Stochastic Bifurcation And Fractional Optimal Control Of Stochastic Systems With Fractional Derivative Damping

The stochastic bifurcation and fractional optimal control of stochastic systems with fractional derivative damping are investigated. The study of stochastic bifurcation consists of two parts:one is the stochastic Hopf bifurcation of quasi integrable Hamiltonian systems with fractional derivative damping. First, the averaged Ito stochastic differential equations for energy integrals of sub-systems are obtained by using the stochastic averaging method for quasi integrable Hamiltonian systems. Then, an expression for the average bifurcation parameter of the averaged system is obtained and a criterion for determining the stochastic Hopf bifurcation of the system by using the average bifurcation parameter is proposed. The other is the stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping. First, the averaged Ito stochastic differential equations for amplitude and phase difference are obtained by using the stochastic averaging method for strongly nonlinear systems under bounded noise excitation. Then, the stationary probability density of amplitude is obtained by solving the stationary Fokker-Planck-Kolmogorov (FPK) equation associated with the averaged Ito equations and the stochastic jump as well as its P-bifurcation as the system parameters change are examined by using the obtained stationary probability density.In the study of fractional stochastic optimal control, first, a set of partially averaged Ito stochastic differential equations of controlled quasi integrable Hamiltonian systems with fractional derivative damping are obtained by applying the stochastic averaging method for quasi integrable Hamiltonian systems and a fractional order performance index is proposed. Then, the stochastic dynamical programming equation of the partially averaged system is established by using the stochastic dynamical programming principle and solved to yield the fractional optimal control law. The fractional stochastic optimal time-delay control and stochastic stabilization of quasi integrable Hamiltonian systems with fractional derivative damping are investigated as a generalization of the above control strategy. For the problem of time-delay control, the time-delayed control forces are approximately replaced by the control forces without time delay. For the problem of stochastic stabilization, it is is formulated as a fractional ergodic control with undetermined cost function. Then, the cost function is determined by the requirement of minimizing the largest Lyapunov exponent.