An Algorithm for Maximizing the Biogas Production in a Chemostat

Abstract

In this work, we deal with the optimal control problem of maximizing biogas production in a chemostat. The dilution rate is the controlled variable, and we study the problem over a fixed finite horizon, for positive initial conditions. We consider the single reaction model and work with a broad class of growth rate functions. With the Pontryagin maximum principle, we construct a one-parameter family of extremal controls of type bang-singular arc. The parameter of these extremal controls is the constant value of the Hamiltonian. Using the Hamilton–Jacobi–Bellman equation, we identify the optimal control as the extremal associated with the value of the Hamiltonian, which satisfies a fixed point equation. We then propose a numerical algorithm to compute the optimal control by solving this fixed point equation. We illustrate this method with the two major types of growth functions of Monod and Haldane.

Appendix: Reduced Model

In this final part, we provide a HJB proof for the optimal synthesis for the reduced model, that is, the case where the initial data satisfy $s_{\text {in}}=x_0+s_0$. In particular, we show how the fixed point characterization of the optimal control can be used analytically in a special case when the dynamics reduces to a single equation. A well-known property of the chemostat model is that the set $I:=\{ (x,s)\in \mathbb {R} :x+s=s_{\text {in}} \}$ is invariant for dynamics (1), and thus, for initial conditions in I, the dynamics reduce to $\dot{s} = \big (D - \mu (s) \big ) (s_{\text {in}} -s).$ This special case was solved in [13], with the following assumptions

• (H1) The function $s \mapsto \mu (s) (s_{\text {in}} -s)$ has a unique maximizer $s^*$ on $[0,s_{\text {in}}]$.

• (H2) The upper bound on the controls is such that $D_{\max } > \mu (s^*)$.

The optimal control is then $D^*(s)= 0$ if $s>s^*$, $D^*(s)= D_{\max }$ if $s<s^*$ and $D^*(s)=\mu (s^*)$ if $s=s^*$. Here, we will give another proof of the optimality of this control, by using the fixed point characterization. First, we can identify the control $D^*$ as a control of type (15) where the singular arc is reduced to $s=s^*$. In other words, it corresponds to the control $\psi _{h^*}$ where $h^*$ satisfies Eq. (9) for the singular arc with $s=s^*$, which in this case is $h^* \mu '(s^*) = \mu (s^*)^2$. Next, since $s^*$ is a maximizer, we have $\mu '(s^*)(s_{\text {in}}-s^*) - \mu (s^*) =0$ and therefore $h^*= \mu (s^*)(s_{\text {in}} - s^* )$.

To prove the optimality of $\psi _{h^*}$, we must now show that $h^*$ is a fixed point of the map** $h \mapsto - \partial _{t_0} J(t_0,x_0,s_0,\psi _{h})$. For this, we first study the trajectories obtained with the feedback control $\psi _{h^*}$. We denote in the remainder of the section the right-hand side of the differential equation for $s(\cdot )$ with control $\psi _{h^*}$ as $f(s):= (\psi _{h^*}(s) - \mu (s) ) (s_{\text {in}} -s)$.

Notice that for $s>s^*$ we have $f(s)=-\, \mu (s) (s_{\text {in}} -s) <0$ and for $s<s^*$ we have $f(s)= (D_{\max } - \mu (s) ) (s_{\text {in}} -s)>0$ from assumption (H2). Thus, $s^*$ is reachable from any initial condition in I with control $\psi _{h^*}$. We define the time $t^*$ when $s^*$ is reached, from a given initial condition $s_0 \in [0,s_{\text {in}}]$ and initial time $t_0$ with control $\psi _{h^*}$, that is, $t^*:= \inf \left\{ t \geqslant t_0 : s(t,t_0,s_0,\psi _{h^*} ) = s^*\right\} .$ Finally, note that with control $D=\mu (s^*)$ the point $s=s^*$ becomes a steady state. Therefore, the trajectories with control $\psi _{h^*}$ are

$$\begin{aligned} s(t) = {\left\{ \begin{array}{ll} s(t,t_0,s_0,\psi _{h^*} ), &{}\quad \text { for } t_0 \leqslant t \leqslant \min (t^*, T), \\ s^*, &{}\quad \text { for } \min (t^*, T)\leqslant t\leqslant T. \end{array}\right. } \end{aligned}$$

We can now compute $\partial _{t_0} J(t_0,x_0,s_0,\psi _{h^*})$, and for this, we need the following.

Lemma A.1

For any initial condition $(x_0,s_0) \in I$, for the trajectories with control $\psi _{h^*}$, we have $\partial _{t_0} s(t) = -\, f( s (t))$ at time $t \in [t_0,t^*]$.

Proof

We can write the differential equation satisfied by $s(\cdot )$ as $s(t) = s_0 + \displaystyle \int _{t_0}^t f(s(\tau )) \, \text {d}\tau$, and differentiating, we get $\partial _{t_0} s(t) = -\, f( s_0) + \displaystyle \int _{t_0}^t f'(s(\tau ))\partial _{t_0} s(\tau ) \, \text {d}\tau .$ This is a linear differential equation, and the solution is $\partial _{t_0} s(t) = -\, f( s_0) \exp \left( \displaystyle \int _{t_0}^t f'(s(\tau )) \, \text {d}\tau \right) .$ Now, as f(s(t)) does not change sign for $t\in [t_0,t^*)$ and since f(s(t)) is the derivative of s(t), we have $\displaystyle \int _{t_0}^t f'(s(\tau )) \, \text {d}\tau = \int _{s_0}^{s(t)} \frac{f'(s)}{f(s)} \text {d}s = \ln \left( \frac{f(s(t))}{f(s_0)} \right) .$$\square$

We are now in a position to prove the optimality of the feedback control proposed earlier.

Proposition A.1

For any initial condition $(x_0,s_0) \in I$ and for any initial time $t_0$ such that $s^*$ is reachable, that is, when $t^* \leqslant T$, we have $\partial _{t_0} J(t_0,x_0,s_0,\psi _{h^*}) = -\mu ( s^* ) (s_{\text {in}}-s^*)$, so that $\psi _{h^*}$ is the optimal control.

Proof

We start by writing the cost as

$$\begin{aligned} J(t_0,x_0,s_0,\psi _{h^*}) = \int _{t_0}^{t^*} \mu (s(t))(s_{\text {in}}-s(t)) \, \text {d}t + (T-t^*) \mu (s^*)(s_{\text {in}}-s^*) \end{aligned}$$

differentiating with respect to $t_0$ we get

$$\begin{aligned} \partial _{t_0} J(t_0,x_0,s_0,\psi _{h^*}) = -\,\mu (s_0)(s_{\text {in}}-s_0) + \int _{t_0}^{t^*} \partial _{s} \big ( \mu (s(t))(s_{\text {in}}-s(t)) \big ) \partial _{t_0} s(t)\, \text {d}t. \end{aligned}$$

Note that the terms with $\partial _{t_0}t^*$ cancel out because $s(t^*)=s^*$. Now, using Lemma A.1 we get

$$\begin{aligned} \partial _{t_0} J(t_0,x_0,s_0,\psi _{h^*})&= -\,\mu (s_0)(s_{\text {in}}-s_0) - \int _{t_0}^{t^*} \partial _{s} \big ( \mu (s(t))(s_{\text {in}}-s(t)) \big ) \dot{s}(t) \, \text {d}t \\&= -\,\mu (s_0)(s_{\text {in}}-s_0) - \int _{t_0}^{t^*} \frac{\text {d}}{\text {d}t} \big ( \mu (s(t))(s_{\text {in}}-s(t)) \big ) \, \text {d}t \\&= -\,\mu ( s^* ) (s_{\text {in}}-s^*). \end{aligned}$$

We conclude by recalling that $h^* = \mu ( s^* ) (s_{\text {in}}-s^*)$, and therefore, $h^*$ is a fixed point of $h \mapsto - \partial _{t_0} J(t_0,x_0,s_0,\psi _{h})$. $\square$