  We investigate the optimal model reduction problem for large-scale
quadratic-bilinear (QB) control systems. Our contributions are threefold.
First, we discuss the variational analysis and the Volterra series formulation
for QB systems. We then define the $\mathcal H_2$-norm for a QB system based on
the kernels of the underlying Volterra series and also propose a truncated
$\mathcal H_2$-norm. Next, we derive first-order necessary conditions for an
optimal approximation, where optimality is measured in term of the truncated
$\mathcal H_2$-norm of the error system. We then propose an iterative model
reduction algorithm, which upon convergence yields a reduced-order system that
approximately satisfies the newly derived optimality conditions. We also
discuss an efficient computation of the reduced Hessian, using the special
Kronecker structure of the Hessian of the system. We illustrate the efficiency
of the proposed method by means of several numerical examples resulting from
semi-discretized nonlinear partial differential equations and show its
competitiveness with the existing model reduction schemes for QB systems such
as moment-matching methods and balanced truncation.
