Optimal control of bacterial growth for the maximization of metabolite production

Abstract

Microorganisms have evolved complex strategies for controlling the distribution of available resources over cellular functions. Biotechnology aims at interfering with these strategies, so as to optimize the production of metabolites and other compounds of interest, by (re)engineering the underlying regulatory networks of the cell. The resulting reallocation of resources can be described by simple so-called self-replicator models and the maximization of the synthesis of a product of interest formulated as a dynamic optimal control problem. Motivated by recent experimental work, we are specifically interested in the maximization of metabolite production in cases where growth can be switched off through an external control signal. We study various optimal control problems for the corresponding self-replicator models by means of a combination of analytical and computational techniques. We show that the optimal solutions for biomass maximization and product maximization are very similar in the case of unlimited nutrient supply, but diverge when nutrients are limited. Moreover, external growth control overrides natural feedback growth control and leads to an optimal scheme consisting of a first phase of growth maximization followed by a second phase of product maximization. This two-phase scheme agrees with strategies that have been proposed in metabolic engineering. More generally, our work shows the potential of optimal control theory for better understanding and improving biotechnological production processes.

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Notes

  1. 1.

    Some of the BOCOP definition files and C++ programs that we prepared in order to conduct numerical simulations for this paper have been uploaded to http://www-sop.inria.fr/members/Jean-Luc.Gouze/JMBcode/code.zip.

  2. 2.

    The code has been uploaded to http://www-sop.inria.fr/members/Jean-Luc.Gouze/JMBcode/code.zip.

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Acknowledgements

This work was supported in part by the PIA project Reset (ANR-11-BINF-0005), ANR project Maximic (ANR-17-CE40-0024-01), Inria IPL AlgaeInSilico, and Labex SIGNALIFE (ANR-11-LABX-0028-01). The authors thank Johannes Geiselmann and Eugenio Cinquemani for discussions and comments on the manuscript.

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Correspondence to Ivan Yegorov.

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I. Yegorov: Also known as I. Egorov.

Appendices

Appendix A: Time-varying bounds for state variables in Model 1

As can be directly verified, the unique solutions to the following Cauchy problems can serve as time-varying bounds (32) for the state variables \( {\hat{p}}, {\hat{x}} \) in Model 1:

$$\begin{aligned} \left\{ \begin{aligned}&\frac{d {\hat{p}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{p}}_0 \right) }{d {\hat{t}}} \, = \, \max _{\rho \, \in \, [0, 1]} \left( \left( k_1 \, \frac{{\hat{p}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{p}}_0 \right) }{K_1 + {\hat{p}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{p}}_0 \right) } \,\, \right. \right. \\&\qquad \left. \left. - \,\, \left( 1 + {\hat{p}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{p}}_0 \right) \right) \, \frac{{\hat{p}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{p}}_0 \right) }{K + {\hat{p}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{p}}_0 \right) } \,\, - \,\, E_M \right) \, \rho \right) \,\, \\&\qquad \, + \,\, E_M \,\, - \,\, k_1 \, \frac{{\hat{p}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{p}}_0 \right) }{K_1 + {\hat{p}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{p}}_0 \right) }, \quad {\hat{t}} \in \left[ 0, {\hat{T}} \right] , \\&{\hat{p}}_{\mathrm {up}} \left( 0; \, {\hat{p}}_0 \right) \, = \, {\hat{p}}_0, \end{aligned}\right. \end{aligned}$$
(72)
$$\begin{aligned} \left\{ \begin{aligned}&\frac{d {\hat{x}}_{\mathrm {low}} \left( {\hat{t}}; \, {\hat{x}}_0 \right) }{d {\hat{t}}} \,= \, -\frac{{\hat{p}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{p}}_0 \right) }{K + {\hat{p}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{p}}_0 \right) } \, {\hat{x}}_{\mathrm {low}} \left( {\hat{t}}; \, {\hat{x}}_0 \right) , \quad {\hat{t}} \in \left[ 0, {\hat{T}} \right] , \\&{\hat{x}}_{\mathrm {low}} \left( 0; \, {\hat{x}}_0 \right) \, = \, {\hat{x}}_0, \end{aligned}\right. \end{aligned}$$
(73)
$$\begin{aligned} \left\{ \begin{aligned}&\frac{d {\hat{x}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{x}}_0 \right) }{d {\hat{t}}} \, = \, k_1 \, \frac{{\hat{p}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{p}}_0 \right) }{K_1 + {\hat{p}}_{\mathrm {up}} \left( {\hat{t}}; \, {\hat{p}}_0 \right) }, \quad {\hat{t}} \in \left[ 0, {\hat{T}} \right] , \\&{\hat{x}}_{\mathrm {up}} \left( 0; \, {\hat{x}}_0 \right) \, = \, {\hat{x}}_0. \end{aligned}\right. \end{aligned}$$
(74)

Although the right-hand side of the system (72) is nonlinear, one can see that it satisfies the sublinear growth property (see, e.g., (Markley 2004, §2.3)) due to the boundedness of the fractions involved. This ensures the extendability of the unique solution of the Cauchy problem to the whole observed time interval. The existence, uniqueness, and extendability of the solution of (72) obviously imply those for (73) and (74).

Appendix B: Proof of Theorem 4.6

Consider an extremal process (46) and a time subinterval where

$$\begin{aligned} \psi _2 \, = \, \frac{d \psi _2}{d {\hat{t}}} \, = \, 0 \end{aligned}$$
(75)

(a singular regime). Assume that \( {\hat{p}} \) and \( {\hat{r}} \) are steady, i.e.,

$$\begin{aligned} \frac{d {\hat{p}}}{d {\hat{t}}} \, = \, \frac{d {\hat{r}}}{d {\hat{t}}} \, = \, 0 \end{aligned}$$
(76)

(\( {\hat{x}} \) and \( {\hat{y}} \) are not necessarily steady). It suffices to show that these conditions lead to \( \, \psi _3 - \psi _0 \equiv 0, \, \) because the latter contradicts with (45). Note that, in line with Property 4.4, the fulfillment of the equality \( \, \psi _3 - \psi _0 = 0 \, \) for at least one instant implies its fulfillment in the whole considered time interval.

The condition (38) and the first state equation in (22) yield

$$\begin{aligned}&E_M \, + \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}}}{K + {\hat{p}}} \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}} \,\,> \,\, E_M \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}} \,\, > \,\, 0, \nonumber \\&\quad {\hat{r}} \,\, = \,\, \frac{E_M \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}}}{E_M \, + \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}}}{K + {\hat{p}}} \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}}} \,\, \in \,\, (0, 1). \end{aligned}$$
(77)

From the second adjoint equation in (43) and the condition (75), one obtains

$$\begin{aligned} \psi _1 \,\, = \,\,&-\frac{1}{E_M \, + \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}}}{K + {\hat{p}}} \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}}} \, \nonumber \\&\cdot \, \left( (\psi _3 - \psi _0) \, \left( \frac{{\hat{p}}}{K + {\hat{p}}} \, + \, k_1 \, e^{-{\hat{y}}} \, \frac{{\hat{p}}}{K_1 + {\hat{p}}} \right) \, + \, \psi _0 \, \frac{{\hat{p}}}{K + {\hat{p}}} \right) . \end{aligned}$$
(78)

The third adjoint equation in (43) and the equation (40) lead to

$$\begin{aligned}&\frac{d}{d {\hat{t}}} \, (\psi _3 - \psi _0) \,\, = \,\, \frac{d \psi _3}{d {\hat{t}}} \,\, = \,\, (\psi _3 - \psi _0) \, k_1 \, e^{-{\hat{y}}} \, \frac{{\hat{p}} \, \left( 1 - {\hat{r}} \right) }{K_1 + {\hat{p}}}, \nonumber \\&\frac{d}{d {\hat{t}}} \left( (\psi _3 - \psi _0) \, e^{-{\hat{y}}} \right) \,\, = \,\, (\psi _3 - \psi _0) \, e^{-{\hat{y}}} \, \left( k_1 \, e^{-{\hat{y}}} \, \frac{{\hat{p}} \, \left( 1 - {\hat{r}} \right) }{K_1 + {\hat{p}}} \, \right. \nonumber \\&\qquad \left. - \, k_1 \, e^{-{\hat{y}}} \, \frac{{\hat{p}} \, \left( 1 - {\hat{r}} \right) }{K_1 + {\hat{p}}} \, + \, \frac{{\hat{p}} \, {\hat{r}}}{K + {\hat{p}}} \right) \,\, = \,\, (\psi _3 - \psi _0) \, e^{-{\hat{y}}} \, \frac{{\hat{p}} \, {\hat{r}}}{K + {\hat{p}}}. \end{aligned}$$
(79)

From (76), (78) and (79), one arrives at

$$\begin{aligned} \begin{aligned} \frac{d \psi _1}{d {\hat{t}}} \,\,&= \,\, -\frac{1}{E_M \, + \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}}}{K + {\hat{p}}} \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}}} \, \left( (\psi _3 - \psi _0) \, k_1 \, e^{-{\hat{y}}} \, \right. \\&\quad \left. \cdot \, \frac{{\hat{p}} \, \left( 1 - {\hat{r}} \right) }{K_1 + {\hat{p}}} \, \frac{{\hat{p}}}{K + {\hat{p}}} \,\, + \,\, (\psi _3 - \psi _0) \, e^{-{\hat{y}}} \frac{{\hat{p}} \, {\hat{r}}}{K + {\hat{p}}} \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}} \right) \,\, \\&= \,\, -\frac{(\psi _3 - \psi _0) \, k_1 \, e^{-{\hat{y}}}}{E_M \, + \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}}}{K + {\hat{p}}} \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}}} \, \frac{{\hat{p}}^2}{\left( K + {\hat{p}} \right) \, \left( K_1 + {\hat{p}} \right) }, \\ \frac{d^2 \psi _1}{d {\hat{t}}^2} \,\,&= \,\, \frac{d \psi _1}{d {\hat{t}}} \, \frac{{\hat{p}} \, {\hat{r}}}{K + {\hat{p}}} \,\, \\&= \,\, -\frac{(\psi _3 - \psi _0) \, k_1 \, e^{-{\hat{y}}}}{E_M \, + \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}}}{K + {\hat{p}}} \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}}} \, \frac{{\hat{p}}^3 \, {\hat{r}}}{\left( K + {\hat{p}} \right) ^2 \, \left( K_1 + {\hat{p}} \right) }. \end{aligned} \end{aligned}$$
(80)

On the other hand, the first adjoint equation in (43) and the condition (75) give

$$\begin{aligned} \begin{aligned} \frac{d \psi _1}{d {\hat{t}}} \,\,&= \,\, \left( \left( K \, \frac{1 + {\hat{p}}}{\left( K + {\hat{p}} \right) ^2} \, + \, \frac{{\hat{p}}}{K + {\hat{p}}} \right) \, {\hat{r}} \,\, + \,\, \frac{k_1 K_1}{\left( K_1 + {\hat{p}} \right) ^2} \, \left( 1 - {\hat{r}} \right) \right) \, \psi _1 \,\, \\&\quad - \left( k_1 K_1 \, e^{-{\hat{y}}} \, \frac{1 - {\hat{r}}}{\left( K_1 + {\hat{p}} \right) ^2} \,\, - \,\, K \, \frac{{\hat{r}}}{\left( K + {\hat{p}} \right) ^2} \right) \, (\psi _3 - \psi _0) \,\, \\&\quad + \psi _0 \, K \, \frac{{\hat{r}}}{\left( K + {\hat{p}} \right) ^2}, \end{aligned} \end{aligned}$$

which implies

$$\begin{aligned}&\frac{d^2 \psi _1}{d {\hat{t}}^2} \,\, - \,\, \left( \left( K \, \frac{1 + {\hat{p}}}{\left( K + {\hat{p}} \right) ^2} \, + \, \frac{{\hat{p}}}{K + {\hat{p}}} \right) \, {\hat{r}} \,\, + \,\, \frac{k_1 K_1}{\left( K_1 + {\hat{p}} \right) ^2} \, \left( 1 - {\hat{r}} \right) \right) \, \frac{d \psi _1}{d {\hat{t}}} \,\, \nonumber \\&\quad = \,\, -k_1 K_1 \, \frac{1 - {\hat{r}}}{\left( K_1 + {\hat{p}} \right) ^2} \, (\psi _3 - \psi _0) \, e^{-{\hat{y}}} \, \frac{{\hat{p}} \, {\hat{r}}}{K + {\hat{p}}} \,\, \nonumber \\&\qquad \,\, + \, K \, \frac{{\hat{r}}}{\left( K + {\hat{p}} \right) ^2} \, (\psi _3 - \psi _0) \, k_1 \, e^{-{\hat{y}}} \, \frac{{\hat{p}} \, \left( 1 - {\hat{r}} \right) }{K_1 + {\hat{p}}} \,\, \nonumber \\&\quad = \,\, \frac{{\hat{p}} \, {\hat{r}} \, \left( 1 - {\hat{r}} \right) }{\left( K + {\hat{p}} \right) \, \left( K_1 + {\hat{p}} \right) } \, (\psi _3 - \psi _0) \, k_1 \, e^{-{\hat{y}}} \, \left( \frac{K}{K + {\hat{p}}} \, - \, \frac{K_1}{K_1 + {\hat{p}}} \right) \end{aligned}$$
(81)

due to (76) and (79). By substituting (80) into (81), one gets

$$\begin{aligned} \begin{aligned}&-\frac{(\psi _3 - \psi _0) \, k_1 \, e^{-{\hat{y}}}}{E_M \, + \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}}}{K + {\hat{p}}} \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}}} \, \frac{{\hat{p}}^2}{\left( K + {\hat{p}} \right) \, \left( K_1 + {\hat{p}} \right) } \, \\&\qquad \quad \cdot \, \left( \frac{{\hat{p}} \, {\hat{r}}}{K + {\hat{p}}} \,\, - \,\, \left( \left( K \, \frac{1 + {\hat{p}}}{\left( K + {\hat{p}} \right) ^2} \, + \, \frac{{\hat{p}}}{K + {\hat{p}}} \right) \, {\hat{r}} \,\, + \,\, \frac{k_1 K_1}{\left( K_1 + {\hat{p}} \right) ^2} \, \left( 1 - {\hat{r}} \right) \right) \right) \,\, \\&\quad \quad = \,\, \frac{{\hat{p}} \, {\hat{r}} \, \left( 1 - {\hat{r}} \right) }{\left( K + {\hat{p}} \right) \, \left( K_1 + {\hat{p}} \right) } \, (\psi _3 - \psi _0) \, k_1 \, e^{-{\hat{y}}} \, \left( \frac{K}{K + {\hat{p}}} \, - \, \frac{K_1}{K_1 + {\hat{p}}} \right) . \end{aligned} \end{aligned}$$

This is simplified to

$$\begin{aligned}&\frac{(\psi _3 - \psi _0) \, {\hat{p}}}{E_M \, + \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}}}{K + {\hat{p}}} \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}}} \, \left( K \, \frac{1 + {\hat{p}}}{\left( K + {\hat{p}} \right) ^2} \, {\hat{r}} \,\, + \,\, \frac{k_1 K_1}{\left( K_1 + {\hat{p}} \right) ^2} \, \left( 1 - {\hat{r}} \right) \right) \,\, \nonumber \\&\qquad = \,\, {\hat{r}} \, \left( 1 - {\hat{r}} \right) \, (\psi _3 - \psi _0) \, \left( \frac{K}{K + {\hat{p}}} \, - \, \frac{K_1}{K_1 + {\hat{p}}} \right) \end{aligned}$$
(82)

and trivially holds for \( \, \psi _3 - \psi _0 = 0 \).

Now let \( \, \psi _3 - \psi _0 \ne 0 \). Then, after substituting (77) into (82) and canceling \( \psi _3 - \psi _0 \), one consecutively derives

$$\begin{aligned} \begin{aligned}&\frac{{\hat{p}}}{\left( E_M \, + \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}}}{K + {\hat{p}}} \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}} \right) ^2} \, \left( K \, \frac{1 + {\hat{p}}}{\left( K + {\hat{p}} \right) ^2} \, \left( E_M \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}} \right) \,\, \right. \\&\qquad \,\,\, \left. + \,\, \frac{k_1 K_1}{\left( K_1 + {\hat{p}} \right) ^2} \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}}}{K + {\hat{p}}} \right) \,\, \\&\quad = \,\, \frac{\left( E_M \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}} \right) \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}}}{K + {\hat{p}}}}{\left( E_M \, + \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}}}{K + {\hat{p}}} \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}} \right) ^2} \, \left( \frac{K}{K + {\hat{p}}} \, - \, \frac{K_1}{K_1 + {\hat{p}}} \right) , \\&\frac{K}{K + {\hat{p}}} \, \left( E_M \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}} \right) \,\, + \,\, {\hat{p}} \, \frac{k_1 K_1}{\left( K_1 + {\hat{p}} \right) ^2} \,\, \\&\quad = \,\, \left( E_M \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}} \right) \, \left( \frac{K}{K + {\hat{p}}} \, - \, \frac{K_1}{K_1 + {\hat{p}}} \right) , \\&{\hat{p}} \, \frac{k_1 K_1}{\left( K_1 + {\hat{p}} \right) ^2} \,\, = \,\, - \left( E_M \, - \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}} \right) \, \frac{K_1}{K_1 + {\hat{p}}}, \\&k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}} \,\, = \,\, -E_M \, + \, k_1 \, \frac{{\hat{p}}}{K_1 + {\hat{p}}}, \\&E_M \, = \, 0. \end{aligned} \end{aligned}$$

The obtained contradiction with \( E_M > 0 \) completes the proof. \(\square \)

Appendix C: Optimal feedback control in the biomass maximization problem for Model 1 under the terminal constraint (55)

In this section, we adopt Assumptions 3.44.2 and consider the biomass maximization problem for Model 1 under the terminal constraint (55).

First, we develop a numerical method for constructing the chattering switching curve \( \Sigma \) of the optimal feedback control law in the space of the state variables \( {\hat{p}}, {\hat{r}} \). A similar approach was proposed in (Yegorov et al. 2018, Section 7) regarding biomass maximization problems for self-replicator models without a heterologous metabolic pathway.

The chattering arcs enter the singular regime at the optimal steady state \( \left( {\hat{p}}^*_{\mathrm {opt}}, {\hat{r}}^*_{\mathrm {opt}} \right) \). The latter divides \( \Sigma \) into two subcurves \( \Sigma _0 \) and \( \Sigma _1 \), so that bang-bang switchings from \( u = u_{\max } \) to \( u = u_{\min } \) (in forward time) occur only on \( \Sigma _0 \) and switchings in the opposite direction occur only on \( \Sigma _1 \). We do not include the optimal steady state in \( \Sigma _0 \) and \( \Sigma _1 \). According to Theorem 4.8, \( \Sigma _0 \) lies in the set (56), and \( \Sigma _1 \) is contained in the set (57). The optimal feedback strategy equals \( u_{\min } \) on the upper side of \( \Sigma \) and \( u_{\max } \) on the lower side of \( \Sigma \). These properties can be seen in Fig. 15.

Since u does not explicitly appear in the first equation of (22), the first component of the corresponding velocity vector field is continuous regardless of u. From

$$\begin{aligned} \left. \frac{d {\hat{p}}}{d {\hat{t}}} \right| _{\small {(22)}, \,\, \left( {\hat{p}}, {\hat{r}} \right) \, = \, \left( {\hat{p}}^*_{\mathrm {opt}}, {\hat{r}}^*_{\mathrm {opt}} \right) } \,\, = \,\, 0, \end{aligned}$$
(83)

one arrives at the hypothesis that the tangent line to \( \Sigma \) at \( \left( {\hat{p}}^*_{\mathrm {opt}}, {\hat{r}}^*_{\mathrm {opt}} \right) \) is unique and vertical (as also observed in Fig. 15). For a rigorous verification, one can use the following property (whose general formulation can be found in (Naumov 2003)): if the directions of the vectors

$$\begin{aligned}&\left. \left( \dfrac{d {\hat{p}}}{d {\hat{t}}}, \, \dfrac{d {\hat{r}}}{d {\hat{t}}} \right) \right| _{\small {(22)}, \,\, \left( {\hat{p}}, {\hat{r}} \right) \, = \, \left( {\hat{p}}^*_{\mathrm {opt}}, {\hat{r}}^*_{\mathrm {opt}} \right) , \,\, u \, = \, u_{\min }}, \\&\left. \left( \dfrac{d {\hat{p}}}{d {\hat{t}}}, \, \dfrac{d {\hat{r}}}{d {\hat{t}}} \right) \right| _{\small {(22)}, \,\, \left( {\hat{p}}, {\hat{r}} \right) \, = \, \left( {\hat{p}}^*_{\mathrm {opt}}, {\hat{r}}^*_{\mathrm {opt}} \right) , \,\, u \, = \, u_{\max }} \end{aligned}$$

coincide with each other, then this is also the direction of the unique tangent line to \( \Sigma \) at \( \left( {\hat{p}}^*_{\mathrm {opt}}, {\hat{r}}^*_{\mathrm {opt}} \right) \). Therefore, the latter is vertical by virtue of (83). Note that such tangent lines are also vertical for optimal chattering regimes in the well-known Fuller and Marshal problems (Schattler and Ledzewicz 2012; Zelikin and Borisov 1994).

Thus, one can choose a sufficiently small \( \delta > 0 \) such that, in the closed neighborhood

$$\begin{aligned} \left\{ \left( {\hat{p}}, {\hat{r}} \right) \, \in \, {\mathbb {R}}^2 \, :\, \left( {\hat{p}} - {\hat{p}}^*_{\mathrm {opt}} \right) ^2 \, + \, \left( {\hat{r}} - {\hat{r}}^*_{\mathrm {opt}} \right) ^2 \, \leqslant \, \delta ^2 \right\} \end{aligned}$$
(84)

of the optimal steady state, it is reasonable to approximate the switching curve \( \Sigma \) as the vertical line segment

$$\begin{aligned} L_{\delta } \,\, {\mathop {=}\limits ^{\mathrm {def}}} \,\, \left\{ \left( {\hat{p}}^*_{\mathrm {opt}}, {\hat{r}} \right) \, :\, \left| {\hat{r}} - {\hat{r}}^*_{\mathrm {opt}} \right| \, \leqslant \, \delta \right\} . \end{aligned}$$
(85)

Consider an extremal process

$$\begin{aligned} \left( u(\cdot ), \, {\hat{p}}(\cdot ), \, {\hat{r}}(\cdot ), \, \eta _0, \, \eta _1(\cdot ), \, \eta _2(\cdot ) \right) . \end{aligned}$$
(86)

It satisfies the adjoint system (54) and the maximum condition

$$\begin{aligned} u \left( {\hat{t}} \right) \,\, = \,\, {\left\{ \begin{array}{ll} u_{\min }, &{} \eta _2 \left( {\hat{t}} \right) \, < \, 0, \\ u_{\max }, &{} \eta _2 \left( {\hat{t}} \right) \, > \, 0, \end{array}\right. } \end{aligned}$$
(87)

for the Hamiltonian

$$\begin{aligned} H_1 \left( {\hat{p}}, {\hat{r}}, u, \eta _0, \eta _1, \eta _2 \right) \,\,= & {} \,\, \left( E_M \, \left( 1 - {\hat{r}} \right) \, - \, \left( 1 + {\hat{p}} \right) \, \frac{{\hat{p}} \, {\hat{r}}}{K + {\hat{p}}} \, - \, k_1 \, \frac{{\hat{p}} \, \left( 1 - {\hat{r}} \right) }{K_1 + {\hat{p}}} \right) \, \eta _1 \,\,\nonumber \\&+ \,\, \left( u - {\hat{r}} \right) \, \frac{{\hat{p}} \, {\hat{r}}}{K + {\hat{p}}} \, \eta _2 \,\, - \,\, \frac{{\hat{p}} \, {\hat{r}}}{K + {\hat{p}}} \, \eta _0. \end{aligned}$$
(88)

Let the process (86) contain a chattering arc that enters the singular regime at the optimal steady state \( \left( {\hat{p}}^*_{\mathrm {opt}}, {\hat{r}}^*_{\mathrm {opt}} \right) \). Following Yegorov et al. (2018), we exclude abnormal extremal processes (with \( \eta _0 \equiv 0 \)) from consideration, since they do not allow for singular regimes (due to Theorem 4.8) and therefore do not admit a clear biological interpretation. Then \( \eta _0 \equiv -1 \), and the conserved Hamiltonian value for the process (86) equals

$$\begin{aligned} H_1 \left| _{\small {(88)}, \,\, \left( {\hat{p}}, {\hat{r}} \right) \, = \, \left( {\hat{p}}^*_{\mathrm {opt}}, {\hat{r}}^*_{\mathrm {opt}} \right) , \,\, \eta _2 \, = \, 0, \,\, \eta _0 \, = \, -1} \right. \,\, = \,\, \frac{{\hat{p}}^*_{\mathrm {opt}} \, {\hat{r}}^*_{\mathrm {opt}}}{K + {\hat{p}}^*_{\mathrm {opt}}}. \end{aligned}$$

Let \( {\hat{t}} = {\hat{t}}_{\mathrm {sw}} \) be one of bang-bang switching instants for a chattering arc of (86). If \( \, d {\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) /d {\hat{t}} \,= \, 0, \, \) then \( \, {\hat{r}} \left( {\hat{t}}_{\mathrm {sw}} \right) \, = \, {\hat{r}}^* \left( {\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) \right) \, \) (the function \( \,{\hat{r}}^* = {\hat{r}}^* \left( {\hat{p}} \right) \, \) is defined by (34)) and, from the Hamiltonian conservation property, one obtains

$$\begin{aligned} \frac{{\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) \, {\hat{r}}^* \left( {\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) \right) }{K + {\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) } \,\, = \,\, \frac{{\hat{p}}^*_{\mathrm {opt}} \, {\hat{r}}^*_{\mathrm {opt}}}{K + {\hat{p}}^*_{\mathrm {opt}}} \,\, = \,\, \frac{{\hat{p}}^*_{\mathrm {opt}} \, {\hat{r}}^* \left( {\hat{p}}^*_{\mathrm {opt}} \right) }{K + {\hat{p}}^*_{\mathrm {opt}}}, \end{aligned}$$

which implies \( \, \left( {\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) , \, {\hat{r}} \left( {\hat{t}}_{\mathrm {sw}} \right) \right) \, = \, \left( {\hat{p}}^*_{\mathrm {opt}}, {\hat{r}}^*_{\mathrm {opt}} \right) \, \) due to Property 4.1 and Theorem 4.3, i.e., this is already a singular arc at \( {\hat{t}} = {\hat{t}}_{\mathrm {sw}} \). Therefore, the case \( \, d {\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) / d {\hat{t}} \, = \, 0 \) can be excluded from the consideration of chattering arcs. Then the switching condition \( \, \eta _2 \left( {\hat{t}}_{\mathrm {sw}} \right) \, = \, 0 \, \) and Hamiltonian conservation property yield a correct representation (without division by zero) for the first adjoint component at \( {\hat{t}} = {\hat{t}}_{\mathrm {sw}} \):

$$\begin{aligned} \begin{aligned} \eta _1 \left( {\hat{t}}_{\mathrm {sw}} \right) \,\,&= \,\, \left( E_M \, \left( 1 - {\hat{r}} \left( {\hat{t}}_{\mathrm {sw}} \right) \right) \,\, - \,\, \left( 1 + {\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) \right) \, \frac{{\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) \, {\hat{r}} \left( {\hat{t}}_{\mathrm {sw}} \right) }{K + {\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) } \,\, \right. \\&\quad \,\, \left. - \,\, k_1 \, \frac{{\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) \, \left( 1 - {\hat{r}} \left( {\hat{t}}_{\mathrm {sw}} \right) \right) }{K_1 + {\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) } \right) ^{-1}\cdot \,\, \left( \frac{{\hat{p}}^*_{\mathrm {opt}} \, {\hat{r}}^*_{\mathrm {opt}}}{K + {\hat{p}}^*_{\mathrm {opt}}} \, - \, \frac{{\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) \, {\hat{r}} \left( {\hat{t}}_{\mathrm {sw}} \right) }{K + {\hat{p}} \left( {\hat{t}}_{\mathrm {sw}} \right) } \right) . \end{aligned} \end{aligned}$$
(89)

The formula (89) can be used for approximate setting of the values of \( \eta _1 \) at the points

$$\begin{aligned} \left( {\hat{p}}, {\hat{r}} \right) \,\, \in \,\, {\mathop {L}\limits ^{\circ }}_{\delta } \,\, {\mathop {=}\limits ^{\mathrm {def}}} \,\, L_{\delta } {\setminus } \left\{ \left( {\hat{p}}^*_{\mathrm {opt}}, {\hat{r}}^*_{\mathrm {opt}} \right) \right\} \end{aligned}$$

(as was mentioned above, \( \Sigma \) is close to \( L_{\delta } \) in the ball (84)). Denote the corresponding representation by \( \, \eta _{1, \delta } \, = \, \eta _{1, \delta } \left( {\hat{p}}, {\hat{r}} \right) \). For computational purposes, it is reasonable to discretize \( {\mathop {L}\limits ^{\circ }}_{\delta } \) by the grid consisting of the points

$$\begin{aligned} Q_i \,\, {\mathop {=}\limits ^{\mathrm {def}}} \,\, \left( {\hat{p}}^*_{\mathrm {opt}}, \, {\hat{r}}^*_{\mathrm {opt}} \, + \, i \, \frac{\delta }{n} \right) , \quad i \, \in \, {\mathcal {I}}_n \, {\mathop {=}\limits ^{\mathrm {def}}} \, \{ -n, \, -n + 1, \, \ldots , \, n - 1, \, n \} {\setminus } \{ 0 \}, \end{aligned}$$

for a sufficiently large \( n \in {\mathbb {N}} \). Let us rewrite the state and adjoint equations under the condition (87) in reverse time. The numerical integration of the four resulting reverse-time equations from the starting positions

$$\begin{aligned} \eta _1 \, = \, \eta _{1, \delta } \left( {\hat{p}}, {\hat{r}} \right) , \quad \eta _2 \, = \, 0, \quad \left( {\hat{p}}, {\hat{r}} \right) \, \in \, \{ Q_i \}_{i \in {\mathcal {I}}_n} \end{aligned}$$

allows to find the bang-bang switching points in the state space (at the time instants when \( \eta _2 \) changes its sign) and thereby to approximate \( \Sigma _0 \) and \( \Sigma _1 \). It remains to recall that \( \, \Sigma \, = \, \Sigma _0 \, \cup \, \left( {\hat{p}}^*_{\mathrm {opt}}, {\hat{r}}^*_{\mathrm {opt}} \right) \, \cup \, \Sigma _1 \).

Fig. 15
figure15

The optimal feedback control law in the biomass maximization problem for Model 1 under the terminal constraint (55). The parameter values are given by (48)–(51)

Note that this method of approximating \( \Sigma \) in principle does not depend on a particular finite time horizon \( {\hat{T}} \), although some upper time bound for integrating the reverse-time system of the state and adjoint equations has to be selected.

Figure 15 shows the switching curve \( \Sigma \) approximated via the described algorithm. The global optimal feedback control law in the studied biomass maximization problem is in fact illustrated (the feedback representation combines all possible initial states together, as opposed to the time-dependent open-loop form). The corresponding parameter values (48)–(51) are the same as in Sects. 4.1.2 and 4.1.3.

Fig. 16
figure16

The chattering switching curves \( \Sigma \) and \( \Sigma _{k_1 \, = \, 0} \) in the biomass maximization problem for Model 1 under the terminal constraint (55), and in the biomass maximization problem of Giordano et al. (2016). Different control constraints are considered, and the other parameter values are given by (49)–(51)

Figure 16 compares \( \Sigma \) with the switching curve \( \Sigma _{k_1 \, = \, 0} \) for the biomass maximization problem of Giordano et al. (2016) (whose dynamical system can be obtained from the first two equations of (21) by taking \( k_1 = 0 \) and \( u \equiv \alpha \)). The parameters (49)–(51) are still the same, while different control constraints are now considered. For \( \Sigma \), the control bounds \( u_{\min } = 0.01 \) and \( u_{\max } = 0.81 \) correspond to the case \( \, I_{\min } = \alpha _{\min } = 0.1 \), \( \, I_{\max } = \alpha _{\max } = 0.9 \). For \( \Sigma _{k_1 \, = \, 0} \), we have \( u \equiv \alpha \) and, hence, \( \, u_{\min } = \alpha _{\min } \), \( u_{\max } = \alpha _{\max } \, \) (here we set \( I \equiv 1 \) for the model of Giordano et al. (2016), because it does not contain an inducer). Moreover, \( \, \left( {\hat{p}}^*_{\mathrm {opt}, \, k_1 \, = \, 0}, \, {\hat{r}}^*_{\mathrm {opt}, \, k_1 \, = \, 0} \right) \, \) denotes the optimal steady state for the problem of Giordano et al. (2016).

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Yegorov, I., Mairet, F., de Jong, H. et al. Optimal control of bacterial growth for the maximization of metabolite production. J. Math. Biol. 78, 985–1032 (2019). https://doi.org/10.1007/s00285-018-1299-6

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Keywords

  • Optimal control
  • Nonlinear dynamical systems
  • Mathematical modelling
  • Bacterial growth
  • Biotechnology