Stochastic Methods in the Dynamics of Satellites: Course by Peter Sagirow

By Peter Sagirow

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By Peter Sagirow

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Extra resources for Stochastic Methods in the Dynamics of Satellites: Course Held at the Department for General Mechanics, October 1970

Example text

Stochastic Dynamical Systems 30 plicitly. w ( 2. 43) with A, B matrices and ~ a Wiener vector process. »(s), to where A + denotes = canst, + the fundamental matrix of A. 45) If A depends on time the fundamental matrix can generally not be given in closed form. For the calculation of f see for instance ~4]. 2. 7 Stochastic Stability [3] (5] I As an explicit solution of stochastic differential equations can be obtained only in the simplest cases, stability investigations of stochastic systems are of high importance.

42) can be replaced by [5q~ -t· 4()J-- 21>'1~- 2}1- 2 ] 2 [ 5'1 40 + 4(JJ. d. 38) reads as 2'1o '1~ + q2 • In fig. 6 the stability domain is shown for diffe- rent values of q 2 • The stability domain decreases with increasing q 2 • I f the satellite has a spherical mass distribution (A torque is zero and For 9 (x 1} = s~n xi = B = C) e =0. 1- ~ t . we have JJt • 1. The corresponding stability domain is shown in Fig. 7.

X. i 0 • ( 2. 66) Then the equilibrium position x = 0 of eq. 65) is stable in probability. 67) Chap. 2. Stochastic Dynamical Systems 42 2. 7. 7 The Operator L The operator L can be obtained as follows. It is = dx v fdt + crd'W' • v-(t,x) and .. 'U'tdt + r. ~ i 1 '~~~. t + ~,i. x~ dx;_ + ... - ~~ ,. + ,. t ·1. e d. '~At e)(ftdt + ... ~d- = Thus, we have finally - L a At + v a Lf~ -a ~ .. x~ n t + 2E r 1 n n ~-~ ~=t (c:ro-*)~t ()2 ax~ a . (2. 67) x~ 2. 7. 'br(t) . 68) and has to satisfy the conditions: too a') '\t( 0) C and w are in- t dependent of of f exist and are continuous at least 0• Theorem 2.

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