On infinite dimensional linear programming approach to stochastic control * *This research is partially supported by M. Kamgarpour’s European Union ERC Starting Grant, CONENE and by T. Summers’ the US National Science Foundation under grant CNS-1566127.

We consider the infinite dimensional linear programming (inf-LP) approach for solving stochastic control problems. The inf-LP corresponding to problems with uncountable state and input spaces is in general computationally intractable. By focusing on linear systems with quadratic cost (LQG), we establish a connection between this approach and the well-known Riccati LMIs. In particular, we show that the semidefinite programs known for the LQG problem can be derived from the pair of primal and dual inf-LPs. Furthermore, we establish a connection between multi-objective and chance constraint criteria and the inf-LP formulation.