Optimal control of membrane filtration systems

TL;DR

This paper formulates and analyzes an optimal control problem for membrane filtration systems, using a mathematical model to determine the best filtration and cleaning cycles for maximizing net production.

Contribution

It introduces a novel optimal control framework for membrane filtration, characterizing the optimal strategies and revealing complex control structures like singular and switching curves.

Findings

Optimal control strategy includes singular arcs.

Existence of a switching curve before the terminal state.

Possibility of a dispersal curve with multiple optimal strategies.

Abstract

This paper studies an optimal control problem related to membrane filtration processes. A simple mathematical model of membrane fouling is used to capture the dynamic behavior of the filtration process which consists in the attachment of matter onto the membrane during the filtration period and the detachment of matter during the cleaning period. We consider the maximization of the net production of a membrane filtration system (i.e. the filtrate) over a finite time horizon, where control variable is the sequence of filtration/backwashing cycles over the operation time of process. Based on the Pontryagin Maximum Principle, we characterize the optimal control strategy and show that it presents a singular arc. Moreover we prove the existence of an additional switching curve before reaching the terminal state, and also the possibility of having a dispersal curve as a locus where two different strategies are both optimal.

Figures

Equations

$$\dot{m}=f_1(m),$$

$$\dot{m}=-f_2(m),$$

$$\dot{m}=\frac{1+u}{2}{f}_1(m)-\frac{1-u}{2}{f}_2(m)\mathrm{with}m(0)=m_0,$$

$$J_T(m_0,u(\cdot ))={\int }_0^{T}u(t)g(m(t))dt.$$

$$\underset{u(\cdot )\in \mathcal{U}}{\max }{J}_T(m_0,u(\cdot )),$$

$$f_{+}(m):=\frac{f_1(m)+f_2(m)}{2},f_{-}(m):=\frac{f_1(m)-f_2(m)}{2},$$

$$\dot{m}=f_{-}(m)+u{f}_{+}(m),u\in [-1,1].$$

$$H(m,\lambda ,u)=\lambda f_{-}(m)+u\left(\lambda f_{+}(m)+g(m)\right).$$

$$\dot{\lambda }(t)$$

$$=-\lambda (t)f_{-}^{\prime }(m(t))-u(t)\left(\lambda (t)f_{+}^{\prime }(m(t))+g^{\prime }(m(t))\right),$$

$$u(t)\in \arg \underset{\omega \in [-1,1]}{\max }H(x(t),\lambda (t),\omega ),a.e.t\in [0,T].$$

$${}u=\left|\begin{array}[]{cl}+1&\mbox{when }\phi(m,\lambda)>0,\\ -1&\mbox{when }\phi(m,\lambda)<0,\\ \in[-1,1]&\mbox{when }\phi(m,\lambda)=0,\end{array}\right.$$

$$\varphi (m,\lambda ):=\lambda f_{+}(m)+g(m).$$

$$\psi (m):=g(m)\left[f_{-}^{\prime }(m)f_{+}(m)-f_{-}(m)f_{+}^{\prime }(m)\right]+g^{\prime }(m)f_{+}(m)f_{-}(m),m\ge 0.$$

$$\gamma (m):=-\frac{g(m)f_{-}(m)}{f_{+}(m)}m\ge 0.$$

$$\bar{u}:=-\frac{f_{-}(\bar{m})}{f_{+}(\bar{m})}.$$

$$\bar{\lambda }=-\frac{g(\bar{m})}{f_{+}(\bar{m})}.$$

$$\begin{array}[]{lll}\dot{\phi}&=&-(\lambda f_{-}^{\prime}+u(\lambda f_{+}^{\prime}+g^{\prime}))+(\lambda f_{+}^{\prime}+g^{\prime})(f_{-}+f_{+}u)\\[5.69054pt] &=&\lambda(f_{+}^{\prime}f_{-}-f_{-}^{\prime}f_{+})+g^{\prime}f_{-}\\[5.69054pt] &=&\displaystyle g(f_{-}^{\prime}-f_{+}^{\prime}f_{-}/f_{+})+g^{\prime}f_{-}+\phi\,\frac{f_{+}^{\prime}f_{-}-f_{-}^{\prime}f_{+}}{f_{+}},\end{array}$$

$$\dot{\varphi }=\frac{\psi }{f_{+}}+\varphi \frac{f_{+}^{\prime }{f}_{-}-f_{-}^{\prime }{f}_{+}}{f_{+}}.$$

$$\bar{T}:=T-{\int }_{\bar{m}}^{{\bar{m}}_T}\frac{dm}{f_1(m)}\text{mbox}with{\bar{m}}_T:=g^{-1}(\gamma (\bar{m})).$$

$$g({\bar{m}}_T)=\bar{\lambda }{f}_{-}(\bar{m})=-\frac{g(\bar{m})f_{-}(\bar{m})}{f_{+}(\bar{m})}=\gamma (\bar{m}),$$

$$\dot{m}=f_1(m),m(T)={\bar{m}}_T,$$

$$\bar{T}=T-{\int }_{\bar{m}}^{{\bar{m}}_T}\frac{dm}{f_1(m)}.$$

$$J_1(t):={\int }_{\bar{m}}^{m(T)}\frac{g(m)}{f_1(m)}dm={\int }_{\bar{m}}^{{\bar{m}}_T}\frac{g(m)}{f_1(m)}+{\int }_{{\bar{m}}_T}^{m(T)}\frac{g(m)}{f_1(m)},$$

$$J_s(t):=-\frac{g(\bar{m})f_{-}(\bar{m})}{f_{+}(\bar{m})}(\bar{T}-t)+{\int }_{\bar{m}}^{{\bar{m}}_T}\frac{g(m)}{f_1(m)}dm.$$

$$\bar{T}-t=(T-t)-{\int }_{\bar{m}}^{{\bar{m}}_T}\frac{dm}{f_1(m)}={\int }_{\bar{m}}^{m(T)}\frac{dm}{f_1(m)}-{\int }_{\bar{m}}^{{\bar{m}}_T}\frac{dm}{f_1(m)}={\int }_{{\bar{m}}_T}^{m(T)}\frac{dm}{f_1(m)}.$$

$$\delta (m(T)):=J_1(t)-J_s(t)={\int }_{{\bar{m}}_T}^{m(T)}\left(g(m)+\frac{g(\bar{m})f_{-}(\bar{m})}{f_{+}(\bar{m})}\right)\frac{dm}{f_1(m)}$$

$$\begin{array}[]{lll}\delta^{\prime}(m)&=&\displaystyle\frac{g(m)+\bar{\alpha}}{f_{1}(m)},\\[11.38109pt] \delta^{\prime\prime}(m)&=&\displaystyle\frac{g^{\prime}(m)f_{1}(m)-(g(m)+\bar{\alpha})f^{\prime}_{1}(m)}{f_{1}^{2}(m)},\end{array}$$

$${\delta }^{\prime }(m)=0⟹{\delta }^{\prime \prime }(m)<0.$$

$${\mathcal{D}}_{-}:=\{(t,m)\in [0,T]\times [0,\bar{m}]\},{\mathcal{D}}_{+}:=\{(t,m)\in [0,T]\times [\bar{m},+\infty )\}.$$

$$u[t,m]=\left|\begin{array}[]{ll}1&\mbox{if }m<\bar{m}\mbox{ or }t\geq\bar{T},\\ \bar{u}&\mbox{if }m=\bar{m}\mbox{ and }t<\bar{T},\end{array}\right.$$

Full text

Key-words. Membrane filtration process, Physical backwash strategy, Optimal Control, Pontryagin Maximum Principle, Singular Arcs.

1 Introduction

Membrane filtration systems are widely used as physical separation techniques in different industrial fields like water desalination, wastewater treatment, food, medicine and biotechnology. The membrane provides a selective barrier that separates substances when a driving force is applied across the membrane.

Different fouling mechanisms are responsible of the flux decline at constant transmembrane pressure (TMP) or the increase of the TMP at a constant flux. Hence, the operation of the membrane filtration process requires to perform regularly cleaning actions like relaxation, aeration, backwashing and chemical cleaning to limit the membrane fouling and maintain a good filtrate production.

Usually, sequences of filtration and membrane cleaning are fixed according to the recommendations of the membrane suppliers or chosen according to the operator’s experience. This leads to high operational cost and to performances (for example, quantities of fluid filtered over a given period of time in a membrane filtration process) that can be far from being optimal. For this reason, it is important to optimize the membrane filtration process functioning in order to maximize system performances while minimizing energy costs.

A variety of control approaches have been proposed to manage filtration processes. In practice such strategies were based on the application of a cleaning action (physical or chemical) when either the flux decline through the membrane or the TMP increase crosses predefined threshold values [9]. Smith et al. developed a control system that monitors the TMP evolution over time and initiates a membrane backwash when the TMP exceeds a given set-point, [17]. In [13] the TMP was also used as the monitoring variable but the control action was the increase or decrease of ‘membrane aeration’. The permeate flux was used in [18] as the control variable to optimize the membrane backwashing and prevent fouling. Moreover, knowledge-based controllers found application in the control of membrane filtration process. In [16], Robles et al. proposed an advanced control system composed of a knowledge-based controller and two classical controllers (on/off and PID) to manage the aeration and backwash sequences. The permeability was used by [8] as a monitoring variable in a knowledge-based control system to control membrane aeration flow.

To date, different available control systems are able to increase significantly the membrane filtration process performances. However, more enhanced optimal control strategies are needed to cope with the dynamic operation of the purifying system and to limit membrane fouling. The majority of the control strategies previously cited address energy consumption, but regulation and control have not being proved to be optimal.

In the present work, we consider the maximization of the net fluid production (i.e. the filtrate) per area of a membrane filtration system over a given operation duration. The control variable is the direction of the flow rate: forward for filtration through the membrane and backward for backwashing attached foulants. This problem is quite generic for various fluids to be filtered Membrane fouling is assumed to be only due to the particle deposition onto the membrane surface while pores blocking is neglected. This problem appears of primer importance for water treatment, especially in the actual context of worldwide scarcity of water of ‘good’ quality.

The modeling of the process then leads to consider an optimal control problem governed by a one-dimensional non-linear dynamics taking into account the filtration and backwash operating modes. This optimal control problem falls into the class of classical Lagrange problems, but in which the Hamiltonian is linear in the control variable. For such problems, it is well known that several singularities may occur such as singular arcs, switching surfaces (cf. [4])… The aim of the present work is to give a complete optimal synthesis of this problem in a quite generic way (i.e. without giving the exact expressions of the functions involved in the model) characterizing the occurrence of such singularities. The analysis of these singularities is important for practical implementation because it gives the structure of the control strategies to be applied (how many switches, where or when to switch…) and the information (i.e. which variable and when) that is needed to be measured. The paper is organized as follows.

• •

In Section 2, we present the model that allows us to state the optimal control problem and we give preliminary results about the structure of the optimal control near to the terminal time.

• •

Section 3 is devoted to the analysis of singular arcs (existence and optimality).

• •

In Section 4, we show that a switching curve may appear and moreover that a phenomenon of ‘dispersion’ may occur. This allows us to provide a complete description of an optimal feedback control of the problem (see Theorem 4.1).

• •

Section 5 depicts the previous analysis on two different models. In the first one, the optimal synthesis involves both a singular arc and a switching curve only whereas the second one example also exhibits a dispersal curve (see e.g. [20]). Such a curve is a locus where the optimal control is non-unique : the corresponding trajectories (in our case in number $2$) reach the terminal state with the same cost.

• •

Finally, several possible extensions of this study are discussed in the conclusion.

2 Model description and preliminary results

To describe the membrane filtration process, we consider a simple form of the model of [3]. In a previous work, it was shown that this model is very generic in the sense that it is able to capture the dynamics of a large number of models available in the literature while simple enough to be used for optimizing and control purposes, see [10]. In the present work, it is assumed that the membrane fouling is only due to the particle deposition onto the membrane surface. Let $m$ be the mass of the cake layer formed during the filtration ($m\geq 0$). One can assume that $m$ follows a differential equation

[TABLE]

where $f_{1}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$. We further assume that the physical cleaning of the membrane is performed by a backwashing which consists in reversing the flow. During this phase, the filtration is stopped and the mass detaches from the membrane surface with a dynamics

[TABLE]

where $f_{2}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$. The considered system is operated by alternating two functioning modes: filtration and backwash. For this reason, we consider a control $u$ that takes values $1$ during filtration and $-1$ during retro washing. Then, the controlled dynamics can be written as follows

[TABLE]

where $m_{0}\geq 0$ is the initial mass. As already mentioned in the introduction, the aim of this work is to determine an optimal switching between the two functioning modes which maximizes the net fluid production of the membrane filtration process during a time interval $[0,T]$. Assuming that the flux that passes through the membrane during forwards and backwards operation is given by a function $g:\mathbb{R}_{+}\rightarrow\mathbb{R}$ that depends on $m$, the net amount of fluid per area of membrane during a time interval $[0,T]$ is then

[TABLE]

Given an initial condition $m_{0}\geq 0$, the objective of the paper is to determine an optimal strategy $u(\cdot)$ that takes values $-1$ or $1$ maximizing $J_{T}(m_{0},u(\cdot))$. Nevertheless, it is well known from the theory of optimal control that the existence of an optimal trajectory cannot be guaranteed when the control set is non convex [14]. Therefore, we shall consider for the mathematical analysis that the control $u(\cdot)$ can take values in the interval $[-1,1]$. Hence, we will focus in this paper on the following optimal control problem:

[TABLE]

where $\mathcal{U}$ denotes the set of measurable functions over $[0,T]$ taking values in $[-1,1]$. The question of practical applicability of a control that takes values different to $-1$ and $1$ relies on approximations with chattering controls [19] and is exposed in [11, 12] (see also [2] in the context of fed-batch bioprocesses).

Next, we consider the following hypotheses on the model.

Hypothesis 2.1**.**

The functions $f_{1}$, $f_{2}$ and $g$ are $C^{1}$ functions such that

1. i.

$f_{1}(m)>0$* and $g(m)>0$ for any $m\geq 0$,* 2. ii.

$f_{2}(0)=0$* and $f_{2}(m)>0$ for $m>0$,* 3. iii.

$f_{1}$* and $g$ are decreasing with $\lim_{m\to+\infty}g(m)=0$,* 4. iv.

$f_{2}$* is increasing.*

Let us comment about these hypotheses:

• •

When a membrane operates in filtration, the resistance to flow is never null and increases according to the mass $m$ of the cake layer formed on the membrane surface, which subsequently decreases the permeate flux. Thus, we assume that the rate $f_{1}$ at which the mass of material adheres to the membrane surface during filtration is a positive decreasing function.

• •

When starting membrane backwash, the cake layer is decomposed and the membrane’s permeability increases again. So, the speed $f_{2}$ of the cake detachment can be described by a positive increasing function. When the membrane is clean ($m=0$), there is nothing to be detached: $f_{2}(0)=0$.

• •

At constant TMP, the permeate flux decreases as the extent of fouling gradually increases. Therefore, the variation of the permeate flux $J$ can be described by a decreasing positive function of the mass of the fouling layer.

Thanks to Hypothesis 2.1, one can straightforwardly check the following property.

Lemma 2.1**.**

The domain $\{m>0\}$ is positively invariant whatever is the control $u(\cdot)$.

For convenience, we define two functions $f_{+}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ and $f_{-}:\mathbb{R}_{+}\rightarrow\mathbb{R}$ defined by

[TABLE]

thus the dynamics can be equivalently written

[TABLE]

We shall use the Maximum Principle of Pontryagin (PMP) [15] in order to determine necessary conditions on optimal trajectories. For this purpose, we introduce the Hamiltonian of the system defined by111As the terminal state is free, no abnormal trajectories will occur, moreover we will write the Hamiltonian condition (2.5) with a maximum.

[TABLE]

According to the Pontryagin Maximum Principle, if $u(\cdot)$ is an optimal control and $m(\cdot)$ the associated trajectory, there exists an absolutely continuous function $\lambda:[0,T]\rightarrow\mathbb{R}$ called adjoint vector satisfying the adjoint equation for a.e. $t\in[0,T]$:

[TABLE]

together with the terminal condition $\lambda(T)=0$. Moreover, the Hamiltonian condition is satisfied:

[TABLE]

Thanks to this expression, an optimal control necessarily satisfies:

[TABLE]

where $\phi$ is the switching function defined by

[TABLE]

The adjoint vector $\lambda$ satisfies the following property.

Proposition 2.1**.**

Under Hypothesis 2.1, the adjoint variable satisfies $\lambda(t)<0$ for any $t\in[0,T[$. Moreover, for any initial condition $m_{0}$ there exists $\bar{t}<T$ such that the control $u(t)=1$ is optimal for $t\in[\bar{t},T]$.

Proof.

At $\lambda=0$, one has $\phi(m,0)=g(m)>0$ and then $u=1$ which implies to have $\dot{\lambda}=-g^{\prime}(m)>0$. If $\lambda(t)=0$ for some $\bar{t}<T$ then one has necessarily $\lambda(t)>0$ for any $t>\bar{t}$ which is in contradiction with $\lambda(T)=0$. Therefore $t\mapsto\lambda(t)$ is non-null and has constant sign on $[0,T[$. As $\lambda$ has to reach [math] at time $T$ with $\dot{\lambda}(T)>0$, we conclude that $\lambda$ has to be negative on $[0,T[$. At the terminal time, one has $\phi(m(T),\lambda(T))=\phi(m(T),0)=g(m(T))>0$. By continuity, the function $t\mapsto\phi(m(t),\lambda(t))$ remains positive on a time interval $[\bar{t},T]$ with $\bar{t}<T$, thus we necessarily have $u=1$ on this interval. ∎

A triple $(x(\cdot),\lambda(\cdot),u(\cdot))$ is called an extremal trajectory if it satisfies (2.1)-(2.3)-(2.5). Since the system and the cost are autonomous (i.e. they do not explicitly depend on the time $t$), the Hamiltonian $H$ is constant along any extremal trajectory.

We call switching time (or switch) an instant $t_{s}\in[0,T]$ where the optimal control is non-constant in any neighborhood of $t_{s}$. It follows that at such an instant $t_{s}$, $\phi$ is necessarily vanishing i.e. $\phi(t_{s})=0$. We then say that the trajectory has a switching point at the time $t_{s}$. As the Hamiltonian $H$ is linear with respect to the control variable, we know that the optimal solution is a combination of bang-bang controls and possible singular arcs. Recall that a singular arc is a time interval on which the switching function $\phi$ is identically equal to zero (see [4, 5] for a thorough study of this notion). Since the Hamiltonian is linear w.r.t. the control $u$, the Hamiltonian condition (2.5) does not imply straightforwardly an expression of the optimal control as in (2.6). In the two coming sections, we study first the possibility of having a singular arc, and then the possibility of having switching points outside the singular arc.

3 Singular arc and first optimality results

In this section, we show that singular arcs may appear in the optimal synthesis of the problem. For convenience, we define a function $\psi:\mathbb{R}_{+}\rightarrow\mathbb{R}$ by:

[TABLE]

It will be also convenient to introduce the function $\gamma:\mathbb{R}_{+}\rightarrow\mathbb{R}$ defined as

[TABLE]

We now consider the following hypothesis:

Hypothesis 3.1**.**

The function $\psi$ admits an unique positive root $\bar{m}$ and is such that $\psi(m)(m-\bar{m})>0$ for any positive $m\neq\bar{m}$.

Under Hypothesis 3.1, one can characterize $m=\bar{m}$ as the unique candidate singular arc.

Lemma 3.1**.**

Consider a singular arc defined over a time interval $[t_{1},t_{2}]$. Then the corresponding extremal singular trajectory $(m(\cdot),\lambda(\cdot),u(\cdot))$ satisfies $m(t)=\bar{m}$ and $u(t)=\bar{u}$, $t\in[t_{1},t_{2}]$, where

[TABLE]

Moreover, $\lambda(\cdot)$ is constant equal to $\bar{\lambda}$ where $\bar{\lambda}\in\mathbb{R}$ is defined by

[TABLE]

Proof.

For simplicity, we write $\dot{\phi}$ the time derivative of $t\mapsto\phi(m(t),\lambda(t))$ and we drop the $m$ dependency of functions $f_{-}$, $f_{+}$ and $g$. Thus, we have:

[TABLE]

or equivalently

[TABLE]

As a singular arc has to fulfill $\phi=0$ and $\dot{\phi}=0$, thus equation (3.3) and Hypothesis 3.1 imply $\psi=0$. Then, the single possibility for having a singular arc on a time interval $[t_{1},t_{2}]$ is to have $m(t)=\bar{m}$ for any $t\in[t_{1},t_{2}]$. From equation (2.1), one then obtains the constant control given in (3.1) for having $\dot{m}=0$ at $m=\bar{m}$. Finally, (3.2) is obtained using that $\phi=\lambda f_{+}+g$ is zero along a singular arc. ∎

We deduce the following optimality results.

Proposition 3.1**.**

Suppose that Hypotheses 2.1 and 3.1 hold true and let $m_{0}>0$ be an initial condition. Then, the following properties are satisfied:

1. (i).

When $m_{0}<\bar{m}$, the control $u=1$ is optimal as long as the corresponding trajectory satisfies $m(t)<\bar{m}$, 2. (ii).

When $m_{0}>\bar{m}$, either the control $u=1$ is optimal until $t=T$, or the control $u=-1$ is optimal until a time $\bar{t}<T$ with $m(\bar{t})\geq\bar{m}$. If $m(\bar{t})>\bar{m}$ then $u=1$ is optimal on $[\bar{t},T]$. 3. (iii).

Suppose that $f_{-}(\bar{m})\geq 0$. Then, for any initial condition $m_{0}\geq\bar{m}$, an optimal control satisfies $u=-1$ over some time interval $[0,\bar{t}]$ with $\bar{t}\in[0,T]$ and $u=+1$ over $[\bar{t},T]$. 4. (iv).

Suppose that $f_{-}(\bar{m})<0$ and let $\bar{T}\in\mathbb{R}$ be defined by

[TABLE]

Then, if $\bar{T}>0$, any singular trajectory is optimal until $t=\bar{T}$.

Proof.

Let us start by stating two properties that will be crucial for reducing the number of possible switching times in the optimal synthesis. From Hypothesis 3.1 and equation (3.3), we can deduce that:

• •

When $\phi(m)=0\mbox{ with }m<\bar{m}$ then $\dot{\phi}<0$. This implies that $\phi$ can change its sign only when decreasing. Therefore only a switching point from $u=1$ to $u=-1$ can be optimal in the domain $\{m<\bar{m}\}$.

• •

When $\phi(m)=0\mbox{ with }m>\bar{m}$ then $\dot{\phi}>0$. This implies that $\phi$ can change its sign only when increasing. Therefore, only a switching point from $u=-1$ to $u=1$ can be optimal in the domain $\{m>\bar{m}\}.$

Let us now prove (i). Take $m_{0}<\bar{m}$, and suppose that the control satisfies $u=-1$. It follows that the trajectory remains in the domain $\{m<\bar{m}\}$. From Proposition 2.1, the trajectory necessarily has a switching point at time $t_{c}$ (otherwise, we would have $u=-1$ until the terminal time $t=T$ and a contradiction) implying $\dot{\phi}(t_{c})\geq 0$. On the other hand, we deduce from (3.3) that $\dot{\phi}(t_{c})=\frac{\psi(m(t_{c}))}{f_{+}(m(t_{c}))}<0$ which is a contradiction. Hence, we must have $u=1$ in the domain $\{m<\bar{m}\}$.

The proof of (ii) is similar utilizing that in the domain $\{m>\bar{m}\}$, any optimal trajectory has at most one switching point from $u=-1$ to $u=+1$. It follows that only three cases may occur: either $u=1$ is optimal over $[0,T]$, or the trajectory reaches $m=\bar{m}$ at some instant $\bar{t}<T$, or finally it has exactly one switching point in the domain $\{m>\bar{m}\}$ from $u=-1$ to $u=+1$.

Let us prove (iii). If one has $u=+1$ at time zero, then the result is proved with $\bar{t}=0$. Suppose now that one has $u=-1$ at time zero. We know that if the trajectory switches at some time $\bar{t}\in[0,T]$ before reaching $m=\bar{m}$, then one has $u=1$ for $t>\bar{t}$ and the result is proved. Suppose now that an optimal trajectory reaches the singular arc before $t=T$ and that one has $m(t)=\bar{m}$ on a time interval of non-null length. Since the Hamiltonian is constant along any extremal trajectory, one must have $H=\bar{\lambda}f_{-}(\bar{m})$. Moreover, as the Hamiltonian at time $T$ is given by $H=g(m(T))$, one should have $\bar{\lambda}f_{-}(\bar{m})=g(m(T))>0$. As $\bar{\lambda}<0$, we conclude that when $f_{-}(\bar{m})\geq 0$, this situation cannot occur. Hence, a singular arc is not optimal.

Finally, let us prove (iv) and suppose that $f_{-}(\bar{m})<0$. Accordingly to Propositions 2.1 and 3.1, any optimal trajectory is such that the corresponding optimal control satisfies $u=1$ in a left neighborhood of $t=T$. Let us compute the last instant $\bar{T}<T$ (if it exists) until a singular arc is possible. From the previous analysis, we necessarily have $u=1$ on $[\bar{T},T]$. This imposes (utilizing that the Hamiltonian is constant) the final state to be $\bar{m}_{T}=m(T)$ as a solution of

[TABLE]

which is uniquely defined as $g$ is decreasing, $\lim_{m\rightarrow+\infty}g(m)=0$, and $-\frac{f_{-}(\bar{m})}{f_{+}(\bar{m})}\in[0,1]$. This also imposes that the switching time $\bar{T}$ can be determined integrating backward the Cauchy problem

[TABLE]

until $m(\bar{T})=\bar{m}$, which amounts to write

[TABLE]

which is exactly the exression (3.4).

We now show that any singular extremal trajectory leaving the singular arc $m=\bar{m}$ at a time $t<\bar{T}$ is not optimal. To do so, consider a trajectory $m(\cdot)$ leaving the singular arc at a time $t<\bar{T}$ (necessarily with $u=1$ until the terminal time $T$). In particular, we have $m(T)>\bar{m}_{T}$. Since the dynamics is $\dot{m}=f_{1}(m)$ with $u=1$, the corresponding cost from time $t$ can be written as follows:

[TABLE]

to be compared with the cost $J_{s}(t)$ of the singular arc strategy from time $t$ (i.e. $u=\bar{u}$ over $[t,\bar{T}]$ and then $u=1$ over $[\bar{T},T]$), which is equal to

[TABLE]

Thanks to (3.4) and using that $T-t=\int_{\bar{m}}^{m(T)}\frac{dm}{f_{1}(m)}$, we get

[TABLE]

The difference of costs $\delta(m(T))$ can be then written as:

[TABLE]

Let us now study the behavior of $\delta$ as a function of $m(T)$. For convenience, we write $m$ in place of $m(T)$ and recall that $m\geq\bar{m}_{T}$ since $m(T)\geq\bar{m}_{T}$. By a direct computation, one has:

[TABLE]

where $\bar{\alpha}:=\frac{g(\bar{m})f_{-}(\bar{m})}{f_{+}(\bar{m})}$. From this last expression, since $g^{\prime}<0$, one has at each $m>0$:

[TABLE]

Now, it is to be observed that $\delta(\bar{m}_{T})=0$ and that $\delta^{\prime}(\bar{m}_{T})=0$ (from (3.5))). The previous analysis then shows that $\delta^{\prime}<0$ on $(\bar{m}_{T},+\infty)$. It follows that $\delta$ is decreasing over $[\bar{m}_{T},+\infty)$. Hence, we obtain that $\delta(m)<0$ for any $m>\bar{m}_{T}$. As a conclusion, we have proved that $J_{1}(t)<J_{s}(t)$ for any time $t\in[0,\bar{T})$, thus any singular trajectory is such that it is optimal to stay on the singular locus until $\bar{T}$ (and then use $u=1$ from $\bar{T}$ to $T$) as was to be proved. ∎

Remark 3.1**.**

*In the proof of Proposition 3.1 $\mathrm{(iv)}$, we have pointed out that any singular extremal trajectory is necessarily optimal until the last possible instant $t=\bar{T}$. It is worth to mention that this point is not a consequence of Pontryagin’s Principle. Although no saturation phenomenon of the singular control appears (indeed $u=\bar{u}$ is constant along the singular locus), singular trajectories must leave the singular locus at the time $t=\bar{T}<T$. *

In the sequel, the notation $u[t,m]$ stands for a (non-autonomous) feedback control depending on both current time $t$ and current state $m$, whereas $u(\cdot)$ denotes a control function in open loop (i.e. a function of time only chosen for a given initial condition). Let us consider the following two sub-domains (that are not disjoined)

[TABLE]

From Proposition 3.1, we obtain the following properties about the optimal control on these two sub-domains.

Corollary 3.1**.**

Under Hypotheses 2.1 and 3.1, one has the following properties:

1. (i).

If $f_{-}(\bar{m})\geq 0$ (where $\bar{T}$ is defined in (3.4) when $f_{-}(\bar{m})<0$), then the control $u[t,m]=1$ is optimal at any $(t,x)\in{\cal D}_{-}$. 2. (ii).

If $f_{-}(\bar{m})<0$ and $\bar{T}\leq 0$ where $\bar{T}$ is defined in (3.4), then the control $u[t,m]=1$ is optimal at any $(t,x)\in{\cal D}_{-}$. 3. (iii).

*If $f_{-}(\bar{m})<0$ and $\bar{T}\in(0,T)$, then the control *

[TABLE]

is optimal at any $(t,x)\in{\cal D}_{-}$. 4. (iv).

The set ${\cal D}_{+}$ is optimally invariant i.e from any initial condition $(t,m)\in{\cal D}_{+}$, an optimal trajectory stays in ${\cal D}_{+}$ for any future time.

Proof.

We have seen that one cannot have $u=-1$ in the domain $\mathcal{D}_{-}$, otherwise an optimal control cannot be equal to one at the terminal time. Moreover in the two cases $f_{-}(\bar{m})\geq 0$ or $f_{-}(\bar{m})<0$ together with $\bar{T}\leq 0$, the previous proposition implies that no singular arc occurs. This proves (i)-(ii). If $f_{-}(\bar{m})<0$ and $\bar{T}\in(0,T)$, we have seen that singular arcs are optimal until the terminal time $t=\bar{T}$. This proves (iii). For proving (iv), we utilize the same argument as for proving (i) and (ii). ∎

4 Switching locus and full synthesis

In this section, we shall provide an optimal synthesis of the problem and we will show in particular that it can exhibit a switching curve depending on the parameter values.

4.1 Study of the switching locus in $\mathcal{D}_{+}$

We start by studying if optimal trajectories can have a switching point. Accordingly to Proposition 3.1, this may only occur in the set ${\cal D}_{+}$ with a switching point from $u=-1$ to $u=1$. We shall then investigate the locus where switching points occur. To do so, in the case where $f_{-}(\bar{m})<0$, consider a parameterized curve $\mathcal{C}$ (possibly empty) contained in $\mathcal{D}_{+}$ defined by

[TABLE]

where $\tilde{T}:[\bar{m},+\infty)\rightarrow\mathbb{R}$ is the function defined by

[TABLE]

The following proposition gives existence and characterization of this locus contained in ${\cal D}_{+}$.

Proposition 4.1**.**

Assume that Hypotheses 2.1 and 3.1 are fulfilled.

1. (i).

*If $f_{-}(\bar{m})\geq 0$, then an optimal feedback control is $u[t,m]=1$ for $(t,m)\in{\cal D}_{+}$. * 2. (ii).

If $f_{-}(\bar{m})<0$, then:

• •

If ${\cal C}$ is empty, an optimal feedback control is $u[t,m]=1$ for $(t,m)\in{\cal D}_{+}$.

• •

If ${\cal C}$ is non-empty, consider the domain

[TABLE]

Then the feedback control in $[0,T]\times(0,+\infty)$

[TABLE]

is optimal. Furthermore, the set ${\cal C}$ is tangent to the trajectory that leaves the singular arc at $(\bar{T},\bar{m})$ with the control $u=1$.

Proof.

Suppose that $f_{-}(\bar{m})\geq 0$ and let us prove (i). We only have to show that any optimal control satisfies $u=1$ in $\mathcal{D}_{+}$. In this case, we know that no singular arc occurs, therefore it is enough to exclude switching points from $u=-1$ to $u=+1$ in $\cal D_{+}$. Also, since one has $u=1$ in a neighborhood of $t=T$, it is enough to consider terminal states $m_{T}\geq\bar{m}$. By integrating backward the dynamics with the control $u=1$, one has $H=g(m_{T})=g(m(t))+\lambda(t)f_{1}(m(t))$ for $t<T$ as long as the switching function

[TABLE]

remains positive. As $f_{-}(m)\geq 0$, one has $\gamma(\bar{m})\leq 0$. Notice also that for $m~\geq 0$, one has

[TABLE]

so that $\gamma$ is increasing over $[0,\bar{m}]$ and decreasing over $[\bar{m},+\infty)$. Since $\gamma$ is decreasing over $[\bar{m},+\infty)$ we deduce that $\gamma(m(t))\leq 0$ for any time $t\in[0,T]$. Consequently, $\phi$ cannot change its sign. Therefore the control $u=1$ is optimal at any time as was to be proved.

Suppose now that $f_{-}(m)<0$ and let us prove (ii). Again, we consider terminal states $m_{T}\geq\bar{m}$ and we consider the dynamics with $u=1$ backward in time. Note that when $m_{T}=\bar{m}_{T}$, then one has $g(\bar{m}_{T})=\gamma(\bar{m})$ by conservation of the Hamiltonian. Consider now an initial state $m_{T}>\bar{m}_{T}$ and the system backward in time with $u=1$. If an optimal control is such that $u=1$ until reaching the singular arc, we deduce (Thanks to (4.5)) that

[TABLE]

(since $g$ is decreasing). Thus, the switching function is negative when $\bar{m}$ is reached backward in time by the trajectory. By the mean value Theorem, we conclude that there exists a switching point that necessarily occurs at some value $\tilde{m}>\bar{m}$ such that $\gamma(\tilde{m})=g(m_{T})$, and accordingly to Proposition 3.1 this switching point (from $u=-1$ to $u=1$) is unique. From the monotonicity of $\gamma$ over $[\bar{m},+\infty)$, for each $m_{T}>\bar{m}$, $\tilde{m}$ is uniquely defined by $\tilde{m}=\gamma^{-1}(g(m_{T}))$, or reciprocally, for any $\tilde{m}\geq\bar{m}$, $m_{T}$ is uniquely defined as a function of $\tilde{m}$: $m_{T}(\tilde{m})=g^{-1}(\gamma(\tilde{m}))$ (as $g$ is also a decreasing invertible function), with

[TABLE]

Then, the corresponding switching time $\tilde{T}(\tilde{m})$ satisfies

[TABLE]

If $\tilde{T}(\tilde{m})\leq 0$ then no switch occurs at $\tilde{m}$ i.e. the constant control $u=1$ is optimal from [math] to $T$. It follows that if $\mathcal{C}$ is empty, then $u=1$ is optimal in $\mathcal{D}_{+}$ as was to be proved.

When switching points occur, that is, when $\mathcal{C}$ is non-empty, the previous analysis shows that switching points indeed occur on the curve of $\mathcal{D}_{+}$ given by (4.1) and the corresponding switching times are given by (4.2) as was to be proved. The optimality of the feedback control (4.4) follows by noting that in $\mathcal{D}_{+}$, optimal trajectories have at most one switching point from $u=-1$ to $u=+1$ or from $u=-1$ to $\bar{u}$.

Finally, the derivative of $\tilde{T}$ with respect to $\tilde{m}$ can be determined from expressions (4.8) and (4.7) as

[TABLE]

At $\tilde{m}=\bar{m}$, one has $\tilde{T}(\bar{m})=\bar{T}$ and $\gamma^{\prime}(\bar{m})=0$ (since $\psi(\bar{m})=0)$, which gives $\tilde{T}^{\prime}(\bar{m})=1/f_{1}(\bar{m})>0$. Thus, the parameterized curve ${\cal C}$ is indeed tangent to the trajectory that leaves the singular arc with $u=1$ at $(\bar{T},\bar{m})$. ∎

Remark 4.1**.**

When $f_{-}(\bar{m})<0$ and $\bar{T}>0$ (where $\bar{T}$ is defined in (3.4)), the point $(\bar{T},\bar{m})$ belongs to the curve ${\cal C}$ which is then non-empty. This curve could be a set of disjoint curves in $[0,T]\times(0,+\infty)$ (for instance if the function $\tilde{T}$ has several changes of sign). However, in the examples we met, it is always a single curve (bounded or not), see Section 5. Notice also that the map $\tilde{m}\mapsto\tilde{T}(\tilde{m})$ has no a priori reason to be monotonic, as one can see in the second example in Section 5.

4.2 Dispersal curve

In the sequel, a switching locus is a set of points where optimal trajectories cross this set by switching from $u=-1$ to $u=+1$. Moreover, the control remains constant equal to one after the switching point and the corresponding trajectory does not reach the singular arc. On the other hand, a dispersal curve will stand for a set of points from which there are exactly two optimal trajectories : in our setting, either the optimal control is $u=+1$ until the terminal time or the optimal control is $u=-1$ in the set $\mathcal{W}$ until the time where the trajectory reaches either the singular locus or the switching locus (both strategies having the same optimal cost).

When the set ${\cal C}$ is non-empty (under the condition $f_{-}(\bar{m})<0$), we introduce the following partition:

[TABLE]

with

[TABLE]

One can then characterize optimal trajectories on these two sets as follows.

Corollary 4.1**.**

Assume that Hypotheses 2.1, 3.1 are fulfilled with $f_{-}(\bar{m})<0$ and that ${\cal C}\neq\emptyset$ (where ${\cal C}$ is defined in Proposition 3.1). One has the following properties.

• •

*The set ${\cal C}_{s}$ is not reduced the singleton $\{(\bar{T},\bar{m})\}$ and it is a switching locus. *

• •

The set ${\cal C}_{d}$ (when it is non-empty) is a dispersion locus i.e. from every state in ${\cal C}_{d}$ the two trajectories

with $u=1$ up to the terminal time, 2. 2.

with $u=-1$ up to reaching the singular arc $m=\bar{m}$ or the set ${\cal C}_{s}$,

are both optimal.

Proof.

The domain ${\cal W}$ (when it is not empty) is exactly the set of points $(t,m)\in\mathcal{D_{+}}$ for which the optimal control satisfies $u=-1$ (see Proposition 3.1). From such a state, the optimal trajectory has to leave the domain ${\cal W}$ (as $\dot{m}$ is bounded from above by $-f_{2}(\bar{m})<0$ in this set) reaching either the singular arc or the set ${\cal C}$. At some point $(t,m)$ in ${\cal C}$, an outward normal $n$ to ${\cal W}$ is then given by

[TABLE]

and the velocity vectors $v_{-1}$, $v_{1}$ for the control $u=-1$ and $u=1$ respectively are

[TABLE]

Notice that by construction of the set ${\cal C}$, the velocity vector $v_{1}$ points outward of ${\cal W}$ at any point $(t,m)\in{\cal C}$. Hence, the velocity vector $v_{-1}$ points outward when the scalar product $n\cdot v_{-1}$ is positive, that is when $(t,m)$ belongs to ${\cal C}_{s}$.

We consider now optimal trajectories that reach ${\cal C}$ from ${\cal W}$ and distinguish two cases.

At states in ${\cal C}_{s}$, the velocity vectors $v_{-1}$, $v_{1}$ both point outward of the set ${\cal W}$. Therefore an optimal trajectory reaching ${\cal C}_{s}$ with $u=-1$ leaves it with $u=1$. Then, accordingly to Proposition 2.1, the optimal control stays equal to $1$ up to the terminal time. 2. 2.

At states in ${\cal C}_{d}$, $v_{-1}$ points inward of ${\cal W}$ while $v_{1}$ points outward. Therefore an optimal trajectory cannot reach a point located on ${\cal C}_{d}$. From states in ${\cal C}_{d}$, there are thus two extremal trajectories: one with $u=1$ up to the terminal time, and another one with $u=-1$ up to the singular arc or to the curve ${\cal C}_{s}$ (accordingly to Proposition 2.1 and 3.1) and then $u=1$ up to the terminal time. As the value function of a Lagrange problem with smooth data is everywhere Lipschitz continuous (see for instance [1]), and that $u=-1$ and $u=+1$ are optimal respectively inside and outside ${\cal W}$, we deduce that these two extremal trajectories should have the same cost i.e. are both optimal.

Finally, let us show that ${\cal C}_{s}$ is not reduced a singleton. The state $(\bar{m},\bar{T})$ belongs to ${\cal C}$ (as it is indeed a point where the switching function vanishes) but it also belongs to the singular locus $m=\bar{m}$. Therefore, there exists a trajectory with $u=-1$ that is crosses ${\cal C}$ transversely at this point. By continuity of the solutions of the system with $u=-1$ w.r.t. the initial condition, we deduce that there exist locally other trajectories that cross the non-empty curve ${\cal C}$ transversely with the control $u=-1$. This proves that ${\cal C}_{s}$ is not reduced to a singleton. ∎

Figure 1 illustrates the two kind of points that can belong to the set ${\cal C}$.

4.3 Full synthesis

We conclude this section by summarizing the results of Corollary 3.1 and Proposition 4.1 that give the optimal synthesis of the problem in the whole domain $[0,T]\times[0,+\infty)$.

Theorem 4.1**.**

Assume that Hypotheses 2.1 and 3.1 are fulfilled.

1. (i).

If $f_{-}(\bar{m})\geq 0$ or $f_{-}(\bar{m})<0$ and the set $\mathcal{C}$ is empty, then, an optimal feedback control in $[0,T]\times(0,+\infty)$ is given by $\mathrm{(}$recall (4.3)$\mathrm{)}$

[TABLE] 2. (ii).

If $f_{-}(\bar{m})<0$ and the set $\mathcal{C}$ non-empty, then, an optimal feedback control in $[0,T]\times(0,+\infty)$ is given by

[TABLE]

The vacuity of $\mathcal{C}$ can be verified thanks to the explicit definitions of $\mathcal{C}$ by (4.1).

5 Two numerical case studies

In this section, we illustrate the previous analysis of optimal trajectories on two classical models of the literature that fulfill Hypotheses 2.1 and 3.1.

5.1 Benyahia et al model

Consider the following functions that have been validated on experimental data [3]:

[TABLE]

where $a$, $b$ and $e$ are positive numbers. One can check that Hypothesis 2.1 is fulfilled. A straightforward computation of the function $\psi$ gives

[TABLE]

A further computation of the derivative of $\psi$ gives

[TABLE]

which allows to conclude that $\psi$ is increasing on $\mathbb{R}_{+}$. As one has $\psi(0)=-(2abe^{2}+b^{2})/(4e^{4})<0$ and $\lim_{m\to+\infty}\psi(m)=+\infty$, we deduce that Hypothesis 3.1 is fulfilled. When $\psi$ is null for $m=\bar{m}$, one has

[TABLE]

Therefore $f_{-}(\bar{m})$ and $d(\bar{m})$ have the same sign. A straightforward computation gives

[TABLE]

and thus one has $d(\bar{m})<0$. Therefore, from Proposition 3.1 and Corollary 4.1, there exists a singular arc when $\bar{T}>0$ and a switching locus when $\tilde{T}(\tilde{m})>0$.

Figure 2 shows the general synthesis of optimal controls with the parameters $a=b=e=1$ and for a time horizon of $10$ hours. In this example one can see that the curve ${\cal C}$ is entirely a switching locus i.e. one has ${\cal C}={\cal C}_{s}$.

5.2 Cogan-Chellam model

We now consider the functions

[TABLE]

where $a$, $b$ and $e$ are positive numbers, as proposed in [6, 7]. Clearly Hypothesis 2.1 is fulfilled. Moreover, one has

[TABLE]

Therefore, the function $\psi$ can have two changes of sign at

[TABLE]

where $\bar{m}_{-}$ and $\bar{m}_{+}$ are respectively negative and positive numbers. One has also

[TABLE]

which is positive. Therefore $\psi$ is an increasing function and Hypothesis 3.1 is fulfilled with $\bar{m}=\bar{m}_{+}$. Moreover one can write

[TABLE]

Then, as for the previous model, Proposition 3.1 and Corollary 4.1 allow to conclude that there exists a singular arc when $\bar{T}>0$ and a switching locus when $\tilde{T}(\tilde{m})>0$.

Figure 3 shows the synthesis of optimal controls for this model with the parameters $a=b=e=1$ and for a time horizon of $40$ hours. In this example one can see that the curve ${\cal C}$ splits into two non-empty subsets ${\cal C}_{s}$ and ${\cal C}_{d}$.

5.3 Discussion

Although the two models are very close and posses similar optimal syntheses, a main difference occurs on the size and on the shape of the domain ${\cal W}$ where backwash has to be applied (see Figures 2,3). In particular, its boundary ${\cal C}$ is entirely a switching curve in one case while most of it is a dispersal curve in the second case. This should give valuable information to the practitioners about when and how long backwashing (i.e. $u=-1$) has to be applied out of the singular arc.

For the practical implementation of the optimal control law (where only the values $u=-1$ and $u=1$ can be applied) it is not possible to stay exactly on the singular arc $m=\bar{m}$. But an approximation by a sequence of filtration/backwashing can be applied to stay on the vicinity of the singular arc. This sequence can be chosen so that the average value of $m$ is $\bar{m}$, which provides a good approximation of the optimal value as it has been tested in [11, 12]. One may argue that the optimal control problem could be reformulated as a discrete time problem where the time step is the smallest period of switching between filtration and backwashing that could be applied in practice. We believe that this approach gives less geometric insights of the nature of the optimal control than the continuous formulation. Moreover, computing the optimal value of the criterion for the continuous time problem gives an upper bound of what could be intrinsically expected from the process, independently of the practical implementations.

6 Conclusion

In this work, the application of the Pontryagin Maximum Principle for the synthesis of optimal control of a switched system shows interesting results for maximizing the net fluid production (per filtrate) of a membrane filtration system. The optimal synthesis exhibits bang-bang controls with a ‘most rapid approach’ to a singular arc and a switching curve before reaching the final time. We have also shown that a dispersal curve may occur, leading to the non-uniqueness of optimal trajectories. Practically, the determination of the singular arc allows to compute a sequence of filtration/backwashing to stay about the singular arc, and the determination of the curve ${\cal C}$ provides the information about the domain where backwashing has to be applied. The synthesis also reveals that if one wants to implement a feedback controller, which is more robust than an open-loop controller, the on-line measurement of the mass deposit $m$ or of any invertible function of $m$, such as the fluid flowrate, is crucial.

The main advantage of the present analysis is to describe an optimal synthesis for a very large class of models relying on simple qualitative properties of the functions $f_{1}$, $f_{2}$ and $g$.

Perspectives of this work are first to implement the optimal synthesis with real process constraints, and compare the fluid production (per filtrate) of the membrane filtration process with the classical operating strategies that are proposed in the literature and currently used. Extensions to other fluids or non constant TMP and consideration of multiple objectives (production and energy consumption) could be also the matter of future works, as well as possibilities of multiple singular arcs.

Acknowledgments

The authors thank the TREASURE research network (cf http://www.inra.fr/treasure) for its financial support. The third author would like to thank the MIA Division of INRA and the MISTEA lab, Montpellier for providing him a half year delegation during the academic year 2017-2018.