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Stochastic Control Part 2

Tham khảo tài liệu 'stochastic control part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 32 Stochastic Control xtt 1 xtt f xtt tk tk 1 - G G xtt tt tk 1 - tkwtt 1 where wt is a standard normal variable. The dimension of the phase space of the stochastic problem of concern here is four since the state vector xt xi X2 X3 x4 r Ộ Vr i y U o R4. The size of the mean state vector is four and the number of entries in the variance matrix of the state is sixteen. Since the state vector is a real-valued vector stochastic process the condition Pj Pi holds. The total number of distinct entries in the variance matrix w d be ten. The initial conditions are chosen as r 0 1 AU 0 1 rad Vr 0 0.01 AU TU ẫ 0 1.1 rad TU p 0 0 r xy for the state variable x and y. The initial conditions considered here are in canonical system of units. Astronomers adopt a normalized system of units i.e. canonical units for the simplification purposes. In canonical units the physical quantities are expressed in terms of Time Unit TU and 3 Astronomical Unit AU . The diffusion parameters ơr 0.0121 TU 2 and -4 AU ƠQ 2.2 X 10 ---------- are chosen for numerical simulations. Here we consider a set of TU 2 deterministic initial conditions which implies that the initial variance matrix w d be zero. Note that random initial conditions lead to the non-zero initial variance matrix. The system is deterministic at t 10 and becomes stochastic at t t0 because of the stochastic perturbation. This makes the contribution to the variance evolution coming from the system non-linearity coupled with initial variance terms will be zero at t t1. The contribution to the variance evolution at t t1 comes from the perturbation term GGT xt t only. For t t1 the contribution to the variance evolution comes from the system non-linearity as well as the perturbation term. This assumption allows to study the effect of random perturbations explicitly on the dynamical system. The values of diffusion parameters are selected so that the contribution to the force coming from the random part is smaller than the force coming from .