Multi-Symmetric Lyapunov Equations

This letter studies multi-symmetric Lyapunov equations and their application to stochastic control. It is shown how Lyapunov recursions can be used to efficiently compute the tensor-valued cumulants of the transient- and limit distributions of stochastic linear systems. This analysis is based on the assumption that the process noise distribution admits a moment expansion but, apart from this, all derivations and numerical algorithms are kept entirely general—without introducing any further assumptions on the distribution. Moreover, it is shown both theoretically and numerically how to exploit these recursions to construct accurate approximations of a rather general class of stochastic optimal control problems for linear discrete-time systems with conditional-value-at-risk constraints.