On the coordinate synchronization problem for dynamical systems (Q1776243)

The paper studies the coordinate synchronization problem for coupled dynamical systems. Particularly, there are studied two coupled systems with control \[ \dot x^i = X (t,x^1,x^2,u), \quad i=1,2, \] where \(x^i = (y^i,z^i)^T\) are phase vectors, \(\dim (y^1) = \dim (y^2)\), and \(u=u(t,x^1,x^2)\) is a control vector. The coordinate synchronization takes place when \(\| y^1(t,t_0,x_0^1,x_0^2) - y^2(t,t_0,x_0^1,x_0^2)\| \to 0\) as \(t\to \infty\) for some set of initial conditions \((x_0^1,x_0^2)\). Using the Lyapunov function method and ideas from the theory of partial stability, the author obtains sufficient conditions for coordinate synchronization.