A note on a parameter depending Datko theorem applied to stochastic systems

The interest on parameter depending stochastic differential equations (SDE) arises in a very natural way, for instance in ergodic control and in adaptive control of stochastic systems. In this framework, it is in some case useful to know that a parameterized class of SDEs have solutions that decay exponentially to zero uniformly on the parameter. The uniform decay of the optimal states of a class of linear, infinite dimensional, stochastic controlled systems is obtained under a uniform detectability assumption. This uniform detectability condition is verified for a particular parameter depending controlled stochastic system coming from ergodic control of affine stochastic differential equations.