### Abstract

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

Pages | 217-224 |

Number of pages | 8 |

Journal | Automatica |

Volume | 91 |

State | Published - 2018 |

### Fingerprint

### Keywords

- monotone systems
- Koopman operator
- isostables
- generalized repressilator
- genetic toggle switch

### Cite this

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*Automatica*, vol. 91, pp. 217-224.

**An 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 - An optimal control formulation of pulse-based control using Koopman operator

AU - Sootla,Aivar

AU - Mauroy,Alexandre

AU - Damien,Ernst

PY - 2018

Y1 - 2018

N2 - In many applications, and in systems/synthetic biology in particular, it is desirable to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point), or in other words to solve the switching problem. It was recently shown that, for monotone bistable systems, this problem admits easyto-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 this problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows to build 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. One of the main advantages of using the Koopman operator is the powerful computational tools developed for this framework. Besides the presence of numerical solutions, the switching/convergence problem can also serve as a building block for solving more complicated control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation. Potentially, our approach can be extended to problems with different parametrizations of control signals since the only fundamental limitation is the finite time application of the control signal.

AB - In many applications, and in systems/synthetic biology in particular, it is desirable to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point), or in other words to solve the switching problem. It was recently shown that, for monotone bistable systems, this problem admits easyto-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 this problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows to build 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. One of the main advantages of using the Koopman operator is the powerful computational tools developed for this framework. Besides the presence of numerical solutions, the switching/convergence problem can also serve as a building block for solving more complicated control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation. Potentially, our approach can be extended to problems with different parametrizations of control signals since the only fundamental limitation is the finite time application of the control signal.

KW - monotone systems

KW - Koopman operator

KW - isostables

KW - generalized repressilator

KW - genetic toggle switch

M3 - Article

VL - 91

SP - 217

EP - 224

JO - Automatica

T2 - Automatica

JF - Automatica

SN - 0005-1098

ER -