2, the trajectory $X(.;t,x)$ crosses the curve $\Lambda$ at $(t',x')$, where $x' = X(t';t,x)$. If we do not exert the impulse control before the time $t'$, the trajectory $X(.;t,x)$ goes into the region $\Lambda^+ \cap \Gamma^+$, where it is better to use the impulse control as soon as possible. If the impulse control is exerted at time $t''<t'$, it is not optimal from the definition of $\Lambda^-$, because $X(t'';t,x)$ is inside the region $\Lambda^-$. Therefore, the impulse control is to be exerted on $\Lambda$ with the magnitude $\xi(t',x')$, and the trajectory jumps to $\tilde{\Lambda}$. In view of the configuration of Fig.II-2, there is no need of further impulse control until the time $T$. Therefore, we obtain the optimal cost function $u(t,x)$ by the expression:

$$ u(t,x) = g(t',x', \xi(t',x')) + \int_t^{t'} h(s,X(s;t,x))ds \\ + u(t',x'-\xi(t',x')) \quad (II.5.2) $$

where $(t,x)$ belongs to the region $P_1 R_4 R_1 P_1$.

The continuous differentiability of $u(t,x)$ defined by (II.5.2) follows from the continuities of the curve $\Lambda$ and $\xi(t,x)$. Since the point $(t',x'-\xi(t',x'))$ belongs to the region $P_1 R_4 R_1 P_1$ by assumption, the term $u(t',x'-\xi(t',x'))$ is calculated by (II.5.1). Differentiating $u(t,x)$ defined by (II.5.2) along the trajectory $X(t';t,x)$, we obtain

$$ \frac{\partial u}{\partial t} + f(t,x) \frac{\partial u}{\partial x} + h(t,x) = 0. \quad (II.5.3) $$
By means of (II.5.3), it is shown that the $u(t,x)$ satisfies (II.2.1) and (II.2.3) of Q.V.I. Thus, we will consider the remaining inequality (II.2.2);

$$ u(t,x) \le \min_{0 \le \xi \le x} \{g(t,x,\xi) + u(t,x-\xi)\}. \qquad (II.5.4) $$

Since $u(t,x) \le \varphi(t,x)$ inside the region $P_1 R_4 R_1 P_1$, there may be a case where $(t,x)$ minimizing the expression:

$$ \min_{0 \le \xi \le x} \{g(t,x,\xi) + \varphi(t,x-\xi)\} \qquad (II.5.5) $$

does not coincide the $\xi$ minimizing the right hand side of (II.5.4).

However, it is easily proved that $u(t,x)$ is monotone increasing in $x$ in the region $P_1 R_4 OTR_3 P_1$. If $g(t,x,\xi) = \text{constant}$, $\xi(t,x) = x$ and $\tilde{\Lambda}$ always coincides with $t$-axis. In this case, there is no need of the modification of $\xi(t,x)$. Also, in the neighborhood of the trajectory $P_1 P_1$, there is no need of such modification. But, in general, there is a possibility of the necessary modification of $\xi(t,x)$, which may cause the modification of $\tilde{\Lambda}$. For the simplicity of discussions, we assume that there is no need of the modification of $\tilde{\Lambda}$. Under this assumption, the inequality (II.5.4) is satisfied inside the closed region $P_1 R_4 R_1 P_1$. Thus, $u(t,x)$ defined by (II.5.2) satisfies Q.V.I. in the region $P_1 R_4 R_1 P_1$ except the boundary $P_1 R_4$ and $R_1 P_1$.

### II.5.3 Optimal Cost $u(t,x)$ in the region

On the boundary $P_1 R_4$, the definition (II.5.2) is reduced to

$$ \begin{aligned} u(t,x) &= g(t,x,\xi(t,x)) + u(t,x-\xi(t,x)) \\ &= \min_{0 \le \xi \le x} \{g(t,x,\xi) + u(t,x-\xi)\} \end{aligned} \qquad (II.5.6) $$

with which (II.2.2) and (II.2.3) are satisfied on the boundary.

In the region $P^+ \cap \tilde{\Lambda}^+$, we define $u(t,x)$ by

$$ u(t,x) = \min_{0 \le \xi \le x} \{g(t,x,\xi) + u(t,x-\xi)\}. \qquad (II.5.7) $$

In view of the definition of $\tilde{\Lambda}^+$, we obtain the inequality:

$$ u(t+\delta t, x(t+\delta t)) + \int_{t}^{t+\delta t} h(s, x(s)) ds > u(t,x) \qquad (II.5.8) $$
$$ \text{or} \qquad \frac{\partial u}{\partial t} + f(t,x) \frac{\partial u}{\partial x} + h(t,x) > 0 \qquad (\text{II.5.9}) $$

which is valid in $\Lambda^+$ without the necessity of the modification of $\xi(t,x)$. (II.5.8) shows that (II.2.1) of Q.V.I. is also satisfied.

Therefore, we have solved Q.V.I. and can determine the Free Boundary of our problem. If the initial point $(0, x_0)$ lies on the part $OR_1$ of the x-axis, there is no need of impulse control. If $(0, x_0)$ lies on the part $R_1 R_4$, it is necessary to exert the impulse control only once when the trajectory $X(\cdot; 0, x_0)$ reaches the curve $\Lambda$. If $(0, x_0) \in \Lambda^+$, it is necessary first to use the impulse control with the magnitude $\xi(0, x_0)$ at $t=0$ and then another impulse control on the curve $\Lambda$ if necessary. The condition (II.1.13) is useful for the derivation of this conclusion. Hence, the optimal trajectory does not go out of the region $P_1 R_4 OTR_2 P_1$ on the interval $(0, T]$. Therefore, the Free Boundary is composed of the curve $R_4 P_1 R_2$. The time and the magnitude of the optimal impulse control is determined only by the part $R_4 P_1$ of $\Lambda$.

It is to be noted that assumptions are all satisfied in the optimal design problem of the aerator systems.
## II.6 Determination of the Optimal Policy (2) Modification of $\Gamma$ and $\Lambda$

The configuration of Fig.II-2 is the basis for the reasoning of the previous section. In Fig.II-2, the time $t_1$ is negative when the trajectory in contact with $\Gamma$ at $P_1$ crosses the curve $\Lambda$. Then, we can solve our problem completely and determine the optimal impulse control corresponding to the arbitrary initial condition $(0, x_0) \in \mathcal{O}$.

In deriving $u(t,x)$ from $\varphi(t,x)$, one of the most important and fundamental assumptions is that it is sufficient for almost all trajectories to exert the impulse control at most only once. If $t_1 > 0$ as in Fig.II-3, there is a trajectory for which it is necessary to use the impulse control twice or three times. In this case, the fundamental assumption is not valid and we must modify our procedure on the interval $[0, t_1]$ of Fig.II-3.

### II.6.1 Analysis of the Procedure to Determine the Free Boundary

If we apply the procedure of the previous section to the configuration of Fig.II-3, we can determine the Free Boundary on the interval $[t_1, T]$ by means of the following steps.

Step 1. Calculation of $\varphi(t,x)$, which becomes the solution $u(t,x)$ of Q.V.I. in the region $P_1S_1TR_3P_1$.

Step 2. Calculation of $\xi(t,x)$, which determines the optimal magnitude of the impulse control to be exerted at $(t,x)$ if necessary.

Step 3. Calculation of the curves $\Gamma$ and $\tilde{\Gamma}$.

Step 4. Calculation of the curves $\Lambda$ and $\tilde{\Lambda}$.

Step 5. Proof of Theorem 1.

Step 6. Calculation of $u(t,x)$ in the region $P_1Q_1R_1P_1$ and the proof of its satisfaction of Q.V.I.

Step 7. Definition of $u(t,x)$ in $\Gamma^+ \cap \Lambda^+$ and the proof of its satisfaction of Q.V.I.

Repeating and modifying these steps, we can determine successively the Free Boundary on $[0, t_1]$.
## II.6.2 Step 1 to Step 3

We start from Step 1. We adopt $\Lambda$ as the temporary Free Boundary of the region $P_1Q_1QOS_1P_1$ and define $\varphi_1(t,x)$ by

$$ \varphi_1(t,x) = \int_t^{t'} h(s, X(s; t, x)) ds + g(t', x', \xi(t', x')) \\ + u(t', x' - \xi(t', x')), \quad t_1 \le t' < T, \quad (II.6.1) $$

$$ \varphi_1(t,x) = \int_t^{t'} h(s, X(s; t, x)) ds + g(t', x', \xi(t', x')) \\ + \varphi_1(t', x' - \xi(t', x')), \quad 0 \le t' < t_1. \quad (II.6.2) $$

In view of Fig.3, it is clear that there is no contradiction in these definitions. It is already shown in II.5.2 that $\varphi_1(t,x)$ defined by (II.6.1) becomes the solution of Q.V.I. in the closed region $P_1Q_1R_1P_1$. But, it is not certain that $\varphi_1(t,x)$ defined by (II.6.1) yields the solution of Q.V.I. even in the region $Q_1S_2S_1R_1Q_1$, because the inequality (II.2.2) of Q.V.I. is to be satisfied.

Hence, we proceed to the Step 2. We define $\xi_1(t,x)$ by $\xi$ which minimizes the expression:

$$ \min_{0 \le \xi \le x} \{g(t, x, \xi) + \varphi_1(t, x - \xi)\} \quad (II.6.3) $$

in the region $P_1QOS_1P_1$. The existence of $\xi_1(t,x)$ easily follows from the continuity of $g(t,x,\xi)$ and $\varphi_1(t,x)$. The uniqueness, continuity and piecewise continuous differentiability are assumed in the following. In the analysis of the previous section, it is assumed that $\xi_1(t,x) = \xi(t,x)$ if $(t,x)$ belongs to the closed region $P_1Q_1R_1P_1$. But, on the interval $[0, t_1]$, the curve $\Lambda$ is in the region $Q_1S_2S_1R_1Q_1$, which shows that $\xi(t,x)$ may be recalculated in that region.

The step 3 is the formal investigation of the inequality (II.2.2) of Q.V.I. The curve $\Gamma_1$ is defined by $x(t)$ satisfying

$$ \bar{\Phi}_1(t, x) = \varphi_1(t, x) - \{g(t, x, \xi_1(t, x)) + \varphi_1(t, x - \xi_1(t, x))\} = 0 \quad (II.6.4) $$

on the interval $[0, T]$, where $(t,x)$ belongs to the closed region $R_3P_1QOTR_3$ of Fig. II-3. In the closed region $R_3P_1S_1T$, $\varphi_1(t,x)$ is to be replaced by $u(t,x)$. Furthermore, the existence and uniqueness of $x(t)$ defining $\Gamma_1$ is assumed on $[0, T]$.
On the interval $[t_1, T]$ of Fig. II-3, $\Gamma_1$ coincides with $\Gamma$, which follows from (II.3.5) and (II.5.1). In view of (II.5.6), it is shown that $\Gamma_1$ coincides with $\Lambda$ on the interval $[t_1, t_2]$. Considering the continuity of $\xi_1(t,x)$ and $\varphi_1(t,x)$, we can conclude that the curve $\Gamma_1$ defined by (II.6.4) on the interval $[0, t_1]$ passes through the point $Q_1$.

Let us define the curve $\tilde{\Gamma}_1$ by $x(t) - \xi_1(t, x(t))$ and the regions $\Gamma_1^+$ and $\Gamma_1^-$ by

$$ \Gamma_1^+ \equiv \{ (t, x) : \Phi_1(t, x) > 0, (t, x) \in \partial \Omega \}, \quad (II.6.5) $$

$$ \Gamma_1^- \equiv \{ (t, x) : \Phi_1(t, x) < 0, (t, x) \in \partial \Omega \}. \quad (II.6.6) $$

### II.6.3 Step 4

If $(t,x) \in \Gamma_1^+$ with $0 \le t \le t_1$, it is better to exert the impulse control at $(t,x)$ with the magnitude $\xi_1(t,x)$. If $(t,x) \in \Gamma_1^-$ with $0 \le t \le t_1$, it is better to let the trajectory $X(.; t, x)$ reach the curve $\Lambda$ where the impulse control is to be exerted. But, if the trajectory $X(.; t, x)$ crosses $\Gamma_1$ from $\Gamma_1^-$ to $\Gamma_1^+$ on $[0, t_1]$, it is not known whether we should exert the impulse control on $\Gamma_1$ or not. This is completely the same problem as was studied in section II.4.

Thus, we proceed to the step 4 and compare

$$ J_t = \mathcal{L}(t, x, \xi_1(t, x)) + \varphi_1(t, x - \xi_1(t, x)) \quad (II.6.7) $$

$$ \begin{aligned} J_{t+\delta t} = & \mathcal{L}(t+\delta t, X(t+\delta t; t, x), \xi_1(t+\delta t, X(t+\delta t; t, x))) \\ & + \varphi_1(t+\delta t, X(t+\delta t; t, x) - \xi_1(t+\delta t, X(t+\delta t; t, x))) \\ & + \int_t^{t+\delta t} h(s, X(s; t, x)) ds \end{aligned} \quad (II.6.8) $$

where $0 < \delta t \ll 1$. In the closed region $P_1 \subset Q_1 \subset R_1 \subset S_1 \subset T R_2 \subset P_1$, $\varphi_1(t, x)$ and $\xi_1(t, x)$ are replaced by $u(t, x)$ and $\xi(t, x)$.

$J_{t+\delta t} - J_t$ is calculated as follows:

$$ J_{t+\delta t} - J_t = \Phi_1(t, x) \delta t + o(\delta t) \quad (II.6.9) $$

$$ \text{with} \quad \Phi_1(t, x) \equiv \frac{D}{Dt} \mathcal{L}(t, x, \xi_1(t, x)) + \frac{D}{Dt} \varphi_1(t, x - \xi_1(t, x)) + h(t, x) \quad (II.6.10) $$
$$ \frac{D}{Dt}g(t,x, \xi_1(t,x)) = \frac{\partial g(t,x, \xi_1(t,x))}{\partial t} + \frac{\partial g(t,x, \xi_1(t,x))}{\partial x} f(t,x) + \frac{\partial g(t,x, \xi_1(t,x))}{\partial \xi} a_1(t,x), \quad (II.6.10)' $$

$$ \frac{D}{Dt} \varphi_1(t, x - \xi_1(t, x)) = \frac{\partial \varphi_1(t, x - \xi_1(t, x))}{\partial t} + \frac{\partial \varphi_1(t, x - \xi_1(t, x))}{\partial x} \{f(t, x) - a_1(t, x)\} $$

where (II.4.3) and the following estimation are used.

$$ \delta \xi_1 \equiv \xi_1(t+\delta t, x(t+\delta t; t, x)) - \xi_1(t, x) = a_1(t, x) \delta t + o(\delta t). \quad (II.6.11) $$

Since $\varphi_1(t,x)$ and $\xi_1(t,x)$ are to be replaced by $\varphi(t,x)$ and $\xi(t,x)$ on $[t_1, T]$,
it is shown from (II.4.6) that $\mathcal{H}_1(t,x) \equiv \mathcal{H}(t,x)$ on $[t_1, T]$.
However, there
are discontinuities of partial derivatives of $\varphi_1(t,x)$ appearing in (II.6.10)'
along the boundary $S_1 \cap R_1 \cap P_1$. Thus, the curve $\Lambda_1$ defined by $x(t)$ satisfying

$$ \mathcal{H}_1(t, x) = 0 \quad (II.6.12) $$

is not continuous on the whole interval $[0, T]$. The existence and uniqueness
of $x(t)$ is to be assumed, though it can be shown directly in some cases.

The regions $\Lambda_1^+$ and $\Lambda_1^-$ are defined as follows:

$$ \Lambda_1^+ = \{(t,x):(t,x) \in \partial \mathcal{H}_1, \mathcal{H}_1(t,x) > 0\}, \quad (II.6.13) $$

$$ \Lambda_1^- = \{(t,x):(t,x) \in \partial \mathcal{H}_1, \mathcal{H}_1(t,x) < 0\}. \quad (II.6.14) $$

The simplest configuration of the curves $\Gamma_1$ and $\Lambda_1$, and the regions $\Gamma_1^+$, $\Gamma_1^-$, $\Lambda_1^+$ and $\Lambda_1^-$ are shown in Fig.II-4.

### II.6.4 Step 5 to Step 7: Determination of the Free Boundary

Theorem 1 is useful to determine the Free Boundary for the configuration
of Fig.II-2. Thus, we try to prove the similar result for the configuration
of Fig.II-4. In other words, we have

**THEOREM 2.** Suppose that $(t,x) \in \Gamma_1$ and $t_1^+ \le t \le t_1^-$. If and only if the trajectory $X(s;t,x)$ ($s \ge t$) goes into $\Gamma_1^-$, $(t,x) \in \Lambda_1^+$. If and only if $X(s;t,x)$ ($s \ge t$) goes into $\Gamma_1^+$, $(t,x) \in \Lambda_1^-$. If and only if $X(s;t,x)$ is in contact with $\Gamma_1$ at $(t,x)$, $\Gamma_1$ crosses $\Lambda_1$ at $(t,x)$.
**Proof** Suppose that $(\bar{t}, \bar{x})$ is on the part $Q_1 \cap Q_2$ of $\Lambda$. In view of (II.5.6) and (II.6.1), we have

$$ \varphi_1(\bar{x}, \bar{x}) = g(\bar{x}, \bar{x}, \xi(\bar{x}, \bar{x})) + \varphi(\bar{x}, \bar{x} - \xi(\bar{x}, \bar{x})). \quad (II.6.15) $$

Since $\Lambda$ is in the region $P_1 \cap Q_1 \cap S_2 \cap S_1$ of Fig.II-3, we have

$$ \varphi_2(\bar{x}, \bar{x} - \xi(\bar{x}, \bar{x})) < \varphi(\bar{x}, \bar{x} - \xi(\bar{x}, \bar{x})). \quad (II.6.16) $$

From (II.6.15) and (II.6.16), we have

$$ \begin{aligned} \varphi_1(\bar{x}, \bar{x}) &> g(\bar{x}, \bar{x}, \xi(\bar{x}, \bar{x})) + \varphi_2(\bar{x}, \bar{x} - \xi(\bar{x}, \bar{x})) \\ &> g(\bar{x}, \bar{x}, \xi_1(\bar{x}, \bar{x})) + \varphi_1(\bar{x}, \bar{x} - \xi_1(\bar{x}, \bar{x})). \end{aligned} \quad (II.6.17) $$

Therefore, we can conclude that $(t,x)$ belongs to $\Gamma_1^\dagger$. And, if the crossing point of $\Gamma_1$ with $\Lambda_1$ exists, it is necessarily between the curve $\Lambda$ and $t$-axis. Hence, the configuration of Fig.II-4 is a reasonable one.

It is obvious that $\varphi_1(t,x)$ defined by (II.6.1) satisfies the
Partial differential equation:

$$ \frac{\partial \varphi_1}{\partial \bar{t}} + f(t, \bar{x}) \frac{\partial \varphi_1}{\partial \bar{x}} + h(t, \bar{x}) = 0 \quad (II.6.18) $$

along the trajectory of the Free System (II.1.3).

Now, we replace $\varphi(t,x)$ and $\xi(t,x)$ with $\varphi_1(t,x)$ and $\xi_1(t,x)$ respectively in the proof of Theorem 1. In addition, we use (II.6.18), (II.6.7), (II.6.8) and (II.6.9) at the place where (II.2.7), (II.4.1), (II.4.2) and (II.4.5) are used respectively.
By means of these substitutions, we can prove Theorem 2 completely in the same manner.
Q.E.D.

Thus, we have solved the problem on $[t_2, T]$ and obtained the Free Boundary composed of the curves $\Lambda, \Lambda_1$ and the trajectories $P_2 P_2^1, P_1 P_1^1$.

The logics and the procedures which yield Theorem 2 can be used also on $[0, t_2]$ repeatedly, in order to obtain the Free Boundary on $[0, t_2]$.
## II.6.5 Modification of the Free Boundary in the General Case

Suppose that we have constructed the Free Boundary up to $(\Gamma_{n-1}, \Lambda_{n-1})$ and that it is necessary to modify it again on the interval $[0, t_n]$ (See Fig.5). In order to obtain the modified Free Boundary on $[0, t_n]$, we can follow the procedure expressed by step 1 to step 7 of II.6.1. Since the detailed explanation of these steps are given in II.6.2 and II.6.3, we only enumerate the necessary formula.

### II.6.5.1 Step 1 to Step 4

#### Step 1. Definition of $\varphi_n(t, x)$ via $\varphi_{n-1}(t, x)$

In the same way as (II.6.1) and (II.6.2), we define $\varphi_n(t, x)$ as follows:

$$ \varphi_n(t, x) = \int_t^x \mathcal{H}(\tau, X(\tau; t, x)) d\tau + g(t', x', \xi_{n-1}(t', x')) \\ + u(t', x' - \xi_{n-1}(t', x')), \quad t_n \le t' \le t_n. \quad (II.6.19) $$

$$ \varphi_n(t, x) = \int_t^{t'} \mathcal{H}(\tau, X(\tau; t, x)) d\tau + g(t', x', \xi_{n-1}(t', x')) \\ + \varphi_n(t', x' - \xi_{n-1}(t', x')), \quad 0 \le t' \le t_n \quad (II.6.20) $$

where $(t, x)$ and $(t', x')$ are shown in Fig.11-5.

#### Step 2. Definition of $\xi_n(t, x)$

$\xi_n(t, x)$ is defined by $\xi$ which minimizes the expression:

$$ g(t, x, \xi) + \varphi_n(t, x - \xi) \quad (II.6.21) $$

where $\varphi_n(t, x)$ is defined by (II.6.19) or (II.6.20).

#### Step 3. Definitions of curves $\Gamma_n^+$, $\tilde{\Gamma}_n$ and regions $\Gamma_n^+$, $\tilde{\Gamma}_n$.

The curve $\Gamma_n$ is defined by $x(t)$ satisfying the equality:

$$ \Phi_n(t, x) = \varphi_n(t, x) - \{g(t, x, \xi_n(t, x)) + \varphi_n(t, x - \xi_n(t, x))\} = 0 \quad (II.6.22) $$

where $0 \le t \le t_n$ and $\varphi_n(t, x)$ and $\tilde{\varphi}_n(t, x)$ are defined in Steps 1 and 2.

The curve $\Gamma_n$ is defined by $x(t) - \xi_n(t, x(t))$ where $(t, x(t)) \in \Gamma_n$.

The regions $\Gamma_n^+$ and $\tilde{\Gamma}_n$ are defined as follows:

$$ \Gamma_n^+ = \{(t, x) : \bar{\Phi}_n(t, x) > 0, (t, x) \in \mathcal{D}, 0 \le t \le t_n\} \quad (II.6.23) $$

$$ \tilde{\Gamma}_n = \{(t, x) : \bar{\Phi}_n(t, x) < 0, (t, x) \in \mathcal{D}, 0 \le t \le t_n\}. \quad (II.6.24) $$
### Sten 4. Definitions of curves $\Lambda_n^+$, $\Lambda_n^-$ and regions $\Lambda_n^+, \Lambda_n^-$.

We compare

$$J_t = g(t, x, \xi_n(t, x)) + \varphi_n(t, x - \xi_n(t, x)) \quad (II.6.25)$$

with

$$J_{t+\delta t} = g(t+\delta t, X(t+\delta t; t, x), \xi_n(t-\delta t, X(t-\delta t; t, x))) \\ + \varphi_n(t+\delta t, X(t+\delta t; t, x) - \xi_n(t-\delta t, X(t-\delta t; t, x))) \\ + \int_{t}^{t+\delta t} \mathcal{H}(\tau, X(\tau; t, x)) d\tau \quad (II.6.26)$$

where $0 < \delta t \ll 1$. Using the estimation

$$\delta \xi_n \equiv \xi_n(t+\delta t, X(t+\delta t, t, x)) - \xi_n(t, x) = a_n(t, x) \delta t + o(\delta t), \quad (II.6.27)$$

we obtain

$$J_{t+\delta t} - J_t = \Phi_n(t, x) \delta t + o(\delta t) \quad (II.6.28)$$

where

$$\Phi_n(t, x) \equiv \frac{D}{Dt} g(t, x, \xi_n(t, x)) + \frac{D}{Dt} \varphi_n(t, x - \xi_n(t, x)) + \mathcal{H}(t, x). \quad (II.6.29)$$

$D/Dt$ denotes the derivative along the trajectory of (II.1.3) which yields

$$\frac{D}{Dt} g(t, x, \xi_n(t, x)) = \frac{\partial}{\partial t} g(t, x, \xi_n(t, x)) + f(t, x) \frac{\partial}{\partial x} g(t, x, \xi_n(t, x)) \\ + a_n(t, x) \frac{\partial}{\partial \xi} g(t, x, \xi_n(t, x)) \quad (II.6.30)$$

and

$$\frac{D}{Dt} \varphi_n(t, x - \xi_n(t, x)) = \frac{\partial}{\partial t} \varphi_n(t, x - \xi_n(t, x)) \\ + \{f(t, x) - a_n(t, x)\} \frac{\partial}{\partial x} \varphi_n(t, x - \xi_n(t, x)), \quad (II.6.31)$$

Then, the definition of the curve $\Lambda_n^+$ is given by the function $x(t)$

satisfying the equality:

$$\Phi_n(t, x(t)) = 0, \quad 0 \le t \le t_n. \quad (II.6.32)$$

The curve $\Lambda_n^+$ is defined by $x(t) - \xi_n(t, x)$ where $(t, x(t)) - \xi_n(t, x)) \in \Lambda_n^+$.

The regions $\Lambda_n^+$ and $\Lambda_n^-$ are defined as follows:

$$\Lambda_n^+ = \{(t, x) : (t, x) \in \partial D, \Phi_n(t, x) > 0, 0 \le t \le t_n\} \quad (II.6.33)$$

and

$$\Lambda_n^- = \{(t, x) : (t, x) \in \partial D, \Phi_n(t, x) < 0, 0 \le t \le t_n\}. \quad (II.6.34)$$
### II.6.5.2 Step 5 to Step 7

From (II.6.19) and (II.6.20), we have

$$ \frac{D}{Dt} \varphi_n(t, x) = \frac{\partial}{\partial t} \varphi_n(t, x) + f(t, x) \frac{\partial}{\partial x} \varphi_n(t, x) = -h(t, x). \quad (II.6.35) $$

Then, we obtain the following theorem:

**THEOREM 3.** Suppose that $(t,x) \in \Gamma_n^+$ and $t_n' \le t \le t_n$. $(t,x) \in \Lambda_n^+$ if and only if $X(s;t,x)$ $(s \ge t)$ goes into $\Gamma_n^-$. $(t,x) \in \Lambda_n^-$ if and only if $X(s;t,x)$ $(s \ge t)$ goes into $\Gamma_n^+$. $\Gamma_n^+$ crosses $\Lambda_n^-$ at $(t,x)$ if and only if $X(s;t,x)$ is in contact with $\Gamma_n^+$ at $(t,x)$.

**Proof.** Suppose that $(t,x) \in \Gamma_n^+$ and $X(s;t,x)$ goes into $\Gamma_n^-$. We will prove that $(t,x) \in \Lambda_n^+$. From the definition (II.6.22) of $\Gamma_n^+$, we have

$$ \varphi_n(t, x) = g(t, x, \xi_n(t, x)) + \varphi_n(t, x - \xi_n(t, x)). \quad (II.6.36) $$

At $(t+\delta t, X(t+\delta t))$ with $0<\delta t \ll 1$, the definition (II.6.34) of $\Gamma_n^-$ yields

$$ \varphi_n(t+\delta t, X(t+\delta t)) < g(t+\delta t, X(t+\delta t), \xi_n(t+\delta t, X(t+\delta t))) \\ + \varphi_n(t+\delta t, X(t+\delta t) - \xi_n(t+\delta t, X(t+\delta t))). \quad (II.6.37) $$

Substituting (II.6.36) from (II.6.37) side by side and using (II.6.35), we obtain the inequality with the left hand side:

$$ \begin{aligned} \varphi_n(t+\delta t, X(t+\delta t)) - \varphi_n(t, x) &= \left\{ \frac{\partial \varphi_n}{\partial t} + f(t, x) \frac{\partial \varphi_n}{\partial x} \right\} \delta t + o(\delta t) \quad (II.6.38) \\ &= -h(t, x) \delta t + o(\delta t). \end{aligned} $$

In view of (II.6.25), (II.6.26) and (II.6.28), we can calculate its right hand side as follows:

$$ \begin{aligned} & g(t+\delta t, X(t+\delta t), \xi_n(t+\delta t, X(t+\delta t))) - g(t, x, \xi_n(t, x)) \\ & + \varphi_n(t+\delta t, X(t+\delta t) - \xi_n(t+\delta t, X(t+\delta t))) - \varphi_n(t, x - \xi_n(t, x)) \\ &= J_{t+\delta t} - J_t - \int_t^{t+\delta t} h(t', x(t', t, x)) dt' \quad (II.6.39) \end{aligned} $$

$$ = \{ \Phi_n(t, x) - h(t, x) \} \delta t + o(\delta t). \quad (II.6.40) $$

Combining (II.6.38) with (II.6.40), we obtain the inequality:
$$ -h(t, x) \delta t < \{ \Phi_n(t, x) - h(t, x) \} \delta t + o(\delta t) $$

$$ \text{or} \quad 0 < \Phi_n(t, x) \delta t + o(\delta t). \quad (II.6.41) $$
Since $0 < \delta t \ll 1$, (II.6.41) is equivalent to the inequality

$$ \mathcal{H}_n(t, x) > 0, \qquad (II.6.42) $$

which means $(t,x) \in \Lambda_n^+.$

In order to prove the "only if" part, it is sufficient to trace back the calculations from (II.6.42) to (II.6.37).

The proof of the case where $X(s;t,x)$ goes into $\Gamma_n^+$ can be done in the same way.

If $X(s;t,x)$ is in contact with $\Gamma_n$ at $(t,x)$, we obtain the equality in (II.6.41) up to the first order term of $\delta t$. That is,

$$ \mathcal{H}_n(t, x) = 0 \qquad (II.6.43) $$

which means that $(t,x)$ belongs to both $\Lambda_n$ and $\Gamma_n$. It is also clear that we can prove the converse.

Q.E.D.

As to the steps 6 and 7, we can follow the same discussion as in II.5.

Using the Theorems 1, 2 and especially the Theorem 3 repeatedly, we can construct the Free Boundary on the whole interval $[0,T]$, which determines the optimal timings and the optimal magnitudes of the impulse control.

The number of timings when the impulse control should be exerted optimally varies with the initial value $x(0)$. If we denote by $N^*$ the maximum number, the case of Fig.II-6 corresponds to the case $N^*=5$.
## II.7 Concluding Remarks to Chapter 1

In this chapter, we presented the procedure by which the O.V.I. is solved directly for the optimal impulse control problem of the 1-dimensional deterministic nonlinear system. Though the configurations of Fig.II-2 and Fig.II-4 are the simplest ones, Theorems 1, 2 and 3 can be applied to more general cases. Related problems are discussed in the following.

### II.7.1 Linear System with Quadratic Performance Index

Many assumptions are necessary in the course of reasoning of this chapter. These assumptions are all satisfied in the optimal design problem of aerator system, which is a special case of the linear system with the quadratic performance index. In such cases, it is easy to obtain the integral $(t,x)$ of (II.2.7) analytically. Thus, the result can be extended to the higher dimensional case. Details are shown in the next chapter.

### II.7.2 Extension to the Higher Dimensional Case

It is easy to extend the theory formally to the general nonlinear problem with the higher dimensional state space. But, the integral $\varphi(t,x_1, \dots, x_n)$ may not be found easily. Therefore, we must be very careful about the expression of nonlinear systems and the suitable selection of the performance index.

### II.7.3 Case where the Deterministic Disturbance $w(t,x)$ exists.

In the reasoning of this chapter, it is assumed that $w(t,x) = 0$. If $w(t,x) \neq 0$, there exist the discontinuities of $x$ at $t = r_1, \dots, r_m$. These discontinuities can be regarded as known in advance. Thus, we divide the interval $O,T$ into subintervals $I_{m+1} = [0, r_1], \dots, I_1 = [r_m, T]$ and solve the problem backwards from $I_1$. In other words, if the problem is completely solved on $I_1$, we can calculate the cost function at $x(r_m-0)$, with which (II.2.7) can be solved on $I_2$. Such problem arises in the optimal design of the aerator system, when the big amount of polluted water flows into the river in the midway between the source and the sea.
### II.7.4 Restriction on the Capacity of Impulse Control

The restriction (II.1.10) on the impulse control can be replaced by more general one, because we have only to calculate $\xi(t,x)$ according to the restriction. For the case where $\xi(t,x)$ is always equal to $x$, the state $x$ is always pulled back to $0$ and $\bar{f}$ and $\bar{A}$ coincide with $t$-axis. This case is easier to study.

### II.7.5 Systems with many inputs

It seems that the study of the case with many inputs has also some practical applications. But, its analysis may be complicated.

### II.7.6 Case where $N \to \infty$

The reasoning of this paper is implicitly based on the assumption that $N$ is finite. It is explicitly expressed by the assumption that there exists a positive number $\alpha$ in (II.1.12). If we can modify and apply our theory to the optimal control problem of chemical reactors, the impulse control scheme can be regarded as the batch process in the chemical engineering. Then, does the case $N \to \infty$ mean that the continuous tubular reactor is better than the batch reactor? From the theoretical viewpoint, this problem is interesting in relation to the optimal singular control problem and the maximum principle type formulation of the impulse control theory.

### II.7.7 Computational Algorithm

If the integral $\varphi(t,x)$ does not exist or has the complicated form, it is necessary to compute it numerically. The process of reasoning, in particular the steps 1 to 7, can be used directly as an algorithm. Thus, it is obvious that the accuracy of approximation depends only on the accuracy of the numerical calculation of $\varphi(t,x)$ and we need not consider the convergence to the optimal solution. However, this algorithm is a little complicated in comparison with the direct difference approximation of Q.V.I. Thus, the comparison of our method with the direct method may be another interesting problem. The extension of our method to the stochastic case may be also interesting.
### III. OPTIMAL IMPULSE CONTROL OF HIGH DIMENSIONAL LINEAR SYSTEM WITH QUADRATIC PERFORMANCE INDEX

In this chapter, we extend the results of chapter II to the high dimensional linear system with quadratic performance index. Using the integral of (II.2.7), we can derive analytically the necessary formula for the application in chapter IV.

#### III.1 Formulation of the Problem

The state of the system is expressed by the n-vector $x=(x_1, ..., x_n)$.
$t$ is the time. The behaviour of the system is described by a system of ordinary differential equations:

$$ \dot{x} = A(t)x + bv + w(t,x), \quad x(0) = x_0, \qquad (\text{III.1.1}) $$

where $A(t)$ is a $n \times n$ matrix with piecewise continuously differentiable elements $A_{ij}(t)$ ($i, j=1, ..., n$) and

$$ b = (1, ..., 0)' \qquad (\text{III.1.2}) $$

$w(t,x)$ is the deterministic disturbance as in chapter II.

Considering the nature of the pollution control problem of the river studied in chapter III, we are concerned only with the behaviour of the system in the region:

$$ \mathcal{O} = \{ (t,x) : 0 \le t \le T, x_1 \ge 0, A_{12}(t)x_2 + \dots + A_{1n}(t)x_n > 0 \} \qquad (\text{III.1.3}) $$

and assume that the condition:

$$ A_{12}(0)x_{20} + A_{13}(t)x_{30} + \dots + A_{1n}(0)x_{no} > 0 \qquad (\text{III.1.4}) $$

is always satisfied and implies the inequality:

$$ A_{12}(t)x_2(t) + A_{13}(t)x_3(t) + \dots + A_{1n}(t)x_n(t) > 0, \quad 0 < t \le T. \qquad (\text{III.1.5}) $$

The restriction (II.1.10) is used as it is. In other words, if we exert the impulse control $v$ with the magnitude $\xi$ on the system, only the $x_1$-coordinate can be changed discontinuously from $x_1$ to $x_1 - \xi$. Hence, we suppose that there is a restriction on the possible magnitude of the impulse control at $(t,x)$: $0 \le \xi \le x_1$. \qquad (III.1.6)
The conditions (III.1.2),..., (III.1.6) yield a reasonable framework of an
impulse control problem stated as follows.

The times: $0 \le T_1 < T_2 < \dots < T_N < T$, and its magnitudes:

$$ \xi_1, \xi_2, \dots, \xi_N \qquad (\text{III.1.7}) $$

of the impulse control exerted on the system are to be determined so as to
minimize the performance index:

$$ J \equiv N + \sum_{i=1}^{N} g(T_i) \xi_i + \frac{1}{2} \int_{0}^{T} \bar{X}(t; 0, x_o)' Q \bar{X}(t; 0, x_o) dt \\ + \frac{1}{2} \bar{X}(T; 0, x_o)' P_o \bar{X}(T; 0, x_o), \qquad (\text{III.1.8}) $$

$$ \text{with} \quad Q(t) \ge 0 \quad \text{and} \quad P_o \ge 0. \qquad (\text{III.1.9}) $$

$\bar{X}(t;0,x_o)$ is the solution of (III.1.1) corresponding to the initial condition
$x(0)=x_o$ and the impulse control (III.1.7). In addition, we denote the
solution of the Free System: $\dot{x} = A(t)x$ as

$$ X(t; t', x') = \bar{\Phi}(t, t')x', \quad X(t'; t', x') = x', \qquad (\text{III.1.10}) $$

where $\bar{\Phi}(t,t')$ is the fundamental matrix of (III.1.4). It is easily seen
that the existence and the uniqueness of $\bar{\Phi}(t,t')$ are assured.

**Remark III.1.1** The above formulation is almost parallel to that of II.1.
The controllability of the system:

$$ \dot{x} = A(t)x + bv \qquad (\text{III.1.11}) $$

is not assumed. This is because the formulation is adapted to the optimization
problem of the aerator system design discussed in the next chapter. If $A(t)$
$= A$ = Constant matrix, the theory of canonical form yields (III.1.2).

**Remark III.1.2** Another possible formulation is to express the system as

$$ \dot{x} = A(t)x + bv + f(t) + w(t, x) \qquad (\text{III.1.12}) $$

where $f(t)$ is the continuously acting disturbance known in advance. This
expression can be more generally used, but more difficult for the purpose
of the analysis. The form of $\mathcal{F}(t,x)$ becomes complicated such as

$$ \mathcal{F}(t,x) = \text{Quadratic term of } x + \text{Linear term of } x \\ \qquad + \text{Constant term (function of } t). \qquad (\text{III.1.13}) $$

### III.2 Verification of Assumptions

As is known from the formulation and the remarks of the previous
section III.1, it is necessary to formulate the problem in the "well-posed"
settings. In the discussion of chapter II, we imposed several conditions on the system and the performance index and used a series of assumptions in the course of reasoning. In the following, we extend the theory developed in chapter II to the problem formulated in III.1, verifying the conditions and the assumptions.

First, the condition (II.1.11) should be interpreted as

$$f_1(t, 0, x_2, \dots, x_n) = A_{12}(t)x_2 + \dots + A_{1n}(t)x_n > 0, \quad 0 \le t \le T, \quad (\text{III.2.1})$$

which is satisfied by assumptions (III.1.4) and (III.1.5) if $(0, x_0) \in \partial \mathcal{Q}$.

As to the condition (II.1.12), we have from (III.1.8)

$$g(t, x, \xi) = 1 + g(t) \xi. \quad (\text{III.2.2})$$

$$\text{Hence, if } g(t) > 0, \quad (\text{III.2.3})$$

(II.1.12) is satisfied with $\alpha=1$. It is easily seen that (II.1.13) is satisfied with the inequality.

About the condition (II.1.14), we take into account the structure (III.1.2) and (III.1.6), and try to make the most of the reasoning of II.2.2. That is, if we put

$$h(t, x) = \frac{1}{2} x' Q(t) x \quad (\text{III.2.4})$$

and suppose that $h(t,x)$ satisfies

$$h(t, 0, x_2, \dots, x_n) = 0, \quad (t, x) \in \partial \mathcal{Q}, \quad (\text{III.2.5})$$

$$h(t, x_1, x_2, \dots, x_n) > 0, \quad (t, x) \in \partial \mathcal{Q}, \quad x_1 \neq 0, \quad (\text{III.2.6})$$

we obtain necessarily

$$h(t, x) = \frac{1}{2} Q_{11}(t) x_1^2, \quad Q_{11}(t) > 0, \quad 0 \le t \le T. \quad (\text{III.2.7})$$

It is obvious that (II.1.15) is satisfied.

Concerning the condition (II.1.16), we put

$$Q(t, x) = \frac{1}{2} x' P_0 x. \quad (\text{III.2.8})$$

Considering in the same way as above, we obtain

$$Q(t, x) = \frac{1}{2} p_0 x_1^2, \quad p_0 > 0, \quad (\text{III.2.8})$$

which satisfies the condition (II.1.16).
**Remark III.2.1** Modifying the reasoning of II.2.2 a little, we can see
that the assumption:

$$Q(t) \ge 0, P_0 \ge 0 \qquad (\text{III.2.10})$$

will do well in place of (III.2.7) and (III.2.9). But, details
are omitted.

### III.3 Q.V.I.

From (II.2.1),..., (II.2.4), (III.1.2) and (III.1.8), we have the
Q.V.I. for our problem as follows.

$$\frac{\partial u}{\partial t} + x' A(t)' \frac{\partial u}{\partial x} + \frac{1}{2} x' Q(t) x \geq 0, \quad (t,x) \in \mathcal{D}, \qquad (\text{III.3.1})$$

$$u(t,x) \leq \min_{0 \leq \xi \leq x} [1 + g(t)\xi + u(t, x_1 - \xi, x_2, \dots, x_n)], \quad (t,x) \in \mathcal{D}, \qquad (\text{III.3.2})$$

$$\left\{ \frac{\partial u}{\partial t} + x' A(t)' \frac{\partial u}{\partial x} + \frac{1}{2} x' Q(t) x \left[ u(t,x) - \min_{0 \le \xi \le x} \left\{ 1 + g(t)\xi + u(t, x_1 - \xi, x_2, \dots, x_m) \right\} \right] \right\} = 0, \quad (t,x) \in \mathcal{D}, \qquad (\text{III.3.3})$$

$$u(T, x) = \frac{1}{2} x' P_0 x, \quad (T, x) \in \mathcal{D} \qquad (\text{III.3.4})$$

where $Q(t)$ and $P_0$ are determined by (III.2.7) and (III.2.9), respectively.

Free Boundary is obtained by the surface satisfying the equality:

$$u(t,x) = \min_{0 \le \xi \le x_1} [1 + g(t)\xi + u(t, x_1 - \xi, x_2, \dots, x_n)], \quad 0 \le t \le T. \qquad (\text{III.3.5})$$

### III.4 Solution in the Neighborhood of (T,0)

Under the conditions (III.2.1), (III.2.7) and (III.2.9), we can
apply the same discussion of II.2.2 to the problem (III.1.2) and (III.1.8).

Substituting (III.1.5) into (II.2.6), we obtain

$$\varphi(t, x) = \frac{1}{2} x' \left[ \int_{t}^{T} \Phi'(t, \cdot) \partial \Phi(\cdot, \cdot) d\tau + \Phi(\cdot, T) P_{0} \Phi(\cdot, T) \right] x, \qquad (\text{III.4.1})$$

Defining $\bar{P}(t,T; P_0)$ and $P(t_1, t_2)$ by the following:

$$\bar{P}(t, T; P_0) \equiv \left\{ \int_{t}^{T} \Phi'(\tau, \cdot) Q(\tau, \cdot) \Phi(\cdot, \tau) d\tau + \Phi'(T, \cdot) P_0 \Phi(\cdot, T) \right\} P_0, \qquad (\text{III.4.2})$$

$$P(t_1, t_2) \equiv \bar{P}(t_1, t_2; 0), \qquad (\text{III.4.3})$$

we can express $\varphi(t,x)$ as

$$\varphi(t, x) = \frac{1}{2} x' \bar{P}(t, T; P_0) x. \qquad (\text{III.4.4})$$
Furthermore, putting

$$ \wp(t_1, t_2, x) \equiv \frac{1}{2} x' \bar{P}(t_1, t_2)x, \qquad (\text{III.4.5}) $$

we can calculate the integral term of the performance index (III.1.8) along the trajectory $X(., t_1, x_1)$ from $t=t_1$ to $t=t_2$.

Substituting (III.4.1) into (II.2.7), we obtain the differential equation to be satisfied by $\bar{P}(t, T; P_0)$:

$$ \frac{\partial \bar{P}}{\partial t} + A(t)' \bar{P} + \bar{P} A(t) + Q = 0, \quad \bar{P}(T, T; P_0) = P_0. \qquad (\text{III.4.6}) $$

As to $P(t_1, t_2)$, we obtain

$$ \frac{\partial P}{\partial t_1} + A(t)'P + P A(t) + Q = 0, \quad P(t_2, t_2) = 0. \qquad (\text{III.4.7}) $$

$$ \frac{\partial P}{\partial t_2} = \Phi(t_2, t_1) Q \Phi(t_2, t_1). \qquad (\text{III.4.8}) $$

Thus, we can use the explicit expression of the integral $\wp(t,x)$ of (II.2.7). This is a great advantage for the linear system with the quadratic performance index.

It is obvious that the same method of reasoning can be made use of in the neighborhood of $(T, 0, x_2, \dots, x_n)$. $U_\delta$ is defined by

$$ U_{\delta} \equiv \{ (t, x) : (t-T)^2 + x_1^2 < \delta^2, (t, x) \in \mathcal{D} \}. \qquad (\text{III.4.9}) $$

If we adopt the assumption (III.2.10) as it is, the definition of $U_\delta$ becomes

$$ U_{\delta} \equiv \{ (t, x) : (t-T)^2 + x_1^2 + x_2^2 + \dots + x_n^2 < \delta^2, (t, x) \in \mathcal{D} \}. \qquad (\text{III.4.10}) $$

But, this change is not necessary. Details are omitted.

Therefore, in view of (II.2.13), we obtain

$$ u(t, x) \equiv \wp(t, x). \qquad (\text{III.4.11}) $$

### III.5 Calculation of $\xi(t,x)$

We proceed to the step 2 of II.6.1.

Since the expression:

$$ 1 + g(t)\xi + \wp(t, x_1 - \xi, x_2, \dots, x_n) \qquad (\text{III.5.1}) $$

is a quadratic form in $\xi$, we know the existence and uniqueness of $\xi(t,x)$ minimizing (III.5.1) on the closed interval: $0 \le \xi \le x_1$.
If $0 \le \xi(t,x) \le x_1$, we obtain from (II.3.2)

$$ \xi(t,x) = \frac{\{\bar{P}_{11}(t)x_1 + \dots + \bar{P}_{1n}(t)x_n - g(t)\}}{\bar{P}_{11}(t)} \quad (\text{III.5.2}) $$

$$ \text{where} \quad \bar{P}_{ij}(t) = \bar{P}_{ij}(t,T; P_0), \quad i,j = 1, \dots, n \quad (\text{III.5.3}) $$

and we used the property

$$ \bar{P}_{11}(t,T; P_0) > 0, \quad t \le T \quad (\text{III.5.4}) $$

which follows from (III.2.7) and (III.2.9).

(III.5.2) can also be derived from the transformation of the formula:

$$ \begin{aligned} & 1 + g(t)\xi + \varphi(t, x_1 - \xi, x_2, \dots, x_n) \\ &= 1 + g(t)\xi + \frac{1}{2}\left\{ \bar{P}_{11}(t)\xi^2 - 2\{\bar{P}_{11}(t)x_1 + \dots + \bar{P}_{1n}(t)x_n\}\xi + 2\varphi(t, x) \right\} \\ &= 1 + \frac{1}{2}\left[ \bar{P}_{11}(t)\left\{\xi - \frac{\bar{P}_{11}(t)x_1 + \dots + \bar{P}_{1n}(t)x_n - g(t)}{\bar{P}_{11}(t)}\right\}^2 - \frac{\{\bar{P}_{11}(t)x_1 + \dots + \bar{P}_{1n}(t)x_n - g(t)\}^2}{\bar{P}_{11}(t)} \right] \\ &\quad + \varphi(t, x). \end{aligned} \quad (\text{III.5.5}) $$

The condition $0 < \xi(t,x) < x_1$ is equivalent to

$$ \bar{P}_{11}(t)x_1 + \dots + \bar{P}_{1n}(t)x_n > g(t) > 0 \quad (\text{III.5.6}) $$

and

$$ \bar{P}_{12}(t)x_2 + \dots + \bar{P}_{1n}(t)x_n < g(t) \quad (\text{III.5.7}) $$

$$ \text{If} \quad \bar{P}_{12}(t)x_2 + \dots + \bar{P}_{1n}(t)x_n \ge g(t), \quad (\text{III.5.8}) $$

$$ \text{we have} \quad \xi(t,x) = x_1 \quad (\text{III.5.9}) $$

$$ \begin{aligned} & \text{and} \\ &= 1 + g(t)\xi(t,x) + \varphi(t, x_1 - \xi(t,x), x_2, \dots, x_n) \\ &= 1 + g(t)x_1 + \varphi(t, 0, x_2, \dots, x_n) \\ &= 1 + g(t)x_1 + \frac{1}{2} \sum_{i,j=2}^{n} \bar{P}_{ij}(t)x_i x_j. \end{aligned} \quad (\text{III.5.10}) $$

$$ \text{If} \quad \bar{P}_{11}(t)x_1 + \dots + \bar{P}_{1n}(t)x_n < g(t), \quad (\text{III.5.11}) $$

$$ \text{we have} \quad \xi(t,x) = 0. \quad (\text{III.5.12}) $$
### III.6 Calculation of $\nabla$

In view of the existence of $\tilde{u}_g$, it is obvious that we can apply the discussion of II.3 to the problem of this chapter.

From (II.3.5), the definition of the surface $\nabla$ is given by

$$ \Phi(t, x) = \varphi(t, x) - \{1 + g(t)\xi(t, x) + \varphi(t, x_1 - \xi(t, x), x_2, \dots, x_n)\} = 0. \quad (\text{III.6.1}) $$

If $0 < \xi(t, x) < x_1$, we can use the expression (III.5.5) and obtain the expression of the surface $\nabla$ as follows:

$$ \varphi(t, x) = 1 + \varphi(t, x) - \left\{ \bar{P}_{11}(t)x_1 + \dots + \bar{P}_{1n}(t)x_n - g(t) \right\}^2 / 2 \bar{P}_{11}(t) \quad (\text{III.6.2}) $$

or

$$ \left\{ \bar{P}_{11}(t)x_1 + \dots + \bar{P}_{1n}(t)x_n - g(t) \right\}^2 = 2\bar{P}_{11}(t) . \quad (\text{III.6.3}) $$

In view of (III.5.4) and (III.5.6), we obtain

$$ \bar{P}_{11}(t)x_1 + \dots + \bar{P}_{1n}(t)x_n = g(t) + \sqrt{2\bar{P}_{11}(t)} \quad (\text{III.6.4}) $$

which shows the surface (plane) $\nabla$.

If $(t, x) \in \nabla$ defined by (III.6.4), we have

$$ \xi(t, x) = \left\{ \frac{2}{\bar{P}_{11}(t)} \right\}^{\frac{1}{2}} \quad (\text{III.6.5}) $$

by substituting (III.6.4) into (III.5.2). Then, the surface $\tilde{\nabla}$ can be expressed as follows:

$$ \bar{P}_{11}(t)x_1 + \dots + \bar{P}_{1n}(t)x_n = g(t). \quad (\text{III.6.6}) $$

If $\xi(t, x) = x_1$, we substitute (III.5.10) into (III.6.1) and obtain

$$ 1 + g(t)x_1 + \frac{1}{2} \sum_{i,j=2}^{n} \bar{P}_{ij}(t)x_i x_j = \frac{1}{2} \sum_{i,j=1}^{n} \bar{P}_{ij}(t)x_i x_j \quad (\text{III.6.7}) $$

$$ \text{or} \quad \frac{1}{2} \left[ \bar{P}_{11}(t) \left\{ x_1 + \frac{\bar{P}_{12}(t)x_2 + \dots + \bar{P}_{1n}(t)x_n - g(t)}{\bar{P}_{11}(t)} \right\}^2 - \frac{\left\{ \bar{P}_{12}(t)x_2 + \dots + \bar{P}_{1n}(t)x_n - g(t) \right\}^2}{\bar{P}_{11}(t)} \right] = 1. $$

$$ (\text{III.6.8}) $$

Considering that $x_1 > 0$, we obtain the expression of $\nabla$:

$$ x_1 = \frac{\sqrt{2 \bar{P}_{11}(t) + \{ \bar{P}_{12}(t)x_2 + \dots + \bar{P}_{1n}(t)x_n - g(t) \}^2 - \{ \bar{P}_{12}(t)x_2 + \dots + \bar{P}_{1n}(t)x_n - g(t) \}}}{\bar{P}_{11}(t)} \quad (\text{III.6.9}) $$

$$ = \frac{2}{\sqrt{2 \bar{P}_{11}(t) + \{ \bar{P}_{12}(t)x_2 + \dots + \bar{P}_{1n}(t)x_n - g(t) \}^2 + \{ \bar{P}_{12}(t)x_2 + \dots + \bar{P}_{1n}(t)x_n - g(t) \}}} \quad (\text{III.6.10}) $$
In this case, the surface $\tilde{\gamma}$ is the plane $x_1=0$.

Moreover, it is clear that we can extend the method of reasoning of II.3 to this case. Then, can we obtain the similar results as Theorem 1 ?

### III.7 Calculation of $\Lambda$

Examining carefully the process of discussion in II.4, it is seen that we can use the same method as in II.4 in order to calculate the surface $\Lambda$.

For the case where $0 < \xi(t,x) < x_1$, we obtain from (II.4.6), (II.4.7) and (III.5.2) the expression of $\mathcal{H}(t,x)$:

$$ 
\begin{aligned}
\mathcal{H}(t,x) = & \frac{\partial \varphi(t, x_1 - \xi(t,x), x_2, \dots, x_n)}{\partial t} + g(t) \xi(t,x) \\
& + \sum_{i,j=1}^{n} \frac{\partial \varphi(t, x_1 - \xi(t,x), x_2, \dots, x_n)}{\partial x_i} A_{ij}(t) x_j \\
& + \frac{1}{2} x' Q(t) x .
\end{aligned} 
\quad (\text{III.7.1}) 
$$

$$ \text{Putting } \bar{\xi} = (\xi(t,x), 0, \dots, 0), \quad (\text{III.7.2}) $$

we can calculate as follows:

$$ 
\begin{aligned}
\frac{\partial \varphi(t, x - \bar{\xi})}{\partial t} &= \frac{1}{2} (x - \bar{\xi}) \cdot \frac{d\bar{P}(t)}{dt} (x - \bar{\xi}) \\
&= \frac{1}{2} (x - \bar{\xi}) \cdot \{ -A(t) \cdot \bar{P}(t) - \bar{P}(t) A(t) - Q(t) \} (x - \bar{\xi})
\end{aligned} 
\quad (\text{III.7.3}) 
$$

and

$$ \frac{\partial \varphi(t, x - \bar{\xi})}{\partial x_i} = \sum_{j=1}^{n} \bar{P}_{ij}(t) (x_j - \bar{\xi}_j) . \quad (\text{III.7.4}) $$

Substituting (III.7.3) and (III.7.4) into (III.7.1) and using (III.2.7), we obtain

$$ 
\begin{aligned}
\mathcal{H}(t,x) = & \frac{1}{2} (x-\bar{\xi}) \cdot \{ -A(t) \cdot \bar{P}(t) - \bar{P}(t) A(t) - Q(t) \} (x-\bar{\xi}) \\
& + x' A(t) \cdot \bar{P}(t) (x-\bar{\xi}) + \frac{1}{2} x' Q(t) x + g(t) \bar{\xi}(t,x)
\end{aligned} 
$$
$$ 
\begin{aligned}
&= -\frac{1}{2}(x - \xi) Q(t) (x - \xi) + \xi' A(t) \bar{P}(t) (x - \xi) + \frac{1}{2} x' Q(t) x + \frac{1}{2} g(t) \xi(t, x) \\
&= \frac{1}{2} Q_{11}(t) x_1^2 - \frac{1}{2} Q_{11}(t) (x_1 - \xi(t, x))^2 + \frac{1}{2} g(t) \xi(t, x) \\
&\quad + (\xi(t, x), 0, \dots, 0) A(t) \bar{P}(t) (x_1 - \xi(t, x), x_2, \dots, x_n)
\end{aligned} 
\quad (III.7.5) $$

According to (III.5.2), we have

$$ x_1 - \xi(t, x) = - \left\{ \bar{P}_{12}(t) x_2 + \dots + \bar{P}_{1n}(t) x_n - g(t) \right\} / \bar{P}_{11}(t) > 0. \quad (III.7.6) $$

Substituting (III.7.6) into (III.7.5), it is easily seen that $\underline{\mathcal{L}}(t,x)$ is a quadratic function of $x_1$. Therefore, from $\underline{\mathcal{L}}(t,x)=0$, we can determine the surface $\Lambda$.

If $\xi(t,x)=x_1$, we modify (II.4.8) and obtain

$$ 
\begin{aligned}
\underline{\mathcal{L}}(t,x) &= \dot{g}(t) \xi(t,x) + g(t) \left\{ \sum_{i=1}^{n} A_{1i}(t) x_i \right\} + \frac{D \varphi(t, 0, x_2, \dots, x_n)}{Dt} \\
&\quad + \frac{1}{2} x' Q(t) x
\end{aligned} 
\quad (III.7.7) $$

$$ 
\begin{aligned}
&= \dot{g}(t) \xi(t,x) + g(t) \left\{ \sum_{i=1}^{n} A_{1i}(t) x_i \right\} + \frac{1}{2} Q_{11}(t) x_1^2 \\
&\quad + \frac{1}{2} \sum_{i,j=2}^{n} \left\{ \bar{P}_{ij} + \sum_{k=2}^{n} (\bar{P}_{kj} A_{ki} + \bar{P}_{ki} A_{kj}) \right\} x_i x_j.
\end{aligned} 
\quad (III.7.8) $$

Thus, $\underline{\mathcal{L}}(t,x)$ is again the quadratic form in $x_1$ and we can determine the surface $\Lambda$ from $\underline{\mathcal{L}}(t,x)=0$.

In the application of these formula, we must use properly (III.7.5) and (III.7.8).

### III.8 Determination of Optimal Policy

In view of the discussions and the calculations up to the previous sections, it is seen that the necessary assumptions and conditions are all satisfied.

If we define the regions $\Gamma^+$, $\Gamma^-$, $\Lambda^+$ and $\Lambda^-$ in the same way as in chapter II, the optimal policy can be determined in the regions $\Gamma \cap \Lambda^+$, $\Gamma^+ \cap \Lambda^-$, $\Gamma^- \cap \Lambda^+$ and $\Gamma^- \cap \Lambda^-$. As to the configuration of these regions and the trajectories, it is easily seen after careful consideration that Theorem 1 is also valid for the problem studied in this chapter.
Therefore, completely in the same way as in chapter II, we can
obtain the Free Boundary and determine the optimal impulse control
at least theoretically. Moreover, we can make use of the same
procedure of modification for the surface $\Gamma$ and $\Lambda$. Using (III.4.5),
we can calculate $\varphi_1(t,x)$ analytically corresponding to (II.6.1).
But, this procedure of modification is a little complicated and it is
better to study the practical problem in detail.

### III.9 Concluding Remarks to Chapter II

The condition (III.2.1) is the most important condition for the
study of "Linear Quadratic" problem of the impulse control theory.
It is an essential condition to be satisfied by the system.

The conditions (III.2.7) and (III.2.9) are not essential. We
can do without them. (III.5.6), (III.5.7) and (III.5.8) are the
conditions on both the system and the performance index. We must
be careful about these conditions in the study of the particular
problem. We can treat in the same way the case where $1 + g(t)\xi$ is
replaced by $1 + g_1(t)\xi + g_2(t)\xi^2$.
# IV. OPTIMAL DESIGN OF THE AERATOR SYSTEM ALONG THE LONG POLLUTED RIVER

## IV.1 Introduction

One of the most important problem of the environmental control is to improve the quality of the water polluted by the waste water from the industries and the cities. Usually, the quality of water is described by the concentrations of DO(Dissolved Oxygen) and BOD(Biochemical Oxygen Demand). If the level of BOD concentration is high, there are much pollutant in the water, which consumes plenty of DO in the course of disintegration by the bacteria in the water. The decrease of DO under a certain level is fatal to the livings in the water. Moreover, such highly polluted water may not be suitable for drinking, agriculture and industry. By blowing the air or the oxygen into the water, we can achieve the effect of artificial aeration, by which DO increases discontinuously. Dams and rapids along the river has the same effect.

Thus, if we construct the aeration system along the river, we can improve the quality of its water and keep the quality better than the standards decided by law. But, it may be expensive to construct and operate the aeration system. Hence, it is reasonable in a sense that we look for the compromising point between the cost and the quality.

Therefore, there arises a problem: how to construct or control the aeration system "optimally" ?

This is the interesting problem for the application of the modern optimization theory. There appeared many studies on this problem in these several years, with different criteria. Some of them assume the availability of the distributed aeration system ($Lin^{2,3}$, $Hullett^4$), which is difficult to realize for the case of the long polluted river.
In this respect, a system of aerators is a more realistic approach (Sayama⁵). But, it seems to the author that their formulation is not satisfactory, because the square term of control is included in the performance index. This square term of control is almost necessary for the application of the optimal control theory but difficult to give the practical meanings.

One of the best way to formulate and solve this problem is to separate the optimal design problem from the optimal control problem. That is, we study the optimal design problem with the performance index:

The construction cost + water quality
so as to determine the optimal locations and the optimal capacities of
the aerators along the polluted river. When the optimal design problem
is solved, we are to study the optimal control problem with the performance
Index: The operation cost + water quality
so as to adapt the system to the varying conditions, supposing that we
cannot change the locations and capacities of the aerators. Nevertheless
to say, it is best to reintegrate the both problems.

In this chapter, we study the optimal design problem. To express
the quality of water along the river, we use the partial differential
equation of the first order with the Streeter-Phelps piston flow model.
But, it is easily seen that the optimal design changes as the variable
conditions of the river. Therefore, for the purpose of the optimal
design, it is necessary to fix a certain steady condition of the river,
industries and so on. For example, the worst condition. As to the
steady state, we can describe the quality of the water along the river
by the ordinary differential equations, which is obtained by considering
a certain mass of water travelling down the river or the characteristic
curves of the partial differential equations. In this formulation,
the effect of the aerator is expressed by the impulsive control,
because the DO concentration of a certain mass of water increases
discontinuously by the aerator.

Therefore, we can formulate and solve the optimal design problem
by means of the Impulse Control Theory. (Bensoussan and Lions¹).

## IV.2 Description of the Problem

### IV.2.1 Description of the river

The river studied in the following is L km long. The water polluted
by the industrial and municipal wastes flows into the river at $n_1$ different
spots. In addition, the water for the service or the agricultural use
is taken from this river at $n_2$ different spots. These spots are expressed
by $x_1, x_2, \dots, x_n$ ($n=n_1+n_2$):

$$x_0=0 < x_1 < x_2 < \dots < x_n < x_{n+1} =L$$

where x expresses the distance along the river. The interval $I=[0,L]$ is
divided into $n+1$ subintervals

$$I_j = [x_j, x_{j+1}), j = 0,1,\dots,n.$$

$F(t,x)$, $D(t,x)$ and $L(t,x)$ are the flow-rate [$m^3/min$], DO deficit [ppm]
and BOD concentration [ppm], respectively. $t$ is the time. DO deficit is
defined by

$$D(t,x) = C_s - \text{DO concentration},$$

where $C_s$ is the saturation value of DO and the function of water temperature.

We assume that the pattern of the river flow can be approximately
expressed by the piston flow and the diffusion effect in the direction of
the flow (x-axis) is negligible. Then, in view of the definition of $x_i$,
it is clear that the "initial" value of the interval $I_j$ does not coincide
with the "terminal" value of the preceding interval $I_{j-1}$. We denote the
flow-rate, DO deficit and BOD concentration of the inflow or outflow at the
spot $x=x_i$ as

$f_i$, $D_i$ and $L_i$, $i = 0, 1, \dots, n$

respectively. Without considering the effects of rainfall, evaporation, underground water and transient phenomena, $F(t,x)$ in the interval $I_j$ is given by

$$F(t, x) = f_0 + f_1 + \dots + f_j \quad (IV.2.1)$$

If we denote the initial values of the interval $I_j$ and the terminal values of the interval $I_{j-1}$ as $D(t, x_j+0)$, $L(t, x_j+0)$ and $D(t, x_j-0)$, $L(t, x_j-0)$, respectively, we have

$$\begin{cases} D(t, \chi_j+0) F_j = D(t, \chi_j-0) F_{j-1} + D_j f_j \\ L(t, \chi_j+0) F_j = L(t, \chi_j-0) F_{j-1} + L_j f_j \end{cases} \quad (IV.2.2)$$

Thus, the initial conditions for the interval $I_j$ are determined as follows:

$$\begin{cases} D(t, \chi_j+0) = D(t, \chi_j-0) + \frac{f_j}{F_j} \{ D_j - D(t, \chi_j-0) \} \\ L(t, \chi_j+0) = L(t, \chi_j-0) + \frac{f_j}{F_j} \{ L_j - L(t, \chi_j-0) \} \end{cases} \quad (IV.2.3)$$

### IV.2.2 Dynamics of the Water Quality

Within the interval $I_j$, we use the Streeter-Phelps model for the piston flow and obtain the partial differential equation:

$$\left\{ \begin{aligned} \left( \frac{\partial}{\partial t} + v_c \frac{\partial}{\partial x} \right) D(t, x) &= -k_1 D(t, x) + k_1 L(t, x) - V \\ \left( \frac{\partial}{\partial t} + v_c \frac{\partial}{\partial x} \right) L(t, x) &= -(k_1 + k_2) L(t, x) \end{aligned} \right. \quad (IV.2.4)$$

$v_j$ is the average velocity of the river flow in the interval $I_j$. $k_1, k_2$ and $k_3$ are constants. $V$ is the control variable and shows the effect of aerator. As is stated in IV.1, the effect of aeration can be expressed by the delta function of $x$ in (IV.2.4).

In reality, $k_1, k_2$ and $k_3$ vary with the temperatures of the air and the water, perturbation by the wind and so on. These coefficients cannot
always be regarded as constants. But, in this report, we study how to
design optimally the aerator system corresponding to the appropriate
steady state flow. The way to adapt to the varying circumstances is to be
studied separately as the optimal control problems.

IV.2.3 Steady State Characteristics

Now, we regard $f_i$, $D_i$, $F_i$, $v_i$, $k_1$, $k_2$ and $k_3$ as constants given.
For this steady state flow, it is easily seen that the partial differential
equation (IV.2.4) can be transformed into the ordinary differential equation:

$$
\left.
\begin{aligned}
\frac{dD}{dt} &= -k_3 D + k_1 L - V \\
\frac{dL}{dt} &= -(k_1 + k_2)L
\end{aligned}
\right\}
\qquad (\text{IV.2.5})
$$

within every interval I_j. (IV.2.5) expresses the behaviour of D and L
along the fundamental characteristics of (IV.2.4) and shows the changing
quality of a certain mass of water running down along the river.

Moreover, the spots x₁, ..., xₙ of inflows and outflows are transformed
into the times t₁, ..., tₙ :

$$
t_0 = 0 < t_1 < \dots < t_n < t_{n+1} = T, \tag{IV.2.6}
$$

where T is easily calculated from L and vᵢ: T = Σ<sub>i=0</sub><sup>n</sup> (xᵢ₊₁ - xᵢ) / vᵢ.

We define the subinterval Iᵢ again:

$$
I_i = [t_i, t_{i+1}), \quad i = 0, 1, \dots, n. \tag{IV.2.7}
$$

At the initial time tᵢ of Iᵢ, D(t) and L(t) have the discontinuities with
the jumps :

$$
\left.
\begin{aligned}
D(t_i + 0) - D(t_i - 0) &= \frac{f_i}{F_i} \{ D_i - D(t_i - 0) \} \\
L(t_i + 0) - L(t_i - 0) &= \frac{f_i}{F_i} \{ L_i - L(t_i - 0) \}
\end{aligned}
\right\}
\quad (\text{IV.2.8})
$$

calculated from (IV.2.3).

For a certain mass of the water travelling down along the river,
the aeration has the effect of reducing its *DO deficit* suddenly with a
certain amount. In other words, the control V in (IV.2.5) can be regarded
as the delta function of t. Thus, the formulation is suitable for the
application of the impulse control theory.

In this chapter, we assume that the DO-deficit is reduced to zero
by the aerator. In other words, the magnitude ξ of the impulse control V
is given by

$$ \xi = D \qquad (IV.2.9) $$

at the point (t,D,L).

**Remark IV.2.1** It is to be noted that we cannot control L(t) by the aeration.

In other words, L(t) is not controllable by the impulse control V.
Even if the DO deficit is reduced to zero, it is easily seen in view
of (IV.2.5) that the DO deficit always increases again so long as
L(t) > 0. Hence, it is necessary to construct the next aerator, which
is often expensive. Therefore, we must look for the harmony between
the water quality and the economy. This is the fundamental structure
of the problem.

**Remark IV.2.2** According to the Streeter-Phelps model (IV.2.5), the BOD concentration L(t) decreases even if there is no dissolved oxygen: the DO deficit = C<sub>s</sub>. It is hardly possible that such phenomenon actually occurs in the water. Therefore, we cannot use the Streeter-Phelps model in such case. But, the optimal design must be studied for the worst condition of the river, that is, for the season when the dissolved oxygen is least. These are incompatible with each other. But, there exists no model for such highly polluted case.

### IV.2.4 Performance Index

The necessity of the aeration system along the river is related with
the standards of the water quality, which is often the product of compromise
between the economy and the environment. In this report, we adopt the policy
to minimize the following performance index:
$$J \equiv N + \frac{1}{2} \int_{0}^{T} k D(t)^2 dt + \frac{1}{2} P_0 D(T)^2 + \frac{1}{2} \sum_{\substack{\text{outflow} \\ \text{spots}}} P_j^2 D(t_j)^2, \quad (\text{IV.2.10})$$

where N is the number of the aerators whose construction cost is normalized as 1. The second term is concerned with the quality of water along the river. The third term expresses the constraint at the outlet to the sea or to the other river. The fourth term shows that at the outflow spots, the quality of water should be kept better. The coefficients k, $p_0$ and $p_j$s are to be determined from the comparison of the obtained optimal solution $D(t)$ and $L(t)$ with the standards of the quality of water.

Therefore, there is a possibility that the best solution is not to construct the aerator.

It is to be noted that the variation of the flow-rate along the river is not explicitly contained in (IV.2.5). Furthermore, it seems to be natural that the power needed to decrease the DO deficit with a certain amount $\xi_i$ varies with the flow-rate of the river. Thus, it is probable that the following performance index is more appropriate than (IV.2.10).

$$J \equiv \alpha N + \sum_{i=1}^{N} \beta(t_i) \xi_i + \frac{1}{2} \int_{0}^{T} k D(t)^2 dt - \frac{1}{2} P_0 D(T)^2 + \frac{1}{2} \sum_{\substack{\text{outflow} \\ \text{spots}}} P_j^2 D(t_j)^2, \quad (\text{IV.2.11})$$

where $\alpha$ is the fixed cost of construction of aerator and $\beta(t)$ changes with the interval $I_k$.

Though (IV.2.11) is more practical than (IV.2.10), we only study (IV.2.10) in this report. Thus, we have formulated the problem to which the theory of Chapters II and III can be applied obviously.

## IV.3 Behaviour of the Free System

The discontinuity (IV.2.8) can be regarded as the deterministic impulsive disturbance $w_D(t,D)$ and $w_L(t,L)$ which act on the system at the times: $t_1, t_2, \dots, t_n$ with the magnitudes given by (IV.2.8). Hence, the system is described as follows.
$$
\begin{equation}
\left.
\begin{aligned}
\dot{D} &= -k_3 D + k_1 L - V + w_D(t, D), \\
\dot{L} &= -(k_1 + k_2)L + w_L(t, L).
\end{aligned}
\right\}
\tag{IV.3.1}
\end{equation}
$$

IV.3.1 Analysis on the (D,L) Phase Plane

In order to apply the general theory of chapters I and III, we
examine whether the necessary assumptions are satisfied or not. In the
first place, we will study the properties of the Free System without V,
$w_D(t,D)$ and $w_L(t,L)$:

$$
\left.
\begin{aligned}
\dot{D} &= -k_3 D + k_1 L, & D(0) &= D_0 \\
\dot{L} &= -k_2 L, & L(0) &= L_0
\end{aligned}
\right\}
\qquad (\text{IV.3.2})
$$

$$
\text{where } k_1 \equiv k_1+k_2, \ k_1, k_2, k_3 > 0. \tag{IV.3.3}
$$

In (IV.3.2), $L(t)$ is calculated as follows:

$$
L(t) = e^{-k_1'(t-t_0)} Z(t_0). \tag{IV.3.4}
$$

Usually, the BOD concentration is positive. Thus, $L_0 > 0$ and $L(t) > 0$.
Therefore, if $D = 0$, we have

$$
\dot{D} = k_1 L > 0 \tag{IV.3.5}
$$

and the condition (III.2.1) is always satisfied. In view of (IV.2.10),
it is shown that (III.2.7) and (III.2.9) are satisfied. Thus, we can
apply the results of chapter III to the optimal design problem of this
chapter.

(0,0) is the only critical point of (IV.3.2). In view of (IV.3.3),
it is seen that the trajectory (D(t), L(t)) approaches to (0,0) as the
time t tends to infinity. That is, after the sufficiently long period,
the BOD concentration and the DO deficit approach to zero and the water
becomes purified.

Since the DO deficit is always positive, we define the region $\mathcal{D}_1$:

$$
\mathcal{D}_1 \equiv \{ (D, L) : D \ge 0, L \ge 0 \}. \tag{IV.3.6}
$$

Considering (IV.3.5) and the fact that $\frac{dI}{dt} = 0$ if $L = 0$, we know that $(D(t), L(t)) \in \mathcal{D}_1$ if $(D_o, L_o) \in \mathcal{D}_1$.
Furthermore, the DO deficit increases in the region: $k_1L > k_3D$, while the BOD concentration always decreases in $\mathcal{O}_1$ (Fig.IV-1). This means the decrease of the dissolved oxygen by the decomposition of the organic pollutants is greater than the increase of the dissolved oxygen by the natural aeration. As is seen from the above discussion, the Streeter-Phelps model expresses the phenomena well.

However, as pointed out in Remark IV.2.2, the BOD concentration decreases with the rate $dL/dt = -(k_1+k_2)L$ according to the Streeter-Phelps model, even under the condition of little dissolved oxygen. In spite of this defect, we will continue the analysis with (IV.2.5), because it is shown that the model is sufficiently good for the small DO deficit.

The effect of inflow is characterized by the discontinuity (IV.2.8) and shown by the trajectory AB of Fig.IV-1. By means of the impulse control, the trajectory is pulled back to the L-axis and shown by EF of Fig.IV-1.

### IV.3.2 Calculation of $\varphi(t,D,L)$

According to the order of steps of II.6.1, the first step to be taken is to calculate $\varphi(t,D,L)$ given by (III.4.1).

$$ \text{Putting} \quad A = \begin{pmatrix} -k_3 & k_1 \\ 0 & -k_1 \end{pmatrix}, \qquad (\text{IV.3.7}) $$

we can calculate the fundamental matrix $\bar{E}(t,t')$ of (IV.3.2) as follows.

$$ \bar{E}(t,t') \equiv \exp A(t-t') = \begin{pmatrix} e^{-k_3(t-t')} & \frac{k_1}{k_3-k_1} \{e^{-k_1(t-t')} - e^{-k_3(t-t')}\} \\ 0 & e^{-k_1(t-t')} \end{pmatrix}. \quad (\text{IV.3.8}) $$

Moreover, putting

$$ Q = \begin{pmatrix} k & 0 \\ 0 & 0 \end{pmatrix} \qquad (\text{IV.3.9}) $$

$$ P_0 = \begin{pmatrix} P_0 & 0 \\ 0 & 0 \end{pmatrix} \qquad (\text{IV.3.10}) $$

and substituting (IV.3.8), (IV.3.9) and (IV.3.10) into (III.4.2), we have
$$ \bar{P}(t, T; P_0) \equiv \begin{pmatrix} \bar{P}_{11}(t, T; P_0) & \bar{P}_{12}(t, T; P_0) \\ \bar{P}_{12}(t, T; P_0) & \bar{P}_{22}(t, T; P_0) \end{pmatrix} \qquad (\text{IV.3.11}) $$

where

$$ \bar{P}_{11}(t, T; P_0) = k \left\{ \frac{1}{2k_3} - \left(\frac{1}{2k_3} - \mu\right) e^{-2k_3(T-t)} \right\}, \qquad (\text{IV.3.12}) $$

$$ \bar{P}_{12}(t, T; P_0) = \frac{\kappa \kappa_1}{\kappa_3 - \kappa_1} \left\{ \frac{\kappa_3 - \kappa_1}{2\kappa_3(\kappa_1 + \kappa_3)} - \left(\frac{1}{\kappa_1 + \kappa_3} - \mu\right) e^{-(\kappa_1 + \kappa_3)(T-t)} \right. \\ \left. + \left(\frac{1}{2\kappa_3} - \mu\right) e^{-2\kappa_3(T-t)} \right\}, \qquad (\text{IV.3.13}) $$

$$ \bar{P}_{22}(t, T; P_0) = k \left( \frac{\kappa_1}{\kappa_3 - \kappa_1} \right)^2 \left\{ \frac{(\kappa_1 - \kappa_3)^2}{\kappa_1 \kappa_3 (\kappa_1 + \kappa_3)} - \left(\frac{1}{2\kappa_1} - \mu\right) e^{-2\kappa_1(T-t)} \right. \\ \left. + 2\left(\frac{1}{\kappa_1 + \kappa_3} - \mu\right) e^{-(\kappa_1 + \kappa_3)(T-t)} - \left(\frac{1}{2\kappa_3} - \mu\right) e^{-2\kappa_3(T-t)} \right\}. \qquad (\text{IV.3.14}) $$

$\mu$ is defined by

$$ \mu = \frac{P_0}{k} \qquad (\text{IV.3.15}) $$

and the performance index is expressed as

$$ J = N + k \left[ \frac{1}{2} \int_{0}^{T} D(t)^2 dt + \frac{1}{2} \mu D(T)^2 \right] \qquad (\text{IV.3.16}) $$

when there is no inflows or outflows. Hence, $k$ shows the ratio of the criteria of water quality to the construction cost of the aerator which is normalized as 1. $\mu$ shows the ratio of the water quality at the outlet to the quality all along the river. Therefore, if we take $k$ sufficiently large, we need many aerators. If $k$ is too small, we need not construct any aerators. Considering these properties, it is clear that $k$ is to be determined by means of the comparison between the obtained optimal trajectory and the standards of the water quality along the river.

Using (IV.3.11) in (III.4.1), we have

$$ y(t, T; D, L) = \frac{1}{2} (D, L) \bar{P}(t, T; P_0) (D, L)' \qquad (\text{IV.3.17}) $$
Suppose that we operate the system from $(t_1, D, L)$ to the time $t_2$ along the trajectory of Free System (IV.3.2). Then, the second term of (IV.3.16) during the interval $[t_1, t_2]$ can be calculated as follows:

$$ \bar{\varphi}(t_1, t_2; D, L) = \frac{1}{2} (D, L) P(t_1, t_2) (D, L)' \quad (IV.3.18) $$

where

$$ P(t_1, t_2) = \bar{P}(t_1, t_2; 0). \quad (IV.3.19) $$

### IV.3.3 Effect of Inflow and Outflow

The effect of inflows and outflows are expressed by $w_D(t, D)$, $w_L(t, L)$ and the fourth term of (IV.2.10). In the section IV.6, we will mention the analysis of the case with inflows and outflows.

#### IV.4 Optimal Design 1. The Case without Inflows and Outflows

In this section, we study how to determine the optimal solution for the simplest case where there is no inflow or outflow and no need of the modification of the Free Boundary.

##### IV.4.1 Calculation of $\bar{\xi}$

According to the procedure of II.6.1, the next step is to calculate $\xi(t, D, L)$, which is given by (IV.2.9):

$$ \xi(t, D, L) = D. \quad (IV.4.1) $$

In view of (IV.3.1) and (IV.3.4), the BOD concentration $L(t)$ is known on the whole interval $[0, T]$, even if there exists the disturbance $w_L(t, L)$. Therefore, we can study our problem on $(t, D)$-space in place of $(t, D, L)$-space.

Now, we proceed to the step 3 and calculate the curve $\bar{\xi}$. We can apply (III.6.9) and (III.6.10) to this case and obtain

$$ D = \frac{\sqrt{2 \bar{P}_{11}(t) + \{\bar{P}_{12}(t)L\}^2}}{\bar{P}_{11}(t)} - \frac{\bar{P}_{12}(t)L}{\bar{P}_{11}(t)} \quad (IV.4.2) $$

$$ = \frac{2}{\sqrt{2 \bar{P}_{11}(t) + \{\bar{P}_{12}(t)L\}^2 + \bar{P}_{12}(t)L}} \quad (IV.4.3) $$

$$ \text{where } \bar{P}_{ij}(t) = \bar{P}_{ij}(t, T; P_0), \quad i, j = 1, 2. \quad (IV.4.4) $$
Substituting (IV.3.4) into (IV.4.3), we know the curve $\tilde{\Gamma}$ expressed in $(t, D)$-space:

$$D = \frac{2}{\sqrt{2 \bar{P}_{11}(t) + \{\bar{P}_{12}(t) \bar{e}^{k_1 t} L_0\}^2 + \bar{P}_{12}(t) \bar{e}^{k_2 t} L_0}} \quad (\text{IV.4.5})$$

If the parameters $k$ and $p_0$ of the performance index satisfy the inequality:

$$\mu \equiv \frac{p_0}{k_2} < \frac{1}{2k_3}, \qquad (\text{IV.4.6})$$

$D(t)$ defined by (IV.4.5) is a monotone increasing function. Thus, for convenience of analysis, we assume (IV.4.6) in the following. (IV.4.6) shows that the water quality at the outlet is rated lower than the water quality along the river, which is natural for the case of the long river.

Finally, the curve $\tilde{\Gamma}$ is t-axis on $(t, D)$-plane.

### IV.4.2 Calculation of $\Lambda$

Next, we proceed to the step 4 of II.6.1. Considering (IV.4.1), we can calculate the curve $\Lambda$ by (III.7.8):

$$\Phi(t, D, L(t)) = \frac{D \varphi(t, 0, L(t))}{Dt} + \frac{1}{2} k D^2 = 0, \quad (\text{IV.4.7})$$

where we used the fact that $g(t) = 0$ in (IV.2.10). Calculating the total derivative of (IV.4.7), we obtain

$$\Phi(t, D, L(t)) = \frac{1}{2} \{ k D^2 - 2 k_1 \bar{P}_{12}(t) L(t)^2 \} = 0. \quad (\text{IV.4.8})$$

Using (IV.4.6) and the fact that $k_3$ is greater than $k_1$ and $k_1'$ in general, it is known that $\bar{P}_{12}(t)$ is monotone decreasing on $[0, T]$:

$$\mu \equiv \frac{p_0}{k_2} < \frac{1}{2k_3} < \frac{1}{k_1' + k_3}, \quad (\text{IV.4.9})$$

$$\bar{P}_{12}(t) > 0, \quad 0 \le t < T \quad \text{and} \quad \bar{P}_{12}(T) = 0. \quad (\text{IV.4.10})$$

In view of (IV.4.10), we obtain the explicit expression of the curve $\Lambda$ from (IV.4.8) as follows:

$$D = K_1(t) L(t), \quad (\text{IV.4.11})$$
where $K_1(t)$ is defined by

$$K_1(t) = \sqrt{\frac{2k_1}{k} \bar{P}_{12}(t)} \quad (IV.4.12)$$

According to (IV.4.1), the curve $\tilde{\Lambda}$ coincides with the t-axis on $(t, D)$-plane.

Moreover, by means of (IV.4.9), it is shown that $\bar{P}_{11}(t)$ is also an monotone decreasing function on $[0, T]$. In addition, (IV.3.4) shows that $L(t)$ is also monotone decreasing. Therefore, $D(t)$ defined by (IV.4.11) is a monotone decreasing function on $[0, T]$ and satisfies

$$D(T) = 0. \quad (IV.4.13)$$

Applying Theorem 1 to the results obtained above, we obtain Fig.IV-2 which shows the configuration of the curves $\tilde{P}$, $\tilde{\Lambda}$ and the trajectories.

### IV.4.3 Determination of Free Boundary

According to the theory of Chapters II and III, we can determine first the Free Boundary on Fig.IV-2, and then the optimal impulse control for the arbitrary initial value $D_0$. Free Boundary is composed of the part $P_1Q$ of $\tilde{\Lambda}$ and the trajectory $P_1R_2$ of the Free System (IV.3.2). The optimal impulse control is exerted only on the interval $[0, T_1]$.

For example, for the initial value $D_0$ of the DO deficit shown in Fig.IV-2, it is optimal to exert the impulse control at time $T_1$, which corresponds to the location $X_1$ easily calculated along the river. Therefore, the optimal design for the steady state $(k_1, k_2, k_3, L_0, D_0)$ and the performance index $(k, \mu)$ is to construct an aerator at the location $x=X_1$.

In the following, we consider the influence of the variations of these parameters on the optimal design. First, if the BOD concentration $L_0$ increases, the curve $\tilde{\Lambda}$ moves upward and the curve $\tilde{P}$ moves downward, which is seen from (IV.4.5) and (IV.4.11). Thus, the crossing point $P_1$ moves to the right. Therefore, the point $R_1$ also moves to the right. If the point $R_1$ becomes situated on the positive t-axis, it is necessary to modify the Free Boundary.
The modification of the Free Boundary is studied in the next section and there arises the case where it is necessary to exert the impulse control more than twice. Thus, the increase of the initial BOD concentration results in the increase of the impulse control, which is the reasonable result from the physical point of view.

If $k$ is taken to be smaller, it is easily seen that only the curve $\Gamma_1$ moves upward and the point $P_1$ moves to the left until $\tau_1 < 0$. That is, if we make little of the quality of water, there is no need of the aerator, which is also the natural conclusion. It is also possible to give the appropriate interpretation to the other parameters.

## IV.5 Optimal Design 2. Modification of Free Boundary

According to the procedure given in the section II.6 of chapter II, we proceed to the calculation for the case where the modification is necessary. We discuss on the $(t,D)$-plane. $s_1$ is supposed to be positive. We calculate $\varphi_1(t,D,L(t))$ as if the curve $\Lambda$ coincide with the Free Boundary on the whole interval $[0, \tau_1]$ of Fig.IV-3. In Fig.IV-3, the trajectory of (IV.3.2) starting from the point $(t',D')$ reaches the curve $\Lambda$ at $(t,D)$ and jumps to the $t$-axis by the impulse control. After that, we follow again the trajectory of the Free System (IV.3.2) until $t=T$. If the time $t$ of the impulse control is before the time $s_1$ of Fig.IV-3, it is necessary to exert the impulse control again or several times more. But, even for such case, we can modify the Free Boundary repeatedly and obtain the optimal one finally, as is stated in chapter II.

Using (IV.3.17) in (II.6.1), we obtain

$$ \varphi_1(t', D', L(t')) \equiv \bar{\varphi}(t', t; D', L(t')) + \varphi(t, T; 0, L(t)). \quad (IV.5.1) $$

Considering (IV.2.9), we have the definition of the curve $\Gamma_1$ corresponding to $\varphi_1$ as follows:

$$ \bar{\Phi}_1(t', D') \equiv \varphi_1(t', D', L(t')) - \{1 + \varphi_1(t', 0, L(t'))\} = 0. \quad (IV.5.2) $$
The definition of the curve $A_1$ is given from (II.6.10):

$$ \Phi_1(t', D') \equiv \frac{D \varphi_1(t', 0, L(t'))}{Dt'} + \frac{1}{2} k D'^2 = 0. \quad (IV.5.3) $$

It is shown that the existence and the uniqueness of these curves are assured in view of the discussions in chapters II and III.

$\frac{D}{Dt'}$ of (IV.5.3) shows the direction derivative along the t-axis.

Since Theorem 2 of chapter II can be applied to the curves $\Gamma_1, A_1$ and the trajectories of (IV.3.2), we have only to follow the modification procedure $(\Gamma_2, A_2) \to (\Gamma_3, A_3) \to \dots$, in order to obtain the Free Boundary and determine the optimal impulse control.

### IV.5.2 Iteration

First, we calculate the direction derivative $\frac{D}{Dt'}$ of (IV.5.3):

$$ \frac{D \varphi_1(t', 0, L(t'))}{Dt'} = \frac{\partial \bar{\varphi}(t', t: 0, L(t'))}{\partial t'} + \frac{\partial \bar{\varphi}(t', t: 0, L(t'))}{\partial L} \frac{dL(t')}{dt'} + \frac{\partial \bar{\varphi}(t', t: 0, L(t'))}{\partial t} \frac{\partial t'}{\partial t'} + \frac{D \varphi_1(t, T: 0, L(t'))}{Dt} \frac{\partial t'}{\partial t'} \quad (IV.5.4) $$

where

$$ \frac{D \varphi_1(t, T: 0, L(t'))}{Dt} = \frac{\partial \varphi_1(t, T: 0, L(t'))}{\partial t} + \frac{\partial \varphi_1(t, T: 0, L(t'))}{\partial L} \frac{dL(t')}{dt'} \quad (IV.5.5) $$

and is equal to the second term of (IV.4.8). Therefore, in order to obtain the explicit formula of the curve $A_1$, it is sufficient to calculate the term $\frac{\partial t'}{\partial t'}$. Suppose that the trajectory of (IV.3.2) passing through the point $(t', 0)$ on the t-axis crosses the curve $A$ at the point $(t, D)$. If $t'$ increases by $\delta t'$ ($0 < \delta t' \ll 1$), $t$ increases by $\delta t$. From the ratio $\delta t' / \delta t$, the term $\frac{\partial t'}{\partial t}$, can be calculated.

Using (IV.3.8), the trajectory passing through $(t', 0)$ is given by

$$ \begin{align} D(t) &= \frac{k_1}{k_3-k_1} L_0 e^{-k_1 t'} \{e^{-k_1(t-t')} - e^{-k_3(t-t')}\} \nonumber \\ &= \frac{k_1}{k_3-k_1} L_0 e^{-k_1 t'} e^{-k_1(t-t')} \{e^{-k_1(t-t')} - e^{-k_3(t-t')}\} \nonumber \\ &= \frac{k_1}{k_3-k_1} \{1 - e^{-(k_3-k_1)(t-t')}\} \mathcal{L}(t). \tag{IV.5.6} \end{align} $$
Putting

$$H(x', t) \equiv \frac{k_1}{k_3 - k'_3} \{1 - e^{-(k_3 - k'_3)(x-x')}\}, \quad (\text{IV.5.7})$$

it is seen that $H(t',t)$ is a monotone increasing function of $t$ with $H(t',t') = 0$ and we can express $D(t)$ as

$$D(t) = H(x', t) \mathcal{L}(t). \quad (\text{IV.5.8})$$

Comparing (IV.5.8) with the expression of $\Lambda:(\text{IV.4.11})$, we have

$$H(x', t) \mathcal{L}(t) = K_1(t) \mathcal{L}(t) \quad (\text{IV.5.9})$$

or

$$H(x', t) = K_1(t) \quad (\text{IV.5.10})$$

at the crossing point $(t,D)$ of $D(t)$ with $\Lambda$.

Differentiating the both side of (IV.5.10) by $t'$, we have

$$\frac{\partial H}{\partial t'} \frac{\partial x}{\partial t'} + \frac{\partial H}{\partial x'} = \frac{dK_1(x)}{dt} \frac{\partial x}{\partial t'} \quad (\text{IV.5.11})$$

Then, we obtain $\frac{\partial x}{\partial t'}$, as follows:

$$\frac{\partial x}{\partial t'} = \frac{\frac{\partial H(x', t)}{\partial t'}}{\frac{dK_1(x)}{dt} - \frac{\partial H(x', t)}{\partial t}} \quad (\text{IV.5.12})$$

Differentiating $H(t',t)$, we have

$$\left. \begin{aligned} \frac{\partial H(x', t)}{\partial t'} &= -k_3 e^{-(k_3 - k_1')(t-t')}, \\ \frac{\partial H(x', t)}{\partial t} &= k_3 e^{-(k_3 - k_1')(t-t')} \end{aligned} \right\} \quad (\text{IV.5.13})$$

Using (III.4.6), (IV.4.12) yields

$$\frac{dK_1(x)}{dt} = \sqrt{\frac{k_3}{2k \bar{P}_{12}(x)}} \left\{ -k_3 \bar{P}_{11}(x) + (k_1' + k_3) \bar{P}_{12}(x) \right\}. \quad (\text{IV.5.14})$$

Substituting (IV.5.13) and (IV.5.14) into (IV.5.12), we obtain

$$\frac{\partial x}{\partial t'} = \frac{k_3 e^{-(k_3 - k_1')(t-t')}}{k_3 \bar{P}_{11}(x) + (k_1' + k_3) \bar{P}_{12}(x)} - \sqrt{\frac{k_3}{2k \bar{P}_{12}(x)}} \left\{ -k_3 \bar{P}_{11}(x) + (k_1' + k_3) \bar{P}_{12}(x) \right\}. \quad (\text{IV.5.15})$$
### IV.5.3 Expression of the Curve $\Lambda$

Using (III.4.6), we can calculate

$$ \frac{\partial \bar{\varphi}(t', t: 0, L(t'))}{\partial t'} + \frac{\partial \bar{\varphi}(t', t: 0, L(t'))}{\partial L} \cdot \frac{dL(t')}{dt} = -k_1 \bar{P}_{12}(t', t) L(t')^2 \quad (\text{IV.5.16}) $$

By use of (III.4.8), the third term of (IV.5.4) is calculated as follows:

$$ \frac{\partial \bar{\varphi}(t', t: 0, L(t'))}{\partial t} = \frac{1}{2} \frac{\partial \bar{P}_{22}(t', t)}{\partial t} L(t')^2 = \frac{1}{2} \left( \frac{k_1}{k_3 - k_1} \right)^2 \int e^{-k_1(t-t')} \bar{\varphi}(t', t) L(t')^2 dt' \quad (\text{IV.5.17}) $$

The fourth term of (IV.5.4) is the same as the second term of (IV.4.8):

$$ \frac{D \varphi(t, T, 0, L(t))}{Dt} = -k_1 \bar{P}_{12}(t) L(t')^2 \quad (\text{IV.5.18}) $$

Hence, using

$$ L(t) = e^{-k_1(t-t')} L(t') \quad (\text{IV.5.19}) $$

we have

$$ \frac{D \varphi(t, T, 0, L(t))}{Dt} = -k_1 \bar{P}_{12}(t) e^{-2k_1(t-t')} L(t')^2 \quad (\text{IV.5.20}) $$

Substituting (IV.5.15), (IV.5.16), (IV.5.17) and (IV.5.20) into (IV.5.4), we can calculate $D \varphi_1(t', 0, L(t'))/Dt'$ as the quadratic form of $L(t')$.
Therefore, in view of (IV.5.3), we obtain the expression of $\Lambda_1$:

$$ D(t') = K_2(t') L(t') \quad (\text{IV.5.21}) $$

where

$$ K_2(t) = \sqrt{\frac{1}{k_2} \left[ 2k_1 \bar{P}_{12}(t', t) + \frac{\partial t'}{\partial t} \left\{ 2k_1 \bar{P}_{12}(t) e^{-2k_1(t-t')} - \frac{\partial \bar{P}_{22}(t', t)}{\partial t} \right\} \right]} \quad (\text{IV.5.22}) $$

$$ = \sqrt{\frac{2k_1}{k_2} \bar{P}_{12}(t, t) + \frac{\partial t'}{\partial t} \left\{ k_1(t')^2 e^{-2k_1(t-t')} - \frac{1}{k_1} \frac{\partial \bar{P}_{22}(t', t)}{\partial t} \right\}} \quad (\text{IV.5.23}) $$
By means of $K_2(t')$ and $K_1(t)$, we can express

$$ \frac{D K_1(t', 0, L(t))}{Dt'} = -\frac{1}{2} \kappa K_2(t')^2 L(t')^2, \qquad (IV.5.24) $$

$$ \frac{D K_1(t, T, 0, L(t))}{Dt} = -\frac{1}{2} \kappa K_1(t')^2 L(t')^2. \qquad (IV.5.25) $$

### IV.5.4 Iteration (2)

Equations (IV.4.11), (IV.4.12), (IV.5.8), (IV.5.10) and (IV.5.25) are used to obtain the curve $\Lambda_1$ from the curve $\Lambda$. If we replace (IV.4.11), (IV.4.12) and (IV.5.25) by (IV.5.21), (IV.5.23) and (IV.5.24), respectively, and use (IV.5.8) as it is, we can calculate the curve $\Lambda_2$ from the curve $\Lambda_1$ by replacing $K_1(t)$ in (IV.5.10) by $K_2(t)$. Thus, continuing the procedure of iteration, we can obtain all the necessary curves $\Lambda, \Gamma, \Lambda_1, \Gamma_1, ...$

That is, in order to determine the curve $\Lambda_i$, we have only to use the relations:

$$ D(t') = K_{i+1}(t') L(t'), \qquad (IV.5.26) $$

$$ K_{i+1}(t')^2 = \frac{2\kappa_1}{\kappa} P_{12}(t',t) + \frac{\partial t'}{\partial t} \left\{ K_i(t')^2 e^{-2\kappa_i(t-t')} - \frac{1}{\kappa} \frac{\partial P_{22}(t',t)}{\partial t} \right\}, \qquad (IV.5.27) $$

$$ H(t',t) = K_i(t), \qquad (IV.5.28) $$

$$ \frac{\partial t}{\partial t'} = \frac{\kappa_i e^{-(\kappa_3 - \kappa_i)(t-t')}}{\kappa_i e^{-(\kappa_3 - \kappa_i)(t-t')} - \frac{dK_i(t)}{dt}}. \qquad (IV.5.29) $$

Since $K_i(t)$ defines the curve $\Lambda_i$ and is known, $t$ is fixed by (IV.5.28) corresponding to $t' \in [t_i', t_i]$. The derivative $\frac{\partial t'}{\partial t}$ can be calculated corresponding to the pair $(t',t)$ by means of (IV.5.29). Then, (IV.5.27) yields $K_{i+1}(t)$, with which the curve $\Lambda_i$ is defined by (IV.5.26). For $i=1, \partial K_1(t)/dt$ is given by (IV.5.14). But, for $i=2, 3, ..., $ it is not easy to calculate $\partial K_i(t)/dt$ analytically. Thus, it is more convenient to use the appropriate approximation formula in case of the numerical calculation.
Moreover, we cannot determine the Free Boundary completely only by the curves $\Lambda, \Lambda_1, \Lambda_2, \dots$. As is shown in Fig.IV-5, in order to determine the Free Boundary on $[s_2:s_1]$, it is necessary to calculate the curve $\Lambda_1$ first, and then the curve $\Pi_1$, and finally the crossing point $P_2$. The Free Boundary is determined by the curves $\Lambda, \Lambda_1$ and the trajectories passing through $P_1$ or $P_2$. Furthermore, the interval $[s_2,s_1]$ itself is determined by the crossing time $s_2$ of the trajectory passing through $P_2$. But, it is fairly difficult to calculate the curve $\Pi_1$ explicitly from (IV.5.2). Practically, it is simpler to examine whether the point on $\Lambda_1$ belongs to $\Pi_1^-$ or not.

In the numerical analysis of the example stated in IV.7, the interval $[s_1, \tau_1]$ is divided into the intervals of same length with the dividing points $t_{1i}$ ($i=0,1,\dots,m$). $t_{10}=s_1$ and $t_{1m}=\tau_1$. Then, we calculate $t_{1i}'$ by means of (IV.5.10) on $[s_1',s_1]$. Using (IV.5.21), we obtain the point $D(t_{1i}')$ on the curve $\Lambda_1$. In the next place, we examine whether the point $D(t_{1i}')$ belongs to $\Pi_1^-$ or not. We repeat this process and obtain the point $D(t_{ki}')$ correctly. Then, we know the approximate position of the points $P_2, P_3, \dots$ and $s_2, s_3, \dots$. It is easily seen that the accuracy of this approximation of $P_i$'s and $s_i'$ can be made more precise by increasing the number $m$ of dividing points. Therefore, we can obtain the Free Boundary and determine the optimal impulse control corresponding to the initial value $D_0$ approximately, but with the arbitrary accuracy.
## IV.6 Optimal Design 3.Cases with Inflows and/or Outflows

### IV.6.1 The Case with Inflows

Suppose that the polluted water flows into the river at the spot corresponding to the time $t_1$ ($0 < t_1 < T$). Then, $D(t)$ and $L(t)$ has the discontinuity corresponding to (IV.2.8). The width of the discontinuity of $D(t)$ depends on $D(t_1-0)$ and $D_1$, and $L(t)$ on $L(t_1-0)$ and $L_1$.

Therefore, we can regard $L(t)$ as known on the both intervals $[0, t_1)$ and $(t_1, T]$. Using $L(t)$, we can solve the problem on $(t_1, T]$. In other words, for the arbitrary $D(t_1+0)$, we can determine the optimal impulse control and calculate the value of the performance index corresponding to it.

If the point $(t_1, D(t_1+0))$ is in $\Lambda_i^+$, we exert the impulse control at the time $t_1+0$ and pull back the state to t-axis. Hence, the optimal cost can be calculated in this case. Thus, we can determine the optimal cost $J(D(t_1+0))$ for every $D(t_1+0) \ge 0$.

On the other hand, $D(t_1+0)$ is determined by $D(t_1-0)$ and $D_1$. Since $D_1$ is known apriori, $D(t_1+0)$ is to be regarded as the function of $D(t_1-0)$. Therefore, we can express the optimal cost $J(D(t_1+0))$ as $J(D(t_1-0))$.

Now, we proceed to the optimal design on $[0, t_1)$ by the aid of $J(D(t_1-0))$ and $L(t)$, which is also known on $[0, t_1)$ with the discontinuity at $t_1$. In the first place, we calculate the value of the performance index for the trajectory passing through $(t, D)$ ($0 \le t < t_1$) and without exerting the impulse control until $t=t_1-0$. It is given by

$$ \mathcal{G}(t, D, L(t)) = \frac{1}{2} (D \cdot L(t)) P(t, t_1) (D \cdot L(t))' + J(D(t_1-0)). \quad (\text{IV.6.1}) $$

By means of this $\mathcal{G}(t, D, L(t))$, we can follow the procedure of IV.4 and IV.5 and determine the Free Boundary theoretically. It may be difficult to calculate all the necessary formula analytically. But, we can apply the same procedure of the numerical computation stated in the last part of
the previous section, with which we can obtain the approximation of the
Free Boundary on the whole interval [0,T] with the arbitrary accuracy.
If there are many inflows, we have only to repeat the procedure.

Moreover, if it is necessary to construct an aerator just after the
inflow spot, the result can be interpreted as the necessity of purification
of the inflow. (IV.2.9) assumes that the aerator can fully (100%) reduce
the DO deficit. In practice, this is only 70 - 80 %. In view of this
fact, the above interpretation can be more justified.

IV.6.2 The Case with Outflows

Suppose that the outflow spot corresponds to the time $t_1$. In this
case, there exist no discontinuities of $D(t)$ and $L(t)$. But, we must
consider the fourth term of the performance index (IV.2.10). Hence, there
is the discontinuity of the optimal cost function at $t_1$. Dividing the
interval $[0,T]$ into $[0,t_1)$ and $[t_1,T]$, we can apply the same procedure
as given in IV.6.1.

If it is necessary to construct an aerator just before the outflow
spot, the result may be interpreted as the necessity of purification
of the outflow only.

IV.6.3 The Case with Outflows and Inflows

In this case, we define the intervals $I_j$ by (IV.2.7) and begin the
optimal design from the interval $I_n = [t_n, T]$ to $I_o = [0, t_1)$.
IV.7 The Numerical Example

The procedure of this report is applied to the following optimal
design problem of aeration system. We assume that there is no inflow
or outflow. The coefficients are given as follows:

$$
\kappa_1 = 0.16, \quad \kappa_2 = 0, \quad \kappa_3 = 0.66 \quad [\text{day}^{-1}] \qquad (\text{IV}.7.1)
$$

which are used by Hullett⁴ and Sayama⁵.

The initial data are as follows:

$$
L_0 = 20.0, \quad C_0 = 5.0 \quad [\text{mg/l}] \tag{IV.7.2}
$$

$$
C_5 = 8.0 \quad [\text{mg/l}] \tag{IV.7.3}
$$

$$
D_0 = 3.0 \quad [\text{mg/l}] \tag{IV.7.4}
$$

As to the performance index, we adopt

$$
T = 6.0 \ [\text{day}], \quad \kappa = 0.5, \quad \mu = 0.1 \tag{IV.7.5}
$$

Corresponding to these data, the Free Boundary shown in Fig.IV-6 is
calculated by means of the procedure of this report. For the initial
data given by (IV.7.4), it is seen from Fig.IV-6 and Fig.IV-7 that we
must exert the impulse control four times: at t=0 and on the curves $\Lambda_2$,
$\Lambda_1$ and $\Lambda$. That is, the number of necessary aerators is four and its
locations can be approximately calculated if necessary. The curves $\Lambda$
and $\Lambda_1$ are calculated exactly, by means of the formula obtained in IV-5.
The approximation formula of $dK_2(t)/dt$ is used in the calculation of the
curve $\Lambda_2$. The curves $\tilde{\Lambda}_2$ are not explicitly calculated.

The points Q.V.I. of Fig;IV-6 are obtained by the direct application
of the difference scheme to the Q.V.I., where $\partial u(t,D)/\partial t$ and $\partial u(t,D)/\partial D$
are approximated by

$$
\frac{u((n-1)\Delta t, i\Delta t) - u(n\Delta t, i\Delta t)}{-\Delta t} \quad (\text{IV}.7.6)
$$
and

$$ \frac{u(\text{inst}, (i+1)h) - u(\text{inst}, (i-1)h)}{2h} \qquad (\text{IV.7.7}) $$

The convergence on the t-axis is not good by this difference scheme. Since the value $u(t,0)$ is very important in this example, this defect of bad convergence on t-axis is fatal to solve the problem exactly. We only obtain the curve $\Gamma$ as the Free Boundary. Therefore, we used the exact value $\phi(t,0)$ corresponding to $\Lambda$ as the boundary condition on t-axis. Then, we obtained the points Q.V.I.($u(t,0)$ given) of Fig.IV-6, which is the better result than the former.

Therefore, we can conclude that the result of the direct application of difference approximation to Q.V.I. is very sensitive to the scheme used. This is because the Q.V.I. is hyperbolic type for the deterministic impulse control problem. Hence, it is not recommended to apply the difference scheme directly to Q.V.I. unless more detailed study on the convergence problem is carried out.
## IV.8 Concluding Remarks to Chapter IV

### IV.8 The River Seine around Rouen

With the hope of testing and applying the theory of this report to the real data, the author studied the data about the water quality of the Seine around Rouen measured in 1975, which was possible through the kindness of Mr.Lesouef and Dr.de Marsily, and by the aid of Professor Bensoussan, Mr.Leguay and Mr.Delebecque. Since it is reasonable to adopt the principle that the aeration system should be designed so as to improve the worst condition of the river, the data of July or September is more appropriate for the design.

According to the data measured on the 28th of July, 1975 and the 25th of September, 1975, there exists a range where there exists little dissolved oxygen. This phenomena results from the decrease of water flow (January 650, July 235 m³/sec) and the higher water temperature (January 10, July 22°C). In addition, according to the opinion of Mr.Lesouef, the dissolved oxygen consumed by the organic nitrogen is not negligible, which originates in the activities of the region around Paris.

Therefore, we cannot use the Streeter-Phelps model to describe the water quality around Rouen. The first reason is that the model has the defect of structure in case of little dissolved oxygen, as is stated in IV.2 and IV.3. Secondly, the organic nitrogen is not considered in the model. Finally, it is probable that the effect of tidal action is not negligible, because Rouen is not far from the sea. Since it is usual to express the tidal effect by the diffusion term, we cannot apply the piston flow model to this case.

Thus, the procedure of chapter IV cannot be applied to the real data of the Seine around Rouen. The first thing to be done is to identify the differential equation which describes the real data well. When the good
model is established, we can apply the theory of chapter II to this problem and determine the optimal location of the aerator. In this respect, the intention of the agency is not favourable for the construction of the aeration system, because it is expensive to construct and control the aeration system. On that point, it is to be noted that the aerator can be interpreted as the dam or the water treatment equipment in some cases, as stated in IV.6.1 and IV.6.2.

According to several specialists of meteorology, we are now in the little ice age, which is characterized by little rainfall in Europe.<sup>10)</sup> If their prediction come true, there will be difficult problems in the basin of the river Seine: difficulty of maintaining the navigability of the canal system (occurred in 1971), insufficient supply of the water and the deterioration of its quality, agriculture depression, and so on. Even without considering such extreme cases, it seems probable that one cannot warrant optimism in view of the industrialization and the concentration of the population in future. Is the aerator system useful in such cases ? IV.8.2 Mathematical Model of the Behaviour of Water Quality

The defect of the Streeter-Phelps model and the necessity of other models are already stated. In addition, it takes five days to measure the BOD concentration, which is too long to adapt and control the system to the varying condition of the river. According to the literature, TOD (Total Oxygen Demand) can be measured without time delay<sup>8)</sup>. Thus, it seems desirable that TOD is adopted as the index of the water quality and the mathematical model using TOD is developed.

Supposing that we have to construct the control system of water quality, it is easily seen that there arise many related problems such as the optimal control, optimal observation and optimal synthesis problems.
### IV.8.3 Influence of Water Temperature

If the temperature of water becomes higher, the saturation value of dissolved oxygen decreases. The coefficients $k_1, k_2$, and $k_3$ are also influenced by the temperature of water. If there is a steam power plant on a riverside, we encounter this problem. If it is possible to know the water temperature sufficiently well, we can treat this problem by the method of this chapter, because we can calculate analytically the fundamental matrix of (IV.3.2) with $C_s(t), k_1(t), k_2(t)$ and $k_3(t)$.

### IV.8.4 Stochastic Model

In addition to the temperature of water, there are many disturbances such as the temperature of the air, perturbation of the water surface by wind, chloride ion and so on. Moreover, we have to deal with the nonlinearity and the diffusion effect under certain circumstances. Therefore, it is possible to study the problem in the stochastic formulation from the beginning.

### IV.8.5 Branched Problem

A long river often has the big branches. Then, there arises the problem of designing optimally the whole aeration system. The result will be useful for the comparison of the aerator with the water treatment equipment. This branched problem is interesting from the practical and theoretical viewpoint.
V. CONCLUDING REMARKS

In this paper, we present the procedure to determine directly the
Free Boundary of Q.V.I. for the deterministic Impulse Control problem.
The procedure is applied to the optimal design problem of aerator system
to be constructed along the long polluted river.

In chapter II, the optimal Impulse Control problem is studied for
the deterministic nonlinear control system, where the state of the system
is expressed by a scalar. In this case, Q.V.I. is of a hyperbolic type.
Its solution is obtained backwardly from the terminal time by the presented
procedure, whose characteristic is the introduction of the curve $\mathcal{A}$ and
whose principal result is the Theorem 1. The Free Boundary consists of
the curves $\mathcal{A}_1, \mathcal{A}_2, ...$ and the trajectories of the "Free System" without
control or disturbances. The times when the impulse control is to be exerted
are determined only by the curves $\mathcal{A}_1, \mathcal{A}_2, ...$ For the case where it is
necessary to use the impulse control many times, the curve $\mathcal{A}$ is to be
modified. The procedure of modification is also given and results in the
curves $\mathcal{A}_1, \mathcal{A}_2, ...$, where the Theorem 2 is indispensable.

In chapter III, the result of chapter II is extended to the "Linear
Quadratic Problem" of higher dimension. Many formula necessary and useful
for the application in chapter IV are given.

In chapter IV, the results of chapters II and III are applied to the
optimal design problem of aerator system along the long polluted river.
We adopted the Streeter-Phelps linear model to describe the behaviour of
water quality along the river and as the performance index, Construction
Cost + Integrated Square DO deficit. Thus, the problem is of the "Linear
Quadratic" type, to which the result of chapter III can be applied. The
problem is formulated so that we can deal with the case where there are
many inflows of the polluted water and several outflows of the service
water. Many calculated formula are given and the basic idea of the
computational algorithm is stated so that the results of chapter IV can be applied to the practical problem easily. A numerical example is given, where it is shown that the direct application of the difference scheme to Q.V.I. does not necessarily gives the exact solution, which is the problem to be studied more carefully.

As to the real data of the river Seine around Rouen, the method of chapter IV cannot be applied, because the Streeter-Phelps model does not describe well the highly polluted case. First thing to be done is the identification of the system. Other related problems are stated in the Concluding Remarks of chapters II and IV.

## ACKNOWLEDGEMENT

I would like to express my sincere thanks to Professor Lions for his warm encouragement in the course of this work and to Professor Bensoussan for instructive discussions, and earnest assistance in obtaining the real data. My hearty thanks are due to Mr. Leguay for his discussion and encouragement, and his indispensable assistance in approaching the real data.

Thanks are also due to Professor Sayama, whose lecture of September 1974 at the meeting of the methodology subgroup of the Environmental Pollution Control Project⁹) motivated the author to the present study.

Finally, our best thanks are extended to Mr. De Marsily and Mr. Lesouef for offering us the opportunity to know the real data.
REFERENCES

1. A.Bensoussan & J.L.Lions, Nouvelles Methodes en Contrôle Impulsionnel, Applied Mathematics and Optimization, Vol.1, No.4, pp.289-312 (1975)

2. S.H.Lin, Feedforward water quality control by artificial aeration of a thermally and organically polluted stream, Int.J.Systems Sci., Vol.6, No.10, pp.919-932(1975)

3. S.H.Lin, Feedforward water quality control by artificial aeration of streams receiving multiple thermal and organic wastes discharges, ibid., Vol.6, No.10, pp.933-943 (1975)

4. W.Hullett, Optimal estuary aeration:application of distributed parameter control theory, Applied Mathematics and Optimization, Vol.1, No.1, pp:20-63 (1975)

5. E.Sayama, Y.Kameyama & H.Nakayama, Nonlinear Programming applied to water quality management, Research Abstract;Environmental Pollution Control, April 1972 through March 1975, pp.25-24 (1975)

6. H.Sayama & Y.Kameyama, Multiplier method and its application to water quality management (in Japanese), Research Report for Environmental Pollution Control, the year 1974, pp.21-28 (1975)

7. R.Courant & D.Hilbert, Method of Mathematical Physics, Wiley, New York, 1953

8. R.Tomono, Kohgai Yohgo Jiten (in Japanese; Dictionary of the Terminology of Pollution), Ohm Co.Ltd, Tokyo, 1972.

9. Research Abstract;Environmental Pollution Control, April 1972 through March 1975, 1975

10. La crainte des spécialistes: Une nouvelle période glaciaire, Le Figaro, le 16 Avril, 1976
<figure><img src="image_4.png" /><figcaption>Fig.II-1. The Region where $\varphi(t,x)$ is the optimal solution</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig.II-2. A Configuration satisfying Theorem 1</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig.II-3. The Case where the Modification is necessary</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig.II-4. The Procedure of Modification of $\vec{\Lambda}$</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig.II-5. Region where optimal trajectories can exist</figcaption></figure>
D

<figure><img src="image_9.png" /><figcaption>Fig.11-5. Modification of the curve $\Lambda_n$.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig.IV-1. Phase-Portrait of Streeter-Phelps model</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig.IV-2. Configuration of the curves $\Gamma, \Lambda$ and trajectories</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig.IV-3. Calculation of $\phi_1(t', D')$</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig.IV-4. Calculation of ∂t/∂t'</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig.IV-5. Modification of Free Boundary</figcaption></figure>
EXAMPLE. OPTIMAL DESIGN OF AERATOR SYSTEMS

<figure><img src="image_15.png" /></figure>

$$ \dot{D} = -k_3 D + k_1 L - V $$

$$ \dot{L} = -k_1 L $$

$$ J = N + \frac{1}{2}k \left\{ \int_{0}^{T} D(t)^2 dt + \mu D(T)^2 \right\} \rightarrow \min. $$

$$ k_1 = 0.16, k_3 = 0.66, k = 0.5, \mu = 0.1 $$

$$ T = 6.0, L(0) = 20.0 $$

Q.V.I.

Q.V.I. (u(t, 0) given)
<figure><img src="image_16.png" /></figure>

$L(0) = 20.0$

$k_1 = 0.16$

$k_2 = 0.66$