Approximate controllability of one-dimensional SDE driven by countably many Brownian motions

Abstract

We are given (Ω, $\mathcal{F}$ , P) as a complete probability space with right continuous complete σ-algebra filtration $\left( {\mathcal{F}_t } \right)_{t \in \left[ {0,T} \right]}$ , generated by the infinite sequence of independent Brownian motions (W ⁱ) _i≥1. Let, for every t ∈ [0, T], L ₂(Ω, $\mathcal{F}_t$ , R) be the Hilbert space of all $\mathcal{F}_t$ -measurable, and square-integrable variables in R, and L ₂(∈, C ([0, T], R)) be the space of all square integrable and a.e. continuous functions on R equipped with the norm |X| = (E sup _t∈[0,T] |X(t)|²)^1/2. $L_2^\mathcal{F}$ ([0, T], R) denotes the Hilbert space of all square-integrable and $\mathcal{F}_t$ -adapted processes with values in R. Define the sequence σ(x) = (σ _i(x)) _i≥1, where for each i ≥ 1, σ _i(x) ∈ C([0, T], R) and that σ(x) ∈ ℓ₂, i.e. |σ(x)|² = $\Sigma _{i = 1}^\infty$ |σ _i(x)|² < ∞. In this paper we study the approximate controllability of the one-dimensional semi-linear stochastic differential equation

$$ \left\{ \begin{gathered} dX\left( t \right) = \left[ {AX\left( t \right) + Bu\left( t \right) + b\left( {X\left( t \right)} \right)} \right]dt + \sum\limits_{i = 1}^\infty {\sigma _i \left( {X\left( t \right)} \right)dW^i \left( t \right)} \hfill \\ X\left( 0 \right) = X_0 ,t \in \left[ {0,T} \right], \hfill \\ \end{gathered} \right. $$

where A, B ∈ R, and u ∈ $L_2^\mathcal{F}$ ([0, T], R) is a control. We obtain sufficient conditions for approximate controllability of the above system when coefficients b, and σ satisfy non-Lipschitz conditions.