Optimal Bounded Control for Minimizing the Response of Quasi Non-Integrable Hamiltonian Systems

A strategy for designing optimal bounded control to minimize theresponse of quasi non-integrable Hamiltonian systems is proposed basedon the stochastic averaging method for quasi non-integrable Hamiltoniansystems and the stochastic dynamical programming principle. Theequations of motion of a controlled quasi non-integrable Hamiltoniansystem are first reduced to an one-dimensional averaged Itô stochasticdifferential equation for the Hamiltonian by using the stochasticaveraging method for quasi non-integrable Hamiltonian systems. Then, thedynamical programming equation for the control problem of minimizing theresponse of the averaged system is formulated based on the dynamicalprogramming principle. The optimal control law is derived from thedynamical programming equation and control constraints without solvingthe equation. The response of optimally controlled systems is predictedthrough solving the Fokker–Planck–Kolmogrov (FPK) equation associatedwith completely averaged Itô equation. Finally, two examples are workedout in detail to illustrate the application and effectiveness of theproposed control strategy.