Térence Bayen’s research while affiliated with French National Centre for Scientific Research and other places

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Publications (69)


Illustration of a classical needle-like perturbation of the control (left) and of the corresponding perturbed trajectory (right)
Illustration of Example 2.1
Illustration of an auxiliary control u~k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{u}_k$$\end{document}. In this illustration, for simplicity, we have chosen a control u that is continuous over each interval between two consecutive crossing times, but this is not mandatory. We only know that u satisfies the continuity properties given in Condition (C2)(i)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal {C}2)\mathrm {(i)}$$\end{document}
Illustration of an auxiliary trajectory z~k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{z}_{k}$$\end{document}
Illustration of a classical needle-like perturbation of the auxiliary control u~q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{u}_q$$\end{document}

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On the Use of Needle-Like Perturbations in Spatially Heterogeneous Control Systems
  • Article
  • Publisher preview available

January 2025

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3 Reads

Journal of Optimization Theory and Applications

Térence Bayen

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Loïc Bourdin

In this paper we consider a general control system involving a spatially heterogeneous dynamics. This means that the state space is partitioned into several disjoint regions and that each region has its own (smooth) control system. As a result, the dynamics discontinuously changes whenever the trajectory crosses an interface between two regions. In that spatially heterogeneous setting (and in contrast with the usual smooth case), a needle-like perturbation of the control may generate a perturbed trajectory that does not uniformly converge towards the nominal one, and may lead to the absence of a corresponding first-order variation vector. The first contribution of this paper is to illustrate this issue by means of a simple counterexample. Our second and main contribution is to provide a modified needle-like perturbation of the control (adapted to the spatially heterogeneous setting) which generates a perturbed trajectory that uniformly converges towards the nominal one, and leads to a corresponding first-order variation vector (which has the particularity of admitting a discontinuity jump at each interface crossing). This is made possible under several assumptions (including transverse crossing conditions), by introducing new tools such as auxiliary trajectories and auxiliary controls and by using a conic version of the implicit function theorem.

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Figure 2: Plot of the kinetics µ i (s) = ais bi+s of Monod type for i = 1, ..., 5 (coefficients are given in Table 1).
Global stability of perturbed chemostat systems

January 2025

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28 Reads

This paper is devoted to the analysis of global stability of the chemostat system with a perturbation term representing any type of exchange between species. This conversion term depends on species and substrate concentrations but also on a positive perturbation parameter. After having written the invariant manifold as a union of a family of compact subsets, our main result states that for each subset in this family, there is a positive threshold for the perturbation parameter below which, the system is globally asymptotically stable in the corresponding subset. Our approach relies on the Malkin-Gorshin Theorem and on a Theorem by Smith and Waltman about perturbations of a globally stable steady state. Properties of steady-states and numerical simulations of the system's asymptotic behavior complete this study for two types of perturbation term between species.


Loss control regions in optimal control problems

October 2024

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38 Reads

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1 Citation

Journal of Differential Equations

This paper addresses optimal control problems with loss control regions. In that context the state space is partitioned into disjoint subsets, referred to as regions, which are classified into two types: control regions and loss control regions. When the state belongs to a control region, the control is permanent (i.e. the control value is authorized to be modified at any time). On the contrary, when the state belongs to a loss control region, the control must remain constant as long as the state belongs to this region. The objective of this paper is twofold. First, we reformulate the above setting into a hybrid optimal control problem (with spatially heterogeneous dynamics) involving moreover a regionally switching parameter, and we prove a corresponding hybrid maximum principle: hence first-order necessary optimality conditions in a Pontryagin form are obtained. Second, this paper proposes a two-steps numerical scheme to solve optimal control problems with loss control regions. It is based on a direct numerical method (applied to a regularized problem) which initializes an indirect numerical method (applied to the original problem and based on the aforementioned necessary optimality conditions). This numerical approach is applied to several illustrative examples.


Approximation of Chattering Arcs in Optimal Control Problems Governed by Mono-Input Affine Control Systems

Set-Valued and Variational Analysis

In this paper, we consider a general Mayer optimal control problem governed by a mono-input affine control system whose optimal solution involves a second-order singular arc (leading to chattering). The objective of the paper is to present a numerical scheme to approach the chattering control by controls with a simpler structure (concatenation of bang-bang controls with a finite number of switching times and first-order singular arcs). Doing so, we consider a sequence of vector fields converging to the drift such that the associated optimal control problems involve only first-order singular arcs (and thus, optimal controls necessarily have a finite number of bang arcs). Up to a subsequence, we prove convergence of the sequence of extremals to an extremal of the original optimal control problem as well as convergence of the value functions. Next, we consider several examples of problems involving chattering. For each of them, we give an explicit family of approximated optimal control problems whose solutions involve bang arcs and first-order singular arcs. This allows us to approximate numerically solutions (with chattering) to these original optimal control problems.


Stabilization of the chemostat system with mutations and application to microbial production

July 2023

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27 Reads

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2 Citations

Optimal Control Applications and Methods

In this article, we consider the chemostat system with species, one limiting substrate, and mutations between species. Our objective is to globally stabilize the corresponding dynamical system around a desired equilibrium point. Doing so, we introduce auxostat feedback controls which are controllers allowing the regulation of the substrate concentration. We prove that such feedback controls globally stabilize the resulting closed‐loop system near the desired equilibrium point. This result is obtained by combining the theory of asymptotically autonomous systems and an explicit computation of solutions to the limit system. The performance of such controllers is illustrated on an optimal control problem of Lagrange type which consists in maximizing the production of species over a given time period w.r.t. the dilution rate chosen as control variable.


Optimal control problems with non-control regions: necessary optimality conditions

September 2022

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16 Reads

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1 Citation

IFAC-PapersOnLine

We consider a smooth control system that is subject to loss of control in the sense that the state space is partitioned into several disjoint regions and, in each region, either the system can be controlled, as usual, in a permanent way (that is, one can change the value of the control at any real time), or, on the contrary, the control has to remain constant from the entry time into the region until the exit time. The latter case corresponds to a non-control region. The objective of this paper is to state the necessary optimality conditions for a Mayer optimal control problem in such a setting of loss of control. Our main result is based on a hybrid maximum principle that takes into account a regionally switching parameter.


Optimal control of microbial production in the chemostat

September 2022

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14 Reads

IFAC-PapersOnLine

In this work, we study the problem of maximizing microbial production within a continuous stirred-tank reactor with n ≥ 1 species and one substrate. The interaction between species is described by the chemostat system including a mutation factor. The optimization problem falls into an optimal control problem of Lagrange type in which the control parameter is the dilution rate of the reactor. Thanks to the Pontryagin Maximum Principle, we obtain necessary conditions on optimal controls that involve a singular arc. These computations are highlighted thanks to numerical simulations via a direct method. We also study the related optimization problem at steady-state which provides an insight into optimal solutions of the optimal control problem in terms of turnpike phenomenon.


Figure 1: Plot of s → λ(A ε,s )(1 − s) over [0, 1] for ε = 0 (left), ε = 0.01 (middle), and ε = 0.1 (right). The number of species is n = 5 and kinetics are arbitrary Monod functions.
Figure 4: Comparison of the criteria between the different strategies, as a function of the time horizon. The curves for the two feedbacks are overlapped.
Stabilization of the chemostat system with mutations and application to microbial production

August 2022

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52 Reads

In this paper, we consider the chemostat system with n ≥ 1 species, one limiting substrate, and mutations between species. Our objective is to globally stabilize the corresponding dynamical system around a desired equilibrium point. Doing so, we introduce auxostat feedback controls which are controllers allowing the regulation of the substrate concentration. We prove that such feedback controls globally stabilize the resulting closed-loop system near the desired equilibrium point. This result is obtained by combining the theory of asymptotically autonomous systems and an explicit computation of solutions to the limit system. The performance of such controllers is illustrated on an optimal control problem of Lagrange type which consists in maximizing the production of species over a given time period w.r.t. the dilution rate chosen as control variable.


Figure 1. The illustrated absorption functions ρ i and growth rates µ i lead to a generic form of the actual growth rates s → µ i (δ i (s)) where both species may win the competition. The maximum dilution rate D max is a fixed constant value above the maximum of the functions s → µ i (δ i (s)) for i = 1, 2, as stated in Assumption 3.
Figure 3. (a) Optimal state in Droop model, (b) the resulting optimal co-state trajectories, which satisfy in particular all the transversality conditions of the PMP.
Figure 4. The optimal control in Example 2 (s 0 = 4, q 0 i = 1.9, x 0 i = 0.3) is of bang(0)-singular type.
Figure 6. Evolution of the quantity ρ 2 (s(t))µ 2 (q 2 (t)) − ρ 1 (s(t))µ 1 (q 1 (t)) along the optimal trajectories given in Figure 3a.
Optimal Darwinian Selection of Microorganisms with Internal Storage

February 2022

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76 Reads

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10 Citations

Processes

In this paper, we investigate the problem of species separation in minimal time. Droop model is considered to describe the evolution of two distinct populations of microorganisms that are in competition for the same resource in a photobioreactor. We focus on an optimal control problem (OCP) subject to a five-dimensional controlled system in which the control represents the dilution rate of the chemostat. The objective is to select the desired species in minimal-time and to synthesize an optimal feedback control. This is a very challenging issue, since we are are dealing with a ten-dimensional optimality system. We provide properties of optimal controls allowing the strain of interest to dominate the population. Our analysis is based on the Pontryagin Maximum Principle (PMP), along with a thorough study of singular arcs that is crucial in the synthesis of optimal controls. These theoretical results are also extensively illustrated and validated using a direct method in optimal control (via the Bocop software for numerically solving optimal control problems). The approach is illustrated with numerical examples with microalgae, reflecting the complexity of the optimal control structure and the richness of the dynamical behavior.


Stability of the chemostat system including a linear coupling between species

January 2022

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18 Reads

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3 Citations

Discrete and Continuous Dynamical Systems - B

In this paper, we consider a resource-consumer model taking into account a linear coupling between species (with constant rate). The corresponding operator is proportional to a discretization of the Laplacian in such a way that the resulting dynamical system can be viewed as a regular perturbation of the classical chemostat system. We prove the existence of a unique locally asymptotically stable steady-state for every value of the transfer-rate and every value of the dilution rate not exceeding a critical value. In addition, we give an expansion of the steady-state in terms of the transfer-rate and we prove a uniform persistence property of the dynamics related to each species. Finally, we show that this equilibrium is globally asymptotically stable for every value of the transfer-rate provided that the dilution rate is with small enough values.


Citations (46)


... In particular, note that ∇ t G q is invertible at (0, t c q ) thanks to the first transverse crossing condition in Condition (C2)(iii). 5. From the construction of the perturbed auxiliary trajectoryz α q , it can be proved thatz α q stays inside X j(q) over (t c q−1 ,t q (α)). ...

Reference:

On the Use of Needle-Like Perturbations in Spatially Heterogeneous Control Systems
Loss control regions in optimal control problems

Journal of Differential Equations

... As a widely used control method, cooperative control has captured tremendous attention and has extensive applications, such as robot cooperation 1 , UAV formation flight 2 , autonomous vehicles 3 , and intelligent logistics 4 . In recent years, the research about cooperative control of MASs mainly pertained to optimization problems 5,6 , consensus control 7 , controllability and stabilizability analysis 8,9 . ...

Stabilization of the chemostat system with mutations and application to microbial production
  • Citing Article
  • July 2023

Optimal Control Applications and Methods

... In presence of this phenomenon, each species is able to convert into neighboring species. Theoretical and numerical approaches to study the chemostat system with a perturbation term as well as related questions can be found in [2,4,5,7,8,10,11,14,19,22] (among others). In this paper, we are interested in addressing the next question : what can we say about the asymptotic behavior of a perturbed chemostat system particularly when the perturbation is "small 2 "? Depending on the data defining the system (such as dilution rate, perturbation parameter, kinetics), we wish to know if there is an invariant subset of the state space in which the perturbed system is globally asymptotically stable around a coexistence 3 steady-state. ...

Stability of the chemostat system including a linear coupling between species
  • Citing Article
  • January 2022

Discrete and Continuous Dynamical Systems - B

... We then investigate problem convexity and, in the same spirit as [10], we numerically illustrate the control tradeoffs and the uniqueness of the solution. For the DOCP, we seek the solution via geometric control theory ( [13], [14]) and the Pontryagin's Maximum Principle (PMP, see e.g., [13], [15] and related applications in [16], [17], [18]). We establish a solution in the form of bang-bang control actions and singular arcs. ...

Optimal Darwinian Selection of Microorganisms with Internal Storage

Processes

... A striking point related to second order singular arcs is that they arise in simple examples from mathematical modeling such as in aerospace [31,33], in biomedicine [16,18,25], in biology [19,20], or in physics [22]. The simplest problem involving chattering is known as Fuller's problem for which the underlying system is the two-dimensional double integrator associated with a quadratic cost. ...

Parameter Estimation for Dynamic Resource Allocation in Microorganisms: A Bi-level Optimization Problem
  • Citing Article
  • January 2020

IFAC-PapersOnLine

... Hereafter, we denote by (x n , u n ) an optimal pair of (2.6). Following [5], we can show that, up to a sub-sequence, the sequence x n converges stronglyweakly 5 to an optimal pair (x * , u * ) of (2.1). Let us stress that in general, (u n ) may not converge point-wise to u * . ...

Penalty function method for the minimal time crisis problem

ESAIM Proceedings and Surveys

... Additionally, it is worth mentioning that the PI curve is highly adaptable and continuously adjusts through photoacclimation by changes in chloroplast pigment concentrations and ratios. This modulation is triggered by factors such as light intensity, temperature changes, or the nutritional status of the culture, and it is specific to each photosynthetic organism [47]. ...

The promise of dawn: Microalgae photoacclimation as an optimal control problem of resource allocation
  • Citing Article
  • January 2021

Journal of Theoretical Biology

... To calculate ψ as well as the control (14) and (17), the biomass content X is needed besides the information of S and P. X is rendered difficult to measure on-line, and OBE is performed. Considering ethanol is the primary metabolite that is closely related to the content of biomass, X could be estimated through the construction of the observer on P. The procedure begins with the linearization of the subsystem in (1), dX dt = −D| S 0 ,X 0 ,P 0 X + r 1 | µ m0 ,S 0 ,P 0 dP dt = −D| S 0 ,X 0 ,P 0 P + r 2 | S 0 ,P 0 (27) When (27) is non-singular, and D satisfies the persistent exciting (PE) condition, the non-biased estimation of P drives X to stably convergent to the real value [36]. In the following, a Luenberger observer is constructed for P, ...

Improvement of performances of the chemostat used for continuous biological water treatment with periodic controls
  • Citing Article
  • November 2020

Automatica

... This means that the only possibility for an optimal control to connect the trajectory to the singular arc is an infinite sequence of bang arcs since the proposition excludes the concatenation of the singular arc to the optimal trajectory by a finite number of bang arcs. Also, singular controls may exceed the largest admissible value leading to saturation [1], that is why, the singular control is supposed to be admissible in Proposition 2.1. In addition to this proposition, let us also recall Legendre-Clebsch's condition [24] in the above context (with a single input). ...

Tangency Property and Prior-Saturation Points in Minimal Time Problems in the Plane

Acta Applicandae Mathematicae