Optimization of a control law to synchronize first-order dynamical systems on Riemannian manifolds by a transverse component

Figure 1.  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 $)

Figure 2.  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

Figure 3.  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 $)

Figure 4.  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

Figure 5.  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 $)

Figure 6.  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

Figure 7.  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

Figure 8.  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)