Stochastic Dynamics And Control Of Strongly Non-linear Systems Under Combined Harmonic And White (Wide-Band) Noise Excitations

The stochastic dynamics and control of single and multi-degree-of-freedom strongly nonlinear systems excited by combined harmonic and Gaussian white-noise or combined harmonic and wide band noise are investigated. For the uncontrolled systems, the motion equations are reduced to a set of averaged Ito stochastic differential equations by using the stochastic averaging method in the resonant cases. Then the FPK equation governing the joint probability density of amplitude and phase, the backward Kolmogorov equation governing the conditional reliability function, and the Pontryagin equation governing the mean first passage time are established, respectively. The stationary joint probability density of amplitude and phase, the conditional reliability function and the conditional probability density function of the first passage time, and the mean first passage time are obtained by solving these equations. The asymptotic stability with probability one of a single-degree-of-freedom strongly nonlinear system is investigated via its linearized system and the largest Lyapunov exponent is obtained by introducing a norm in terms of response amplitude. For multi-degree-of-freedom strongly nonlinear systems, introducing a new norm in terms of the square root of the total energy in the definitions of stochastic stability and Lyapunov exponent, the asymptotic stability with probability one for external resonance case and for both external and internal resonances case is investigated via the largest Lyapunov exponents.For the controlled systems, the dynamical programming equations for the control problems of minimizing response, maximizing the reliability and maximizing mean first passage time are formulated from the partially averaged Ito stochastic differential equations based on the stochastic averaging method and the dynamic programming principle. The optimal control law is derived from the dynamical programming equation and control constraint. It is shown that the optimal control law is Bang-bang control if the control is bounded. Then the stationary joint probability density, the conditional reliability function, the mean first-passage time of the optimally controlled system are obtained from solving the reduced FPK equation, the backward Kolmogorov equation and the Pontryagin equation, respectively of fully averaged systems. The representative strongly nonlinear systems such as Duffing oscillator, Duffing-van der Pol oscillator, Duffing-Rayleigh-Mathieu oscillator are taken as the examples to illustrate the proposed procedures and all the theoretical results are verified by Monte-Carlo simulation.