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Approximation and existence of periodic solutions for controlled diffusion equations. (English) Zbl 0703.60052
The author proved the existence of bounded T-periodic controls \(u_ i(t,x)\), \(i=1,...,m\), acting in the same directions as the Wiener process, such that the equation with T-periodic coefficients \[ dx=[f(t,x)+\sum^{m}_{i=1}u_ i(t,x)g_ i(t,x)]dt+\sum^{m}_{i=1}g_ i(t,x)dw_ i(t), \] \(t\geq 0\), \(x\in {\mathbb{R}}^ n\), has a solution T-periodic in distribution. The main assumption is a nondegeneracy condition on the diffusion part: There exist \(h_ 1,...,h_ n\in {\mathcal L}(g_ 1,...,g_ m)\) and \(K>0\) such that \[ \lambda^*H(t,x)H^*(t,x)\lambda \geq K| \lambda |^ 2\text{ for } every\quad \lambda \in {\mathbb{R}}^ n,\quad t\in [0,T]\text{ and } x\in {\mathbb{R}}^ n, \] where \(H(t,x)=(h_ 1(t,x),...,h_ n(t,x))\) and \({\mathcal L}(g_ 1,...,g_ m)\) is the Lie algebra generated by \(g_ 1,...,g_ m\). The method of the author is based on an approximation theorem for solutions of SDEs.
Reviewer: Tran Van Nhung

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
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