  In this paper, we provide for the first time an automated,
correct-by-construction, controller synthesis scheme for a class of infinite
dimensional stochastic systems, namely, retarded jump-diffusion systems. First,
we construct finite abstractions approximately bisimilar to non-probabilistic
retarded systems corresponding to the original systems having some stability
property, namely, incremental input-to-state stability. Then, we provide a
result on quantifying the distance between output trajectory of the obtained
finite abstraction and that of the original retarded jump-diffusion system in a
probabilistic setting. Using the proposed result, one can refine the control
policy synthesized using finite abstractions to the original systems while
providing guarantee on the probability of satisfaction of high-level
requirements. Moreover, we provide sufficient conditions for the proposed
notion of incremental stability in terms of the existence of incremental
Lyapunov functions which reduce to some matrix inequalities for the linear
systems. Finally, the effectiveness of the proposed results is illustrated by
synthesizing a controller regulating the temperatures in a ten-room building
modelled as a delayed jump-diffusion system.
