# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021213
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## Optimization of a control law to synchronize first-order dynamical systems on Riemannian manifolds by a transverse component

 1 School of Information and Automation Engineering, Università Politecnica delle Marche, Via Brecce Bianche, I-60131 Ancona, Italy 2 Dipartimento di Ingegneria dell'Informazione, Università Politecnica delle Marche, Via Brecce Bianche, I-60131 Ancona, Italy

* Corresponding author: Simone Fiori

Received  March 2021 Revised  June 2021 Early access August 2021

The present paper builds on the previous contribution by the second author, S. Fiori, Synchronization of first-order autonomous oscillators on Riemannian manifolds, Discrete and Continuous Dynamical Systems – Series B, Vol. 24, No. 4, pp. 1725 – 1741, April 2019. The aim of the present paper is to optimize a previously-developed control law to achieve synchronization of first-order non-linear oscillators whose state evolves on a Riemannian manifold. The optimization of such control law has been achieved by introducing a transverse control field, which guarantees reduced control effort without affecting the synchronization speed of the oscillators. The developed non-linear control theory has been analyzed from a theoretical point of view as well as through a comprehensive series of numerical experiments.

Citation: Adolfo Damiano Cafaro, Simone Fiori. Optimization of a control law to synchronize first-order dynamical systems on Riemannian manifolds by a transverse component. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021213
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##### References:
Simulation of the evolution of the master system (in green color), controlled systems $\Sigma_L$ (26) (in red color) and $\Sigma_G$ (27) (in blue color) on the sphere $\mathbb{S}^2$, displayed in terms of state components, together with the values taken by the Lyapunov function (7) during evolution (in blue color for $\Sigma_G$ and in red color for $\Sigma_L$)
Simulation of the evolution of the controlled systems $\Sigma_L$ (26) (in red color) and $\Sigma_G$ (27) (in blue color) on the sphere $\mathbb{S}^2$, displayed in terms of transverse control field components, together with the values taken by the control effort during evolution
Synchronization of a master/slave pair oscillators on the sphere $\mathbb{S}^7$ illustrated in terms of state components as well as kinetic energy (in green color for the master oscillator, red color for the system $\Sigma_L$ and blue color for the system $\Sigma_G$)
Synchronization of a master/slave pair oscillators on the sphere $\mathbb{S}^7$ – with and without the transverse field $\tau_G$ – illustrated in terms of control efforts and Lypunov function values (in red color for the system without transverse component and blue color for the system with transverse component). The left-bottom panes shows the course of the difference $\|u\|_{z_s}^2-\|u+\tau_G\|_{z_s}^2$ which takes non-negative values
Synchronization of a master/slave pair oscillators on the sphere $\mathbb{S}^7$ illustrated in terms of transverse field components as well as control efforts and Lypunov function values (in red color for the system $\Sigma_L$ and blue color for the system $\Sigma_G$)
Synchronization of two master/slave oscillators on $\mathbb{SO}(3)$ by the control field (4). In the top panel, the evolution of the squared Riemannian distance $d^2(z^s, z^m)$ is represented versus time. In the bottom panel, the evolution of the squared control effort related to the control law $u$ is represented over times
Synchronization of two master/slave oscillators on $\mathbb{SO}(3)$ by the control field $\tilde{u}$ with (48) as transverse control field. In the top panel, the evolution of the squared Riemannian distance $d^2(z^s, z^m)$ is represented versus time. In the bottom panel, the evolution of the squared control effort associated to the control field $\tilde{u}$ is represented
Synchronization of two master/slave oscillators on $\mathbb{SO}(3)$: Comparison of the squared control effort resulting from the application of the control laws $\tilde{u}(t)$ (in blue color) and $u(t)$ (in red color)
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