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We investigate the optimal model reduction problem for large-scalequadratic-bilinear (QB) control systems. Our contributions are threefold.First, we discuss the variational analysis and the Volterra series formulationfor QB systems. We then define the $\mathcal H_2$-norm for a QB system based onthe kernels of the underlying Volterra series and also propose a truncated$\mathcal H_2$-norm. Next, we derive first-order necessary conditions for anoptimal approximation, where optimality is measured in term of the truncated$\mathcal H_2$-norm of the error system. We then propose an iterative modelreduction algorithm, which upon convergence yields a reduced-order system thatapproximately satisfies the newly derived optimality conditions. We alsodiscuss an efficient computation of the reduced Hessian, using the specialKronecker structure of the Hessian of the system. We illustrate the efficiencyof the proposed method by means of several numerical examples resulting fromsemi-discretized nonlinear partial differential equations and show itscompetitiveness with the existing model reduction schemes for QB systems suchas moment-matching methods and balanced truncation.

$\mathcal H_2$-Quasi-Optimal Model Order Reduction for Quadratic-Bilinear Control Systems