RECORD
An Optimal Control Formulation of Pulse-Based Control Using Koopman Operator
Optimization and Control · Submitted 26 Jul 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that this problem can be solved using a static optimization problem in the case of monotone systems.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
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 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 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 1.00 | 0.06 | 0.04 | asserted 0.90 |
| s1 | 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). | background 1.00 | 0.05 | 0.06 | asserted 1.00 |
| s2 | In this paper, we develop this idea further and formulate a problem of convergence to an equilibrium from an arbitrary initial point. | contribution 0.98 | 0.90 | 0.10 | asserted 0.96 |
| s3 | We show that this problem can be solved using a static optimization problem in the case of monotone systems. | result 0.84 | 0.94 | 0.41 | asserted 1.00 |
| s4 | Changing the initial point to an arbitrary state allows to build closed-loop, event-based or open-loop policies for the switching/convergence problems. | interpretation 0.73 | 0.79 | 0.21 | asserted 0.77 |
| s5 | In our derivations we exploit the Koopman operator, which offers a linear infinite-dimensional representation of an autonomous nonlinear system. | method 0.96 | 0.33 | 0.05 | asserted 1.00 |
| s6 | One of the main advantages of using the Koopman operator is the powerful computational tools developed for this framework. | background 0.39 | 0.13 | 0.05 | asserted 0.99 |
| s7 | 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. | interpretation 0.75 | 0.37 | 0.16 | hedged possibility 0.81 |
| s8 | We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation. | contribution 0.52 | 0.87 | 0.06 | asserted 1.00 |
| s9 | 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. | future work 0.95 | 0.23 | 0.08 | hedged possibility 0.97 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
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 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 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.
Human review
How the literature uses it
3 arXiv papers cite this in 10 sentences
Papers citing it · 6 sentences without a confident label
Builds on it 1 paper
-
In this technical note, we further develop the results of [ref] and [ref] .
Pulse-Based Control Using Koopman Operator Under Parametric Uncertainty · Introduction · 0.99
Uses its method or tool 1 paper
-
The computation of the function $r$ can be performed using the methods to compute $s_1$ [ref] , which may not be numerically cheap for large-scale and/or stiff systems.
Pulse-Based Control Using Koopman Operator Under Parametric Uncertainty · Convergence to an Isostable · 0.93
Cites it as background 2 papers
-
Recently, Koopman theory has been applied for system identification [ref] , estimation [ref] and control [ref] of nonlinear systems.
Data-driven discovery of Koopman eigenfunctions for control · Introduction · 0.98
-
See [ref] for application to pulse-based control of monotone systems as well as [ref] for system identification and [ref] for state estimation.
A data-driven Koopman model predictive control framework for nonlinear flows · Review of related work · 0.99
What it cites · 4 sentences without a confident label
Builds on 1 paper
-
In this paper, we present an optimization program for solving the convergence problem, which is a generalization of the switching problem considered by [ref] and [ref] .
Shaping Pulses to Control Bistable Biological Systems · Introduction · 1.00
Compares with 1 paper
-
In this case, the closed-loop solution allows the switch, while the open-loop (i.e. [ref] ) does not.
Shaping Pulses to Control Bistable Biological Systems · Closed-Loop Switching in Generalized Repressilator · 1.00
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.