# Optimal feedback synthesis and minimal time function for the bioremediation of water resources with two patches

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## Deterministic optimal control for continuous bioremediation pro-cesses

In the first part of this thesis we study minimal time strategies for the treatment of pollution in large water volumes, such as lakes or natural reservoirs, using a single continuous bioreactor that operates in a quasi-steady state. The control consists of feeding the bioreactor from the resource, with clean output returning to the resource with the same flow rate. We drop the hypothesis of homogeneity of the pollutant concentration in the water resource by proposing three spatially structured models, each consisting of configurations of two homogeneous zones where the nature of the connections between them as well as the differentiation of the zones may depend on the structure of the problem; for instance, if most of the water stream is pass-ing from one zone to another, this effect can be seen as a transport term between two zones interconnected in a series configuration; this type of configuration is also suitable to model gradients of concentration. Other type of connection that models the effect of (mostly) pure diffusion between two zones is a configuration in parallel. The confinement of one part of the resource due to geography or reactor design also defines clearly differentiated zones that lead to strong inhomogeneity. Naturally, in a highly inomogeneous resource, in which many zones can be differentiated, a compartimental model can be implemented as an approximation of the dynamics of the system. As it was mentioned in Section 1.3, the problem of the de-pollution of water resources consists in treating the resource by means of a bioreactor leading the pollutant concentration in the resource to decrease under a certain level s > 0 considered safe in environmental terms. In any of the three studied spatial configurations the target of the process will be leading the substrate concentration in both zones under s. If we define as s = (s1; s2) 2 R2+ the spatial variable that denotes the pollutant concentrations in the first and second zones respectively, the target set will be T := fs = (s1; s2) 2 R+2 j s1 s ; s2 s g: (1.27).

Study of stochastic modeling of sequencing batch reactors

The second part of this thesis concerns the modelling and study of a stochastic model of se-quencing batch reactor. This model is obtained as a limit of a sequence of continuous time Markov processes whose jump rates and jump laws are prescribed following the rules of repli-cation and death of microorganisms, input and consumption of substrate, and input flow of water to the tank, that is, processes whose jump law depend only on the current state of the system. The obtained model is given by the controlled stochastic differential equation 8 dxt = (st) vt xtdt + ~ r dWt;x(0) = y; vt ut dvt = utdt; v(0) = w; > ut xt > dst = (st)xt + vt (sin st) dt; s(0) = z; (1.36).
where ut 2 u u = 0 if v t [0;:max] is a control process that satisfies a state constraint: t ( ) vmax, W = (Wt)t 0 is a one-dimensional Brownian motion, and 0 is an individual death rate; this term is usually omitted in the deterministic model (see Section 1.2.3). We study the existence and uniqueness of solutions of equation (1.36) with null death rate = 0 and yield coefficient Y = 1. The coefficients of this equation do not satisfy the usual assumptions of Lipschitz and sublinear growth; nevertheless, we prove that for every initial condition = (y; z; w) 2 (0; 1) [0; sin] [vmin; vmax] and admissible control u with respect to the brownian W , that is, progressively measurable with respect to the Brownian filtration, there exists a solution of equation (1.36) up to the extinction time E = infft 0 j xt = 0g. We prove the following result:
Proposition 1.7 (Chapter 5, Proposition 5.7) Let u = (ut)t 0 be an admissible control with respect to W = (Wt)t 0 and Xu = (xu; su; vu) the solution of (1.36). The probability that xu = (xut)t 0 hits 0 at some time instant is positive.

Minimal time optimal control problem

In this section we describe our problem as a minimal time optimal control problem. For this, first note that, thanks to Assumption 2.3 and Proposition 2.4, to know the time of depollution it is enough to measure the dead zone pollutant’s concentration s2. Consequently, our target set can be defined as T = f(s1; s2) 2 R+2 j s2 s g = R+ [0; s ]: (2.8).
On the other hand, due to Proposition 2.5, we expect to deal with admissible bounded con-trols. Since we are interested in controls s?r( ) 2 A that satisfy s?r(t) 2 [0; s1(t)] for all t 0, our admissible control set will be restricted to AU := fs?r 2 A j s?r(t) 2 [0; s1(t)]; for all t 0g.
Hence, in this work, we consider the following optimization problem: (P) inf fT j s(0) = z; s(T ) 2 T g: s?r2AU.
In what follows will show that the optimal solution s?r of (P) is given by a feedback control that coincides with the solution of the problem with the homogeneous zone.
Proof. The dynamics (2.4) are continuous with respect to (s; s?r). Then, for fixed s 2 R2+, the velocity set generated as the image of the compact and connected set [0; s1] by dynamics (2.4) is a compact and connected set and, since the control variable s?r only acts in the first component, the velocity set is a segment and thus convex. This result, along with Propositions 2.4 and 2.5, assure that hypotheses of Filippov’s theorem [17, Theorem 9.2.i] are satisfied, thus proving the existence of an optimal control for problem (P).

### Application of Pontryagin maximum principle

The Hamiltonian of problem (P) is the mapping H : R2+ R2 R+ ! R given by H(s; ; s?r) = 1 + 1 (s?r)(s?r s1) + ( 1 1 + 2 2)(s1 s2).
According to the Pontryagin maximum principle , if s?r( ) is a solution of (P) with corresponding optimal trajectory s( ) = (s1( ); s2( )), then there exists an adjoint state ( ) = ( 1( ); 2( )), with no both components equal to zero at the same time, such that ( _2 = 11+2 2 ; (2.9) _1 = ( (sr?) + 1) 1 2 2 ; and H(s(t); (t); sr?(t)) = max H(s(t); (t); sr): (2.10) sr2[0;s1(t)].
With these dynamics we can associate the transversality conditions 1(T) = 0 and 2(T) < 0; (2.11). where T denotes the optimal time (this notation is consistent with the definition of problem (P)). Remark 2.2 It is possible to write the maximum principle even though the control set depends on the state s1(t), at each time instant t 0. Indeed, it suffices to do a change of variable u(t) 2 [0; 1] to have a control u( ) whose control set is compact and convex and whose associated s?r(t) = u(t)s1(t) is an admissible control, rewrite the dynamics accordingly, and restate the maximum principle. The result is equivalent to our approach.

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#### Stochastic SBR model

In what follows, we denote by R the set of real numbers, N and Z the sets of natural and integer numbers respectively, R+ and R?+ the sets of non-negative and positive real numbers respectively. For a set A and n 2 N we denote An := f x = (x ; : : : ; x ) x 2 A; i = 1; : : : ; n g. For f g 1 n mj i n C k R n k n ^ . For A R (A; ) two real numbers a; b we define a b := min a; b , we denote by (Cb (A; R ) ) the set of k times differentiable (bounded) functions f : A ! R with contin- k (A)=C k (A; R) k k n ) denotes uous k th derivatives, and C (Cb (A) n= Cb (A; R)); D([0; 1); R the set of right continuous functions f : [0;1) ! R with left limits everywhere. For a topological space (X; ) we denote by B(X) its Borel algebra, and for a measurable space (S; S) we define P(S; S) (or simply P(S)) the set of probability measures on (S; S).
Denote by x, s, and v the biomass and substrate concentration, and culture volume inside the bioreactor vessel. We suppose that water with a constant concentration of nutrient sin and without bacteria is poured into the bioreactor at an inflow rate u 2 [0; umax]. We adopt the usual assumptions that the growth rate of microorganisms is proportional to the mass of microorganisms and it depends of the substrate concentration by means of the uptake function ( ) . We introduce an individual microbial death rate 0. We suppose that the yield coefficient of the reaction, which is the amount of biomass produced by the bacterial specie when one unit of substrate is consumed by the reaction, is a constant Y > 0. We suppose that the culture volume is bounded between a minimum volume vmin assumed to be the lower level to which the tank is emptied during the draw mode, and a maximum volume vmax given by the maximum operative capacity of the tank. For the growth function ( ) we make the following assumption:
Assumption 5.1 The growth function ( ) is defined in [0; 1), (0) = 0, is non-negative, bounded by above by a constant max > 0, and is Lipschitz continuous.
Our aim in this section is to develop a stochastic population process describing the dy-namics of a bacterial specie, substrate, and volume inside the SBR. For this, we consider an individual-based tridimensional birth and death process that represents the discretized total number of microbial cells, substrate molecules, and the water molecules, scaled by scale param-eters Kx; Ks; Kv > 0 that represent the change of units from number of molecules into grams for a large population. We suppose that two of the following events cannot occur at the same time: the division of a microbial cell, the death of a cell, the entry of a unit of substrate into the tank (and at the same time the entry of unit of water), and the consumption of a unit of substrate by a cell. Let us define x^Kt the amount of cells composing the biomass scaled by the parameter Kx, s^Kt the amount of substrate molecules scaled by the parameter Ks, and v^tK the amount of water molecules scaled by the parameter Kv at the time instant t. At the same time, we intro-duce demographic randomness by introducing a perturbation parameter 0 in the birth and death rates of biomass, following . This procedure generates a pure jump Markov process tK := (^xKt ; s^Kt ; v^tK ) that takes values in the state space DK3 := (Z=Kx) (Z=Ks) (Z=Kv).

1 Introduction
1.1 Bioprocesses and bioremediation
1.2 Mathematical models of bioreactors and classical results
1.2.1 Mathematical models
1.2.2 Classical results on chemostats
1.2.3 Classical results on SBRs
1.2.4 Stochastic models of bioreactors
1.3 Model of inhomogeneous lake
1.4 Singular perturbations
1.5 Contributions of the thesis
1.5.1 Deterministic optimal control for continuous bioremediation processes
1.5.2 Study of stochastic modeling of sequencing batch reactors
2 Bioremediation of natural water resources via Optimal Control techniques
2.1 Introduction
2.2 Mathematical model
2.2.1 Description of the dynamics
2.2.2 Preliminary results on the dynamics
2.2.3 Minimal time optimal control problem
2.3 Application of Pontryagin maximum principle
2.4 The effect of recirculation
2.5 Numerical Simulations
2.6 Conclusion
3 Optimal feedback synthesis and minimal time function for the bioremediation of water resources with two patches
3.1 Introduction
3.2 Definitions and preliminaries
3.3 Study of the relaxed problem
3.4 Synthesis of the optimal strategy
3.5 Study of the minimal-time function
3.6 Numerical illustrations
3.7 Conclusion
4 Minimal-time bioremediation of natural water resources with gradient of polluxiitant
4.1 Introduction
4.2 Definitions and preliminaries
4.3 Optimal control problem
4.4 Numerical simulations
4.5 Conclusions
5 Stochastic modelling of sequencing batch reactors for wastewater treatment
5.1 Introduction
5.2 Stochastic SBR model
5.3 Existence of solutions of the controlled stochastic model
5.4 The optimal reach-avoid problem
5.5 Numerical simulations and conclusions
6 Conclusions and perspectives
Bibliography

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