Lyapounov exponent of linear stochastic systems with large diffusion term

Almost sure exponential stabilisation of stochastic systems by state-feedback control

“…The problems of stabilisation of differential equations by noise have been studied by many authors and we here mention Arnold, Crauel & Wihstutz (1983); Pardoux & Wihstutz (1992); Mao (1994); Kwiecińska (2002); Appleby & Mao (2005); Caraballo & Robinson (2004); Yuan & Mao (2004). The results demonstrate that after adding a stochastic term to a deterministic differential equation the top Lyapunov exponent becomes smaller, i.e.…”

“…Yuan & Mao (2004); Yuan & Lygeros (2005)) or in the diffusion part (see e.g. Arnold, Crauel & Wihstutz (1983); Pardoux & Wihstutz (1992); Mao (1994)). The reader may wonder if this is a simplification of the problem, since the more control power you have the easier it is to achieve the a.s. stabilising controller.…”

Random Dynamical Systems

“…From the vast literature we just mention Ariaratnam and Xie [7], Arnold, Eizenberg and Wihstutz [17], Kao and Wihstutz [196], Pardoux and Wihstutz [270,271], Pinsky [278], and Pinsky and Wihstutz [279]. 9), etc.…”

Stabilization by noise for a class of stochastic reaction-diffusion equations

“…Thus, we are considering a situation in which, without any non-degeneracy assumption on the noise, the stochastic part plays a crucial role in stabilizing an equation which is not necessarily asymptotically stable, trying to extend in some sense to an infinite dimensional setting some results on stabilization by noise proved in finite dimension (see e.g. [1], [2], [18] and [19]; see also [3], [4], [12] and [15] for some results in infinite dimension).…”