# Iterated Stochastic Integrals in Non Linear Control Theory

• R. Schott
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 29)

## Abstract

In non linear control theory many systems are governed by an equation of the type $$q(t) = ({A_o} + \sum\limits_1^n {{u_i}(t)} {A_i})q(t)$$ (1), where q ∈ IRN and A0, A1,..., An are N×N square matrices, (u1,...,un) is a multidimensional input. We are interested here by the case of a white noise input. The stochastic equation (1) can be solved for each path. The solution has an expression in terms of iterated stochastic integrals of the following type :
$${a_n}(t) = \int_0^t {d{B_{{{j_n}}}} \ldots d{B_{{{j_1}}}} = \int_0^t {d{B_{{{j_n}}}}(s)\int_0^s {d{B_{{{j_{{n - 1}}}}}} \ldots d{B_{{{j_1}}}}} } }$$
$${B_0}(t) = t,{B_i}(t) = \int_0^t {{u_i}(s)ds} \quad i = 1,2, \ldots, n$$

In order to control the stochastic process solution of (1) we need to etablish some properties of these iterated integrals. In this paper we give an approximation for the density of an(t).

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