Minimization of Pollution Concentration on a Given Time Interval for the Waste Water Cleaning Plant

Abstract

A model of a waste water treatment plant is investigated. The model is described by a nonlinear system of two differential equations with one bounded control. An optimal control problem of minimizing concentration of the polluted water on the given time interval is stated and solved analytically with the use of the Pontryagin Maximum Principle and Green's Theorem. Computer simulations of a model of an industrial waste water treatment plant show the advantage of using our optimal strategy. Possible applications are discussed.

1. Introduction

The activated sludge process (ASP) is a biochemical process for treating sewage and industrial waste water that uses air (or oxygen) and microorganisms to biologically oxidize organic pollutants, producing a waste sludge (or floc) containing the oxidized material. In general, an activated sludge process has an aeration tank where air (or oxygen) is injected and thoroughly mixed into the waste water and a settling tank (usually referred to as a “clarifier” or “settler”). Flocculation-agglomeration is a process where a solute comes out of solution in the form of floc or flakes. Part of the waste sludge is recycled to the aeration tank where the remaining waste sludge is removed for further treatment and ultimate disposal. A diagram of the process is shown in Figure 1.

Figure 1 

Diagram of flocculation-agglomeration process.

The optimal operation of the waste water processes with biological treatment is challenging because of the strong effluent requirements, the complexity of these processes as an object of control, and the need to reduce the operation cost [1].

During the last 30 years various control strategies for the ASP have been developed [2, 3]. Establishing optimal working conditions and control strategies is frequently accomplished with the aid of mathematical models [1, 4–7]. Relevant work in the investigation and comparison of control strategies was done by [8–13]. Obviously, the solution depends on the model. What unites all these papers is that either the considered models are so complex that they cannot be solved analytically or that the controls are not bounded and therefore the realism of the model is questionable. The model proposed in [4] is simple enough that it can be investigated analytically. On the other hand, it properly corresponds to the main steps of the ASP and water cleaning control process. In [4] an optimal control problem of the minimization of the waste concentration in the ASP was stated and the Pontryagin Maximum Principle [14] was offered for its solution. However, the complete analysis of the corresponding boundary value problem for the Maximum Principle was not conducted. The author simply offered a numerical solution to the problem at different piecewise constant controls.

The traditional method for solution of optimal control problems is the Pontryagin Maximum Principle (PMP). Moreover, there exist optimal control problems, for which the PMP gives a complete analytical solution. It occurs when the two-point boundary value problem for the Maximum Principle has a unique solution, or when the PMP is not only a necessary, but also a sufficient optimality condition [15]. There exist other optimal control problems, for which the approach based on Green's Theorem is an effective tool for obtaining the complete analytical solutions [16, 17]. In [7] there is an example of such an optimal control problem.

In practical applications one has to take into account the presence of several perturbations, which may degrade the performance of obtained optimal control law. For example, the parameters of the model may not be precisely known. The evolution of the model may be affected by random external perturbations. At last, the state variables of the model may not be accurately measured. Therefore, it is important to design a robust feedback optimal control, which still accomplishes the desired task in the presence of appointed perturbations. Different approaches to the solution of this problem for the ASP are considered in [7, 18–24].

The optimal control problem stated in [4] does not fit either the first or the second types of the optimal control problems. In fact, on one hand, using the PMP shows a possibility of the existence of singular arcs at corresponding optimal trajectories [25]. The presence of them greatly complicates the analytical analysis of the considered problem. On the other hand, using the approach based on Green's Theorem proves the absence of singular arcs at optimal trajectories. Simultaneously, this approach overestimates the number of switchings of optimal controls. Lastly, only a combination of results from these different approaches allows us to solve the considered optimal control problem analytically.

Our paper deals with the complete analysis of the model and its corresponding optimal control problem proposed by [4]. It is organized as follows. In Section 2, we discuss the model. In Section 3, we state optimal control problem of minimizing water pollution concentrations on a given time interval. In Section 4, we establish properties of the state variables. Then we find optimal solutions of the considered problem in two stages. At first, in Section 5 we investigate the optimal control problem using the Pontryagin Maximum Principle. Then, in Section 6 we apply to the considered problem the approach based on Green's Theorem. As a final result of these investigations we obtain possible types of optimal solutions. They allow us in Section 7 to reduce the optimal control problem to problem of the finite dimensional optimization. The last problem is solved numerically. A numerical simulation of the ASP for different parameters of the model is conducted in Section 8. Finally, Section 9 presents our conclusions.

2. The Model

Let us consider the model of an activated sludge process. A simplified diagram can be shown in Figure 2.

Figure 2 

Simplified diagram of an activated sludge process.

The process has a reaction tank and a settling tank. Polluted water is introduced into the reaction tank at rate 𝑏 and concentration 𝑎₂. Simultaneously, the reaction tank receives biomass 𝑥 at a rate 𝑢 with concentration 𝑎₁. Once in the reaction tank, biomass feast on contaminants. The less-polluted water now leaves the reaction tank and proceeds to a filter. The growth rate of the biomass is assumed to be of the Michaelis-Menten type. Flocculation-agglomeration allows a solute to come out of solution in the form of flock or flakes. Part of the waste sludge is recycled to the aeration tank where the remaining waste sludge is removed for further treatment.

This process can be described by the following system of differential equations:𝑥̇(𝑡)=𝑢(𝑡)𝑎₁+𝜇₀𝑥(𝑡)𝑠(𝑡)̇𝑘+𝑠(𝑡)−(𝑏+𝑢(𝑡))𝑥(𝑡),𝑠(𝑡)=𝑏𝑎₂−𝜇₀𝑌𝑥(𝑡)𝑠(𝑡)𝑘+𝑠(𝑡)−(𝑏+𝑢(𝑡))𝑠(𝑡),𝑡∈[0,𝑇],𝑥(0)=𝑥₀>0,𝑠(0)=𝑠₀>0. (1)

Here 𝑥(𝑡) is the concentration of biomass, 𝑠(𝑡) is the concentration of polluted water, 𝑌 is the substrate utilization-yield coefficient, 𝜇₀ is the maximal-specific rate of bacteria growth, 𝑘 the saturation coefficient.

We consider the value 𝑢 as a control function 𝑢(𝑡) and the set 𝑃(𝑇) is the set of all Lebegue measurable functions 𝑢(𝑡) such that 0<𝑢₁≤𝑢(𝑡)≤𝑢₂ for almost all 𝑡∈[0,𝑇]. The recycle sludge rate 𝑢(𝑡) is not allowed to take values below a certain lower limit 𝑢₁ in order to prevent the biomass from being swept out of the aeration tank. An upper limit 𝑢₂ for 𝑢(𝑡) is given by the limited power of the recycle pump.

For the model (1) assume 𝑘≫𝑠. This case is realistic since one normally tries to keep the substrate-to-biomass ratio comparatively low. Denoting 𝜇=𝜇₀/𝑘 the simplified system can be written as𝑥̇(𝑡)=𝑢(𝑡)𝑎₁̇+𝜇𝑥(𝑡)𝑠(𝑡)−(𝑏+𝑢(𝑡))𝑥(𝑡),𝑠(𝑡)=𝑏𝑎₂−𝜇𝑌𝑥(𝑡)𝑠(𝑡)−(𝑏+𝑢(𝑡))𝑠(𝑡),𝑡∈[0,𝑇],𝑥(0)=𝑥₀>0,𝑠(0)=𝑠₀>0. (2)

Numerical modeling conducted in [4] shows that the second term in the first equation of (2) can be ignored. Finally we obtain the following system:𝑥̇(𝑡)=𝑢(𝑡)𝑎₁̇−(𝑏+𝑢(𝑡))𝑥(𝑡),𝑡∈[0,𝑇],𝑠(𝑡)=𝑏𝑎₂−𝜇𝑌𝑥(𝑡)𝑠(𝑡)−(𝑏+𝑢(𝑡))𝑠(𝑡),𝑥(0)=𝑥₀>0,𝑠(0)=𝑠₀>0. (3) This model will be considered further.

3. Optimal Control Problem of Minimizing Pollution Concentration on a Given Time Interval

Let 𝑠(𝑡) be the pollution concentration at moment 𝑡. For system (3) we will consider an optimal control problem of minimizing the total concentration of pollution over the time interval [0,𝑇]. The problem can be written as𝐼(𝑢)=^𝑇₀𝑠(𝑡)𝑑𝑡→min_{𝑢(⋅)∈𝑃(𝑇)}. (4)

Let us rewrite the problem (3), (4) in the form, which is more convenient for the subsequent analysis. For this we will use the following substitutions of the state variables: 𝑥₁=𝑥𝑎₁,𝑥₂=𝑠𝑎₂ (5) and the control 𝑣=𝑏+𝑢. Moreover, we denote 𝜎=𝜇𝑎₁/𝑌.

Further, instead of the set 𝑃(𝑇) we will consider the set 𝐷(𝑇), which consists of all Lebegue measurable functions 𝑣(𝑡) such that the inequalities𝑏<𝑣₁≤𝑣(𝑡)≤𝑣₂ (6) are valid for almost all 𝑡∈[0,𝑇]. Here 𝑣_𝑖=𝑏+𝑢_𝑖, 𝑖=1,2.

Then problem (3), (4) can be rewritten as the following optimal control problem:𝑥̇₁(𝑡)=−𝑏−𝑣(𝑡)(𝑥₁𝑥̇(𝑡)−1),𝑡∈[0,𝑇],₂(𝑡)=𝑏−𝜎𝑥₁(𝑡)𝑥₂(𝑡)−𝑣(𝑡)𝑥₂𝑥(𝑡),₁(0)=𝑥⁰₁>0,𝑥₂(0)=𝑥⁰₂>0,𝐽(𝑣)=^𝑇₀𝑥₂(𝑡)𝑑𝑡→min_{𝑣(⋅)∈𝐷(𝑇)}, (7)(8) where 𝑥⁰₁=𝑥₀/𝑎₁, 𝑥⁰₂=𝑠₀/𝑎₂.

The existence of the optimal control 𝑣_∗(𝑡) and its corresponding optimal solutions 𝑥^∗₁(𝑡), 𝑥^∗₂(𝑡) for the optimal control problems (7), and (8) follows from [26].

4. Properties of the State Variables

We have the following statement, which can be easily proven using direct integration of the system (7).

Lemma 1. Let 𝑣(⋅)∈𝐷(𝑇) be some control function. Then there exist corresponding to this control, 𝑣(𝑡), solutions 𝑥₁(𝑡), 𝑥₂(𝑡) to the system (7), which on the closed interval [0,𝑇] satisfy the inequalities 𝑥₁(𝑡)>0,𝑥₂(𝑡)>0. (9)

Analyzing system of (7) with the use of Lemma 1 we can conclude that if at some moment of time 𝑡∈[0,𝑇] we have that 𝑥₁(𝑡)=1, then 𝑥̇₁(𝑡)=−𝑏<0. (10) By analogy if at some moment 𝑡 we have 𝑥₂(𝑡)=1, then from (6) we obtain relationship 𝑥̇₂(𝑡)=𝑏−𝜎𝑥₁(𝑡)−𝑣(𝑡)<0. (11) The validity of these relationships leads to the following statement.

Lemma 2. Let 𝑣(⋅)∈𝐷(𝑇) be some control function. Suppose that at some moments of time 𝜏₁,𝜏₂∈[0,𝑇) the following relationships hold: 𝑥₁(𝜏₁)≤1,𝑥₂(𝜏₂)≤1, (12) then one has 𝑥₁(𝑡)<1 for any 𝑡∈(𝜏₁,𝑇] and 𝑥₂(𝑡)<1 for any 𝑡∈(𝜏₂,𝑇].

From results of Lemma 2 the following statement is true.

Lemma 3. Let 𝑣(⋅)∈𝐷(𝑇) be some control function. Suppose that at some moments of time 𝜂₁,𝜂₂∈(0,𝑇) the following relationships hold: 𝑥₁(𝜂₁)>1,𝑥₂(𝜂₂)>1, (13) then one has 𝑥₁(𝑡)>1 for any 𝑡∈[0,𝜂₁) and 𝑥₂(𝑡)>1 for any 𝑡∈[0,𝜂₂).

Moreover, we have the statement.

Lemma 4. Let 𝑣(⋅)∈𝐷(𝑇) be some control function. Suppose that at some moments of time 𝜃₁,𝜃₂∈(0,𝑇) the following relationships hold: 𝑥₁(𝜃₁)≥1,𝑥₂(𝜃₂)≥1, (14) then one has inequalities 𝑥̇₁(𝜃₁)<0,𝑥̇₂(𝜃₂)<0, (15) respectively.

The validity of Lemma 4 follows from the relationships (6), and (7).

5. Pontryagin Maximum Principle

In order to solve the optimal control problems (7), and (8) at first we will apply the Pontryagin Maximum Principle [14, 15]. For the optimal control 𝑣_∗(𝑡) and corresponding optimal trajectories 𝑥^∗₁(𝑡), 𝑥^∗₂(𝑡) there exist nontrivial solutions 𝜓^∗₁(𝑡), 𝜓^∗₂(𝑡) of the adjoint system𝜓̇^∗₁(𝑡)=𝑣_∗(𝑡)𝜓^∗₁(𝑡)+𝜎𝑥^∗₂(𝑡)𝜓^∗₂𝜓̇(𝑡),𝑡∈[0,𝑇],_∗2(𝑡)=(𝜎𝑥^∗₁(𝑡)+𝑣_∗(𝑡))𝜓^∗₂𝜓(𝑡)+1,_∗1(𝑇)=0,𝜓^∗₂(𝑇)=0, (16) for which the control 𝑣_∗(𝑡) is given by𝑣_∗𝑣(𝑡)=²,if𝐿(𝑡)>0,∀𝑣∈[𝑣₁,𝑣₂𝑣],if𝐿(𝑡)=0,₁,if𝐿(𝑡)<0, (17) where𝐿(𝑡)=−(𝑥^∗₁(𝑡)−1)𝜓^∗₁(𝑡)−𝑥^∗₂(𝑡)𝜓^∗₂(𝑡),𝑡∈[0,𝑇] (18) is the switching function. As it follows from (17), the function 𝐿(𝑡) determines the type of the optimal control 𝑣_∗(𝑡).

Systems of (7), (16) and relationships (17) form the two point boundary value problem for the Maximum Principle. Let us study this problem.

For the functions 𝜓^∗₁(𝑡), 𝜓^∗₂(𝑡) the following statement is true.

Lemma 5. Let 𝑣_∗(𝑡) be the optimal control, 𝑥^∗₁(𝑡) and 𝑥^∗₂(𝑡) are optimal trajectories and 𝜓^∗₁(𝑡), 𝜓^∗₂(𝑡) are the solutions of the adjoint system (16) corresponding to them. Then the following inequalities hold: 𝜓^∗₁(𝑡)>0,𝜓^∗₂(𝑡)<0 (19) for all 𝑡∈[0,𝑇).

The proof of this statement is based on integration of the system (16) with the use of Lemma 1.

Using equations of the systems (7), and (16) let us find the derivative of the switching function 𝐿(𝑡). We have the expression𝐿̇(𝑡)=𝑏(𝜓^∗₁(𝑡)−𝜓^∗₂(𝑡))−𝜎(𝑥^∗₁(𝑡)−1)𝑥^∗₂(𝑡)𝜓^∗₂(𝑡)−𝑥^∗₂(𝑡),𝑡∈[0,𝑇]. (20)

Now, let us study the properties of the optimal control 𝑣_∗(𝑡). The following statements are true.

Lemma 6. There exists such time 𝜃∈[0,𝑇) that for the optimal control 𝑣_∗(𝑡) the equality 𝑣_∗(𝑡)=𝑣₂ is valid for all 𝑡∈(𝜃,𝑇].

Proof. For the switching function 𝐿(𝑡) from the terminal conditions of the system (16) and formulas (18), and (20) we have that 𝐿(𝑇)=0,𝐿̇(𝑇)=−𝑥^∗₂(𝑇). (21) From Lemma 1 we obtain the inequality 𝐿̇(𝑇)<0. Since 𝐿(𝑡) is continuous function, there exists value 𝜃∈[0,𝑇) such that 𝐿(𝑡)>0 for all 𝑡∈(𝜃,𝑇]. Furthermore, from (17) we have 𝑣_∗(𝑡)=𝑣₂ for 𝑡∈(𝜃,𝑇].

Lemma 7. Let there exists such moment of time 𝜏∈[0,𝑇) that 𝑥^∗₁(𝜏)≤1. Then for the optimal control 𝑣_∗(𝑡) the relationship 𝑣_∗(𝑡)=𝑣₂ holds for all 𝑡∈[𝜏,𝑇].

Proof. From results of Lemmas 1, 2, and 5 for the switching function 𝐿(𝑡) the inequality 𝐿(𝑡)>0 is true for all 𝑡∈[𝜏,𝑇]. Therefore, the desired result follows from (17).

Corollary 8. If 𝑥⁰₁≤1, then for the optimal control 𝑣_∗(𝑡) the relationship 𝑣_∗(𝑡)=𝑣₂ holds for all 𝑡∈[0,𝑇].

From Lemma 7 it immediately follows that if the optimal trajectory (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) gets into the region 𝑄={(𝑥₁,𝑥₂)∈ℝ²∶0<𝑥₁≤1,𝑥₂>0}, (22) then the corresponding optimal control 𝑣_∗(𝑡) takes value 𝑣₂.

Therefore, the following investigations will be conducted under the condition 𝑥₁>1.

Let us strengthen the result from Lemma 7. We have the statement.

Lemma 9. Let for the optimal trajectories 𝑥^∗₁(𝑡), 𝑥^∗₂(𝑡) there exists a moment of time 𝜏∈(0,𝑇) such that 𝑥^∗₁(𝜏)=1. Then the positive value 𝛿 exists so that for the corresponding optimal control 𝑣_∗(𝑡) the relationship 𝑣_∗(𝑡)=𝑣₂ holds for all 𝑡∈[𝜏,𝜏+𝛿).

The validity of this fact follows from the inequality 𝐿(𝜏)>0, the continuity of the switching function 𝐿(𝑡), and the formula (17) for the optimal control 𝑣_∗(𝑡).

From this statement we have the existence of the region located in the set 𝑊={(𝑥₁,𝑥₂)∈ℝ²∶𝑥₁>1,𝑥₂>0} (23) and attached from the right to the set 𝑄, in which the optimal control 𝑣_∗(𝑡) also takes value 𝑣₂. At Figure 3 this region is shown hatched.

Figure 3 

Region, which is guaranteed by Lemma 9.

6. Green's Theorem

For the subsequent analysis of the optimal control problems (7), and (8) we will use the approach, based on the applying of Green's Theorem [16, 17].

Following from arguments in [16], at first we will conduct auxiliary investigations.

For all (𝑥₁,𝑥₂)∈𝑊 we consider the function: Φ(𝑥₁,𝑥₂)=𝑥₂(𝜎(𝑥₁−1)²+𝜎(𝑥₁−1)−𝑏)−𝑏(𝑥₁−1). (24)

Let us study the curve Φ(𝑥₁,𝑥₂)=0. We have the following function 𝑥₂=𝑓(𝑥₁)=𝑏(𝑥₁−1)𝜎(𝑥₁−1)²+𝜎(𝑥₁.−1)−𝑏 (25) We see that it is defined only by 𝑥₁>𝑥₁, where 𝑥₁=𝜎+√𝜎²+4𝑏𝜎2𝜎>1. (26)

Moreover, the relationships hold lim_𝑥1→+∞𝑓(𝑥₁)=0,lim_𝑥1→𝑥₁𝑓(𝑥₁)=+∞. (27)

Note that for the derivatives 𝑓(̇𝑥₁), 𝑓(̈𝑥₁) of the function 𝑥₂=𝑓(𝑥₁) at 𝑥₁>𝑥₁ we have the expressions𝑓(̇𝑥₁)=−𝑏⋅𝜎(𝑥₁−1)²+𝑏(𝜎(𝑥₁−1)²+𝜎(𝑥₁−1)−𝑏)²̈<0,𝑓(𝑥₁)=2𝑏𝜎⋅𝜎(𝑥₁−1)³+3(𝑥₁−1)+𝑏(𝜎(𝑥₁−1)²+𝜎(𝑥₁−1)−𝑏)³>0. (28)

Then the function 𝑥₂=𝑓(𝑥₁) is decreasing and concave up. The graph of this function and the allocation of signs of the function Φ(𝑥₁,𝑥₂) are shown in Figure 4.

Figure 4 

Allocation of signs of the function Φ(𝑥1,𝑥2).

For the curve Φ(𝑥₁,𝑥₂)=0 we have the following statement.

Lemma 10. In the region 𝑊 there are no interval Δ⊂[0,𝑇] and values of control 𝑣(⋅)∈𝐷(𝑇) such that the curve Φ(𝑥₁,𝑥₂)=0 is a trajectory of the system (7). Moreover, if there exists a moment of time 𝜏∈(0,𝑇) such that for the trajectories 𝑥₁(𝑡), 𝑥₂(𝑡) of the system (7), corresponding to the control 𝑣(⋅)∈𝐷(𝑇), the equality Φ(𝑥₁(𝜏),𝑥₂(𝜏))=0 is valid, then the following inequalities hold: 𝑥̇₁(𝜏)<0,𝑥̇₂(𝜏)<0. (29)

Proof. It is sufficient to show the validity of inequalities (29). The first inequality follows from Lemma 4. In order to prove the second inequality we substitute the function 𝑥₂=𝑓(𝑥₁) into the second equation of the system (7). After necessary transformations we find the expression 𝑥̇₂𝑏𝑥̇(𝜏)=¹(𝜏)𝜎(𝑥₁(𝜏)−1)²+𝜎(𝑥₁(𝜏)−1)−𝑏<0. (30) The statement is proven.

Remark 11. Lemma 10 is valid for the optimal control 𝑣_∗(𝑡) and its corresponding optimal trajectories 𝑥^∗₁(𝑡), 𝑥^∗₂(𝑡), 𝑡∈[0,𝑇].

Now, for all (𝑥₁,𝑥₂)∈𝑊 for which Φ(𝑥₁,𝑥₂)≠0 we consider the function Ω(𝑥₁,𝑥₂𝜕)=−(𝜕𝑥₁((𝑥₁−1)𝑥₂Φ(𝑥₁,𝑥₂)𝜕)+𝜕𝑥₂(𝑥²₂Φ(𝑥₁,𝑥₂)𝑥))=−²(Φ(𝑥₁,𝑥₂))²⋅(𝑥₂(𝜎(𝑥₁−1)−2𝑏)−2𝑏(𝑥₁−1)). (31)

Let us study the curve Ω(𝑥₁,𝑥₂)=0. We have the following function: 𝑥₂=𝑔(𝑥₁)=2𝑏(𝑥₁−1)𝜎(𝑥₁.−1)−2𝑏 (32) We see that it is defined only by 𝑥₁>𝑥₁, where 𝑥₁=1+2𝑏𝜎>𝑥₁. (33)

Moreover, the relationships hold lim_𝑥1→+∞𝑔(𝑥₁)=𝑥₂,lim_𝑥1→𝑥₁𝑔(𝑥₁)=+∞, (34) where 𝑥₂=2𝑏/𝜎.

Note that for the derivatives 𝑔(̇𝑥₁), 𝑔(̈𝑥₁) of the function 𝑥₂=𝑔(𝑥₁) at 𝑥₁>𝑥₁ we have the expressions𝑔(̇𝑥₁)=−4𝑏²(𝜎(𝑥₁−1)−2𝑏)²̈<0,𝑔(𝑥₁)=8𝑏²𝜎(𝜎(𝑥₁−1)−2𝑏)³>0. (35)

Therefore, the function 𝑥₂=𝑔(𝑥₁) is decreasing and concave up. The graph of this function and the allocation of signs of the function Ω(𝑥₁,𝑥₂) are shown in Figure 5.

Figure 5 

Allocation of signs of the function Ω(𝑥1,𝑥2).

For the curves Φ(𝑥₁,𝑥₂)=0, Ω(𝑥₁,𝑥₂)=0 we have the following statements.

Lemma 12. In the region 𝑊 the curves Φ(𝑥₁,𝑥₂)=0, Ω(𝑥₁,𝑥₂)=0 have no common points.

The validity of this fact follows from the easily checked inequality 𝑓(𝑥₁)<𝑥₂ and the properties of the functions 𝑥₂=𝑓(𝑥₁), 𝑥₂=𝑔(𝑥₁) obtained above.

Lemma 13. In the region 𝑊 there are no interval Δ⊂[0,𝑇] and values of control 𝑣(⋅)∈𝐷(𝑇) such that the curve Ω(𝑥₁,𝑥₂)=0 is a trajectory of the system (7). Moreover, if there exists a moment of time 𝜃∈(0,𝑇) such that for the trajectories 𝑥₁(𝑡), 𝑥₂(𝑡) of the system (7), corresponding to the control 𝑣(⋅)∈𝐷(𝑇), the equality Ω(𝑥₁(𝜃),𝑥₂(𝜃))=0 is valid, then the following inequalities hold: 𝑥̇₁(𝜃)<0,𝑥̇₂(𝜃)<0. (36)

Proof. It is sufficient to show the validity of inequalities (36). The first inequality follows from Lemma 4. In order to prove the second inequality we substitute the function 𝑥₂=𝑔(𝑥₁) into the second equation of the system (7). After necessary transformations we find the expression 𝑥̇₂⋅(𝜃)=−𝑏2𝜎(𝑥₁(𝜃)−1)²+(𝜎+2𝑣(𝜃))(𝑥₁(𝜃)−1)+2𝑏𝜎(𝑥₁(𝜃)−1)−2𝑏<0. (37) The statement is proven.

Remark 14. Lemma 13 is valid for the optimal control 𝑣_∗(𝑡) and its corresponding optimal trajectories 𝑥^∗₁(𝑡), 𝑥^∗₂(𝑡), 𝑡∈[0,𝑇].

Next, using the preceding properties of the functions Φ(𝑥₁,𝑥₂), Ω(𝑥₁,𝑥₂) we conduct the arguments, which are similar to the arguments presented in [16]. One of these arguments is the use of Green's Theorem. In this way, we establish the validity of the bang-bang principle for the optimal control problems (7), and (8).

Theorem 15. Let 𝑣_∗(𝑡) be the optimal control, and 𝑥^∗₁(𝑡), 𝑥^∗₂(𝑡) are its corresponding optimal trajectories in the problems (7), and (8). Then at every moment of time 𝑡∈(0,𝑇), for which in the region 𝑊 the relationships hold Φ(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡))≠0,Ω(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡))≠0, (38) the control 𝑣_∗(𝑡) takes only values 𝑣₁ and 𝑣₂.

Now, using the definitions of the sets 𝑄, 𝑊 and results from Lemmas 7, 10, and 13 we will strengthen Theorem 15. As a result, we will formulate the final statement about the type of the optimal control 𝑣_∗(𝑡).

Theorem 16. At the problems (7), and (8) the optimal control 𝑣_∗(𝑡) is the piecewise constant function, which takes only values 𝑣₁ and 𝑣₂.

Next, we study the question about the number of switchings of the optimal control 𝑣_∗(𝑡).

At first, we note that as it follows from Lemma 7, if the optimal trajectory (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) gets into the region 𝑄, then the corresponding optimal control 𝑣_∗(𝑡) has no switchings and takes only value 𝑣₂.

Now, let us consider the region 𝑊. We conduct the arguments, which are similar to the arguments presented in [16]. As in Theorem 15 one of these arguments is the use of Green's Theorem. In this way, we obtain the following statements.

Lemma 17. Let 𝑣_∗(𝑡) be the optimal control, and 𝑥^∗₁(𝑡), 𝑥^∗₂(𝑡) are its corresponding optimal trajectories in the problems (7), and (8). Suppose there exists such interval Δ⊂[0,𝑇], on which the following inequalities hold: Φ(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡))>0,Ω(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡))<0. (39) Then on this interval Δ the control 𝑣_∗(𝑡) takes either the constant value from {𝑣₁;𝑣₂}, or it has the moment of switching 𝜏∈Δ, for which 𝑣_∗(𝜏−0)=𝑣₂,𝑣_∗(𝜏+0)=𝑣₁. (40)

Lemma 18. Let 𝑣_∗(𝑡) be the optimal control, and 𝑥^∗₁(𝑡), 𝑥^∗₂(𝑡) are its corresponding optimal trajectories in the problem (7), (8). Suppose there exists such interval Δ⊂[0,𝑇], on which the following inequalities hold: Φ(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡))>0,Ω(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡))>0. (41) Then on this interval Δ the control 𝑣_∗(𝑡) takes either the constant value from {𝑣₁;𝑣₂}, or it has the moment of switching 𝜃∈Δ, for which 𝑣_∗(𝜃−0)=𝑣₁,𝑣_∗(𝜃+0)=𝑣₂. (42)

Lemma 19. Let 𝑣_∗(𝑡) be the optimal control, and 𝑥^∗₁(𝑡), 𝑥^∗₂(𝑡) are its corresponding optimal trajectories in the problems (7), and (8). Suppose there exists such interval Δ⊂[0,𝑇], on which the following inequalities hold: Φ(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡))<0,Ω(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡))>0. (43) Then on this interval Δ the control 𝑣_∗(𝑡) takes either the constant value from {𝑣₁;𝑣₂}, or it has the moment of switching 𝜂∈Δ, for which 𝑣_∗(𝜂−0)=𝑣₂,𝑣_∗(𝜂+0)=𝑣₁. (44)

Lemma 20. Let there exists the moment of time 𝜏∈(0,𝑇) such that for the optimal trajectories 𝑥^∗₁(𝑡), 𝑥^∗₂(𝑡) the equality Ω(𝑥^∗₁(𝜏),𝑥^∗₂(𝜏))=0 is valid. Then the value 𝜏 is not the moment of switching for the corresponding optimal control 𝑣_∗(𝑡).

From Lemmas 17–19 we make the following conclusion.

Corollary 21. Let 𝑣_∗(𝑡) be the optimal control, and 𝑥^∗₁(𝑡), 𝑥^∗₂(𝑡) are its corresponding optimal trajectories. Let one suppose that 𝑉⊂[0,𝑇] is the interval of the maximum length, on which the functions Φ(𝑥₁,𝑥₂), Ω(𝑥₁,𝑥₂) are sign defined. Then on this interval 𝑉 the control 𝑣_∗(𝑡) is piecewise constant function with values from {𝑣₁;𝑣₂}, which has at most one moment of switching.

The possible types of switchings of the optimal control 𝑣_∗(𝑡) in dependence on signs of the functions Φ(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)), Ω(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) are presented by Lemmas 17–19.

Let us introduce the setsΠ={(𝑥₁,𝑥₂)∈𝑊∶𝑥₁≤𝑥₁}{(𝑥₁,𝑥₂)∈𝑊∶𝑥₁>𝑥₁,𝑥₂<𝑓(𝑥₁)},Ψ={(𝑥₁,𝑥₂)∈𝑊∶𝑥₁<𝑥₁≤𝑥₁,𝑥₂≥𝑓(𝑥₁)}{(𝑥₁,𝑥₂)∈𝑊∶𝑥₁>𝑥₁,𝑓(𝑥₁)≤𝑥₂<𝑔(𝑥₁)},𝐺={(𝑥₁,𝑥₂)∈𝑊∶𝑥₁>𝑥₁,𝑥₂≥𝑔(𝑥₁)}. (45) It is clear that 𝑊=Π⋃Ψ⋃𝐺. Sets Π, Ψ, 𝐺 are shown in Figure 6.

Figure 6 

Signs of functions Φ(𝑥1,𝑥2), Ω(𝑥1,𝑥2) and sets Π, Ψ, and 𝐺.

Summarizing the preceding arguments we conclude the following.

If the optimal trajectory (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) gets into the region 𝐺 (higher than the curve Ω(𝑥₁,𝑥₂)=0), then the corresponding optimal control 𝑣_∗(𝑡) is either constant with values from {𝑣₁;𝑣₂}, or it has the switching from 𝑣₂ to 𝑣₁.

If the optimal trajectory (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) gets into the region Ψ (between the curves Φ(𝑥₁,𝑥₂)=0 and Ω(𝑥₁,𝑥₂)=0), then the corresponding optimal control 𝑣_∗(𝑡) is either constant with values from {𝑣₁;𝑣₂}, or it has the switching from 𝑣₁ to 𝑣₂.

If the optimal trajectory (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) gets into the region Π (to the left of the curve Φ(𝑥₁,𝑥₂)=0 and to the right of the line 𝑥₁=1), then the corresponding optimal control 𝑣_∗(𝑡) is either constant with values from {𝑣₁;𝑣₂}, or it has the switching from 𝑣₂ to 𝑣₁.

All these situations for the optimal trajectory (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) are shown in Figure 7.

Figure 7 

Possible position of optimal trajectory (𝑥∗1(𝑡),𝑥∗2(𝑡)).

Now, let us strengthen Lemma 19. We have the statement.

Lemma 22. Let 𝑣_∗(𝑡) be the optimal control, and 𝑥^∗₁(𝑡), 𝑥^∗₂(𝑡) are its corresponding optimal trajectories. Suppose that on the some interval Δ⊂[0,𝑇] the trajectory (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) gets into the region Π. Then on this interval Δ the control 𝑣_∗(𝑡) is the constant function with value 𝑣₂.

Proof. Let us assume the contradiction. Then from Lemma 19 it follows that the optimal control 𝑣_∗(𝑡) has the moment of switching 𝜏∈Δ, for which the equalities (44) hold.
Next, we consider two possible cases.
Case 23. For the trajectory (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) the inclusion (𝑥^∗₁(𝑇),𝑥^∗₂(𝑇))∈Π (46) is valid. Then from Lemma 6 it follows the existence of the interval (𝜃,𝑇], 𝜃>𝜏, on which the control 𝑣_∗(𝑡) takes value 𝑣₂. Therefore, on the interval (𝜏,𝜃) there exists the moment of time 𝜂, in which the switching of the control 𝑣_∗(𝑡) from 𝑣₁ to 𝑣₂ occurs. This fact contradicts Lemma 19.Case 24. For the trajectory (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) the inclusion (𝑥^∗₁(𝑇),𝑥^∗₂(𝑇))∈𝑄 (47) is valid. From Lemmas 6 and 7 we again have the existence of the interval (𝜃,𝑇], 𝜃>𝜏, on which the control 𝑣_∗(𝑡) takes value 𝑣₂. Moreover, the inclusion (𝑥^∗₁(𝜃),𝑥^∗₂(𝜃))∈Π (48) is true. Therefore, as in the previous case, on the interval (𝜏,𝜃) there exists the moment of time 𝜂, in which the switching of the control 𝑣_∗(𝑡) from 𝑣₁ to 𝑣₂ occurs. This fact again contradicts Lemma 19.
Finally, we have the contradiction. Our assumption is wrong. The control 𝑣_∗(𝑡) is constant on the interval Δ with values from {𝑣₁;𝑣₂}.
Using analogues arguments we conclude that the control 𝑣_∗(𝑡) does not take value 𝑣₁. Then the optimal control 𝑣_∗(𝑡) on the interval Δ takes only value 𝑣₂. This completes the proof.

Now, for the optimal trajectory (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) we analyze into the regions 𝑄, Π, Ψ, 𝐺 the possible mutual dispositions of the initial point (𝑥⁰₁,𝑥⁰₂) and the final point (𝑥^∗₁(𝑇),𝑥^∗₂(𝑇)), comparing the results from Lemmas 17, 18, 20, and 22. Based on such analysis we obtain the possible types of the corresponding optimal control 𝑣_∗(𝑡).

Theorem 25. For the problem (7), (8) the optimal control 𝑣_∗(𝑡) is either a constant function 𝑣_∗(𝑡)=𝑣₂,𝑡∈[0,𝑇], (49)
or a piecewise constant function with one switching𝑣_∗𝑣(𝑡)=¹,if0≤𝑡≤𝜃_∗,𝑣₂,if𝜃_∗<𝑡≤𝑇, (50) where 𝜃_∗∈(0,𝑇) is the moment of switching, or a piecewise constant function with two switchings 𝑣_∗𝑣(𝑡)=²,if0≤𝑡≤𝜃^∗₁,𝑣₁,if𝜃^∗₁<𝑡≤𝜃^∗₂,𝑣₂,if𝜃^∗₂<𝑡≤𝑇, (51) where 𝜃^∗₁,𝜃^∗₂∈(0,𝑇) is the moments of switching.

7. Solution of the Optimal Control Problem

We will conduct the subsequent solution of the optimal control problems (7), and (8) in the following way.

Let us introduce the set 𝑆={(𝜃₁,𝜃₂)∈ℝ²∶0≤𝜃₁≤𝜃₂≤𝑇}. (52)

Next, for any point (𝜃₁,𝜃₂)∈𝑆 we define the control of the type 𝑣_𝜃𝑣(𝑡)=²,if0≤𝑡≤𝜃₁,𝑣₁,if𝜃₁<𝑡≤𝜃₂,𝑣₂if𝜃₂<𝑡≤𝑇. (53)

It is easy to see that the control 𝑣_𝜃(𝑡) includes all possible types (49)–(51) of the optimal control 𝑣_∗(𝑡) from Theorem 25 at the corresponding values of switchings 𝜃₁, 𝜃₂.

Next, we substitute the control 𝑣_𝜃(𝑡) into equations of the system (7) and integrate it on the interval [0,𝑇]. Then we substitute the corresponding formula for the function 𝑥₂(𝑡) into the objective function (8).

Hence, we have the function of two variables 𝐹(𝜃₁,𝜃₂)=𝐽(𝑣_𝜃),(𝜃₁,𝜃₂)∈𝑆. (54)

Therefore, the optimal control problems (7), and (8) can be restated as a problem of the finite dimensional optimization𝐹(𝜃₁,𝜃₂)→min_{(𝜃1,𝜃2)∈𝑆}. (55) The methods of the numerical solution of the problem (55) are well developed [27, 28].

8. Computer Simulation

Our theoretical results obtained in the previous sections allow us to select an optimal successful strategy of ASP. In [4] parameter estimation and verification of the model measurement values from a waste water plant were obtained for every hour of an operating period of one week.

Values of 𝑥₂(𝑡) will be determined by total organic carbon content in the influent and 𝑥₁(𝑡) by the concentration of the suspended solid in the aeration tank.

Let us show our results for the following model parameters: 𝑢₁=0.1lb/min,𝑢₂𝑎=1.0lb/min,𝑇=20hours,₁=0.7lb/gal,𝑎₂𝑥=0.9lb/gal,𝑌=3.0,₀=2.5lb/gal,𝑠₀=2.0lb/gal,𝜇=0.1. (56)

The optimal control 𝑣_∗(𝑡) has the type (50) with one moment of switching at 𝜃_∗=6.2 hours, which was obtained numerically. So, if we select such optimal policy 𝑣₁→𝑣₂, then the concentration of the polluted water 𝑥₂(𝑡) will be minimized on the interval [0,𝑇], where 𝑇 is final operation time (see Figure 8).

Figure 8 

Optimal trajectory 𝑥∗2(𝑡).

9. Conclusions

Activated sludge process involves complex and subtle relationships among a relatively large number of variables. The model investigated in our paper is not intended to be the best ASP model. However, it is nonlinear and it has a bounded control, which makes it quite interesting from the mathematical point of view.

We found the type of optimal control by means of the combination of the Pontryagin Maximum Principle and Green's Theorem. This allowed us to reduce a complex optimal control problem (7), (8) to one of finite dimensional optimization (55).

Our mathematical investigation of the activated sludge process can be summarized by the following components:(1)Complete investigation of a model (7) of the activated sludge process with one bounded control and its corresponding optimal control problem.(2)Development of an optimal control strategy for the recycle flow rate analytically.(3)Computer simulation of the controlled activated sludge process for different model parameters.

Finally, it should be noted that the ideas presented in this study can be applied to other control systems with similar properties.

Acknowledgment

The authors thank Dr. Paul Deignan for his helpful suggestions and style recommendations.