Control Of Two Classes Of Stochastic Systems

With the continuous development of science and technology and modern industry,the requirements on control problems have been continuously raised.In the current society,stochastic optimal control theory has wide range of applications,mainly used in economics,especially with financial issue.In other fields,the theory of stochastic optimal control also has a very wide range of applications,such as biology,physics,engineering,management and so on.In this paper,the optimal control problems of two classes of stochastic systems are studied.Namely,the reliability control of internally resonant multi-degrees-of-freedom(MDOF)system under Gaussian white noise excitation or wide-band colored noise excitation,and the control for stationary probability density of single-degrees-of-freedom(SDOF)system under combined Gaussian white noise excitation and wide-band colored noise excitations.For the problem of reliability control,by means of stochastic averaging method of generalized harmonic function,the equations of motion of the controlled system are simplified to partially-averaged It? equations.Associated with the principle of dynamic programming,a dynamic programming equation determining the maximal dynamical reliability is established.The optimal control force is obtained.The It? equations of the optimal control system are obtained.The conditional dynamical reliability of the optimal system is governed by the backward Kolmogorov equation,and the mean first-cross time is governed by the Pontryagin equation.The boundary conditions and the initial conditions of these two partial differential equations(PDEs)are also given.For the problem of response control,also the stochastic averaging method is applied associated with the dynamic programming principle to yield the optimal control law.The Fokker-PlanckKolmogorov(FPK)equation of the optimal system is established.The exact solution of the stationary probability density is obtained.Monte Carlo simulation is performed to verify the theoretical results.