Lyapunov exponents of controlled SDE's and stabilizability property : Some examples
Abstract
We consider a stochastic differential equation with linear feedback control~: \begindisplaymath dX_t = (A+B\,K)\,X_t\, dt + \sum_k=1^r(A_k+B_k\,K)\,X_t\,\circ\! dW_k(t) \enddisplaymath where $K$ is the feedback gain matrix. For each value of $K$, let $\lambda_K$ be the Lyapunov exponent associated with the solution of the SDE. The set of $\lambda_K$, as $K$ describe the set of matrices, is a connected interval of $\R$. We present some examples where $-\infty$ is the lower bound of this set. For these cases, we say that the corresponding EDS is stabilizable.