Stochastic differential equations and their application to randomly time varying control systems
Sibul, Leon H.
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Let a control system be represented by a system of n first-order stochastic differential equations / ;(t, (I))=a(t, (I))y(t, (1))+ Ux(t, (I)) wEn on (n,!E, P) and tET / where the coefficient matrix a(t, m) is a sum of a deterministic matrix and a. stochastic matrix a(t, 0»). It is assumed that the deterministic part of the system is stable in the sense of Lyapunov. The stochastic differential equation becomes now an integral equation with a stochastic kernel. This integral equation is solved by constructing a resolvent kernel by means of a Neumann series expansion. Sufficient conditions for almost sure unifonn convergence of the Neumann series expansion are given. Integral expressions for the expectation and covariance matrix of the stochastic vector y(t. m) are found. These results have many applications in control systems theory.
The article of record as published may be found at https://doi.org/10.1080/00207177708922260
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