### Abstract

In this paper, we further develop a recently proposed control method to switch a bistable system between its steady states using temporal pulses. The motivation for using pulses comes from biomedical and biological applications (e.g. synthetic biology), where it is generally difficult to build feedback control systems due to technical limitations in sensing and actuation. The original framework was derived for monotone systems and all the extensions relied on monotone systems theory. In contrast, we introduce the concept of switching function which is related to eigenfunctions of the so-called Koopman operator subject to a fixed control pulse. Using the level sets of the switching function we can (i) compute the set of all pulses that drive the system toward the steady state in a synchronous way and (ii) estimate the time needed by the flow to reach an epsilon neighborhood of the target steady state. Additionally, we show that for monotone systems the switching function is also monotone in some sense, a property that can yield efficient algorithms to compute it. This observation recovers and further extends the results of the original framework, which we illustrate on numerical examples inspired by biological applications.

Original language | English |
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Title of host publication | Proceedings of the 10th IFAC Symposium on Nonlinear Control Systems (NOLCOS) |

Pages | 698-703 |

Number of pages | 6 |

Volume | 49 |

Edition | 18 |

DOIs | |

Publication status | Published - 2016 |

Externally published | Yes |

### Publication series

Name | IFAC-PapersOnLine |
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Publisher | IFAC Secretariat |

### Fingerprint

### Cite this

*Proceedings of the 10th IFAC Symposium on Nonlinear Control Systems (NOLCOS)*(18 ed., Vol. 49, pp. 698-703). (IFAC-PapersOnLine). https://doi.org/10.1016/j.ifacol.2016.10.247

}

*Proceedings of the 10th IFAC Symposium on Nonlinear Control Systems (NOLCOS).*18 edn, vol. 49, IFAC-PapersOnLine, pp. 698-703. https://doi.org/10.1016/j.ifacol.2016.10.247

**Shaping Pulses to Control Bistable Monotone Systems Using Koopman Operator.** / Sootla, Aivar; Mauroy, Alexandre; Goncalves, Jorge.

Research output: Contribution in Book/Catalog/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Shaping Pulses to Control Bistable Monotone Systems Using Koopman Operator

AU - Sootla, Aivar

AU - Mauroy, Alexandre

AU - Goncalves, Jorge

PY - 2016

Y1 - 2016

N2 - In this paper, we further develop a recently proposed control method to switch a bistable system between its steady states using temporal pulses. The motivation for using pulses comes from biomedical and biological applications (e.g. synthetic biology), where it is generally difficult to build feedback control systems due to technical limitations in sensing and actuation. The original framework was derived for monotone systems and all the extensions relied on monotone systems theory. In contrast, we introduce the concept of switching function which is related to eigenfunctions of the so-called Koopman operator subject to a fixed control pulse. Using the level sets of the switching function we can (i) compute the set of all pulses that drive the system toward the steady state in a synchronous way and (ii) estimate the time needed by the flow to reach an epsilon neighborhood of the target steady state. Additionally, we show that for monotone systems the switching function is also monotone in some sense, a property that can yield efficient algorithms to compute it. This observation recovers and further extends the results of the original framework, which we illustrate on numerical examples inspired by biological applications.

AB - In this paper, we further develop a recently proposed control method to switch a bistable system between its steady states using temporal pulses. The motivation for using pulses comes from biomedical and biological applications (e.g. synthetic biology), where it is generally difficult to build feedback control systems due to technical limitations in sensing and actuation. The original framework was derived for monotone systems and all the extensions relied on monotone systems theory. In contrast, we introduce the concept of switching function which is related to eigenfunctions of the so-called Koopman operator subject to a fixed control pulse. Using the level sets of the switching function we can (i) compute the set of all pulses that drive the system toward the steady state in a synchronous way and (ii) estimate the time needed by the flow to reach an epsilon neighborhood of the target steady state. Additionally, we show that for monotone systems the switching function is also monotone in some sense, a property that can yield efficient algorithms to compute it. This observation recovers and further extends the results of the original framework, which we illustrate on numerical examples inspired by biological applications.

UR - http://www.scopus.com/inward/record.url?scp=85009188232&partnerID=8YFLogxK

U2 - 10.1016/j.ifacol.2016.10.247

DO - 10.1016/j.ifacol.2016.10.247

M3 - Conference contribution

AN - SCOPUS:85009188232

VL - 49

T3 - IFAC-PapersOnLine

SP - 698

EP - 703

BT - Proceedings of the 10th IFAC Symposium on Nonlinear Control Systems (NOLCOS)

ER -