Control algorithms for oscillations in Hamiltonian systems (Q1594150)

The paper studies control algorithms for oscillations in Hamiltonian systems. The authors consider the Hamiltonian system with the phase vector \(x= (q,p)\) and with the Hamiltonian \[ H(q,p,u)= H_0(q,p)+ \sum^m_{j=1} H_j(q,p)u_j,\quad j= 1,\dots, m \] which is linear with respect to the control actions \(u= (q,p)\). They study a general class of problems where the set of integrals of motion \(F_i\), \(i=1,\dots, k\), of the uncontrolled (free) system is given, and the control objective is to reach their arbitrary prescribed values \(f_i\). Control algorithms ensuring this objective are obtained by the speed gradient method used for the adaptive control of nonlinear systems. The results are applied in oscillation studies of the simple spherical pendulum. The results obtained establish the compatibility and limitations of the speed gradient algorithm applied to the control of oscillatory motion in nonlinear systems.