### Abstract

In many applications, and in systems/synthetic biology, in particular, it is desirable to solve the switching problem, i.e., to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point). It was recently shown that for monotone bistable systems, this problem admits easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step functions of fixed length and fixed magnitude). In this paper, we develop this idea further and formulate a problem of convergence to an equilibrium from an arbitrary initial point. We show that the convergence problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows building closed-loop, event-based or open-loop policies for the switching/convergence problems. In our derivations, we exploit the Koopman operator, which offers a linear infinite-dimensional representation of an autonomous nonlinear system and powerful computational tools for their analysis. Our solutions to the switching/convergence problems can serve as building blocks for other control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation.

Language | English |
---|---|

Pages | 217-224 |

Number of pages | 8 |

Journal | Automatica |

Volume | 91 |

DOIs | |

Publication status | Published - 1 May 2018 |

### Fingerprint

### Keywords

- monotone systems
- Koopman operator
- isostables
- generalized repressilator
- genetic toggle switch
- Genetic toggle switch
- Monotone systems
- Isostables
- Generalized repressilator

### Cite this

*Automatica*,

*91*, 217-224. https://doi.org/10.1016/j.automatica.2018.01.036

}

*Automatica*, vol. 91, pp. 217-224. https://doi.org/10.1016/j.automatica.2018.01.036

**Optimal control formulation of pulse-based control using Koopman operator.** / Sootla, Aivar; Mauroy, Alexandre; Damien, Ernst.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Optimal control formulation of pulse-based control using Koopman operator

AU - Sootla, Aivar

AU - Mauroy, Alexandre

AU - Damien, Ernst

PY - 2018/5/1

Y1 - 2018/5/1

N2 - In many applications, and in systems/synthetic biology, in particular, it is desirable to solve the switching problem, i.e., to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point). It was recently shown that for monotone bistable systems, this problem admits easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step functions of fixed length and fixed magnitude). In this paper, we develop this idea further and formulate a problem of convergence to an equilibrium from an arbitrary initial point. We show that the convergence problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows building closed-loop, event-based or open-loop policies for the switching/convergence problems. In our derivations, we exploit the Koopman operator, which offers a linear infinite-dimensional representation of an autonomous nonlinear system and powerful computational tools for their analysis. Our solutions to the switching/convergence problems can serve as building blocks for other control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation.

AB - In many applications, and in systems/synthetic biology, in particular, it is desirable to solve the switching problem, i.e., to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point). It was recently shown that for monotone bistable systems, this problem admits easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step functions of fixed length and fixed magnitude). In this paper, we develop this idea further and formulate a problem of convergence to an equilibrium from an arbitrary initial point. We show that the convergence problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows building closed-loop, event-based or open-loop policies for the switching/convergence problems. In our derivations, we exploit the Koopman operator, which offers a linear infinite-dimensional representation of an autonomous nonlinear system and powerful computational tools for their analysis. Our solutions to the switching/convergence problems can serve as building blocks for other control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation.

KW - monotone systems

KW - Koopman operator

KW - isostables

KW - generalized repressilator

KW - genetic toggle switch

KW - Genetic toggle switch

KW - Monotone systems

KW - Isostables

KW - Generalized repressilator

UR - http://www.scopus.com/inward/record.url?scp=85042140784&partnerID=8YFLogxK

U2 - 10.1016/j.automatica.2018.01.036

DO - 10.1016/j.automatica.2018.01.036

M3 - Article

VL - 91

SP - 217

EP - 224

JO - Automatica

T2 - Automatica

JF - Automatica

SN - 0005-1098

ER -