**In minisymposium: Matrix Equations**

Lyapunov equations in model order reduction of stochastic systems

Tobias Damm (University of Kaiserslautern)

We consider various approaches to balanced truncation of stochastic linear systems of the form

\begin{align*}
dx = Ax\,dt+ Nx\,dw + Bu\,dt, \quad
y = Cx.
\end{align*}

To this end, we introduce different generalizations of the
reachability Gramian. In particular we analyse the following two pairs of matrix inequalities
\begin{align*}
A^TQ+QA+N^TQN&\le-C^TC,\\
AP+PA^T+NPN^T&\le-BB^T
\end{align*}

and
\begin{align*}
A^TQ+QA+N^TQN&\le-C^TC,\\
A^TP^{-1}+P^{-1}A+N^TP^{-1}N&\le-P^{-1}BB^TP^{-1}.
\end{align*}

Performing truncation based on balancing of $Q$ and $P$ in the two
different cases, we observe that both approaches preserve asymptotic
stability, but only the second leads to a stochastic $H^\infty$-type
bound for the approximation error of the truncated system. Further
properties and numerical issues shall be discussed in the talk.