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We formulate here an approach to model reduction that is well-suited forlinear time-invariant control systems that are stabilizable and detectable butmay otherwise be unstable. We introduce a modified $\mathcal{H}_2$-errormetric, the $\mathcal{H}_2$-gap, that provides an effective measure of modelfidelity in this setting. While the direct evaluation of the$\mathcal{H}_2$-gap requires the solutions of a pair of algebraic Riccatiequations associated with related closed-loop systems, we are able to workentirely within an interpolatory framework, developing algorithms andsupporting analysis that do not reference full-order closed-loop Gramians. Thisleads to a computationally effective strategy yielding reduced models designedso that the corresponding reduced closed-loop systems will interpolate thefull-order closed-loop system at specially adapted interpolation points,without requiring evaluation of the full-order closed-loop system nor evencomputation of the feedback law that determines it. The analytical frameworkand computational algorithm presented here provides an effective new approachtoward constructing reduced-order models for unstable systems. Numericalexamples for an unstable convection diffusion equation and a linearizedincompressible Navier-Stokes equation illustrate the effectiveness of thisapproach.

$\mathcal{H}_2$-gap model reduction for stabilizable and detectable systems