Optimal bounded control for minimizing the response of quasi non-integrable Hamiltonian systems

A strategy for designing optimal bounded control to minimize the response of quasi non-integrable Hamiltonian systems is proposed based on the stochastic averaging method for quasi non-integrable Hamiltonian systems and the stochastic dynamical programming principle. The equations of motion of a controlled quasi nonintegrable Hamiltonian system are first reduced to an one-dimensional averaged Ito stochastic differential equation for the Hamiltonian by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. Then, the dynamical programming equation for the control problem of minimizing the response of the averaged system is formulated based on the dynamical programming principle. The optimal control law is derived from the dynamical programming equation and control constraints without solving the equation. The response of optimally controlled systems is predicted through solving the Fokker-Planck-Kolmogrov (FPK) equation associated with completely averaged Ito equation. Finally, two examples are worked out in detail to illustrate the application and effectiveness of the proposed control strategy.