Abstract. 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 H 2 -norm for a QB system based on the kernels of the underlying Volterra series and also propose a truncated H 2 -norm. Next, we derive first-order necessary conditions for an optimal approximation, where optimality is measured in term of the truncated 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. Key words. Model order reduction, quadratic-bilinear control systems, tensor calculus, H 2 -norm, Sylvester equations, nonlinear partial differential equations. AMS subject classifications. 15A69, 34C20, 41A05, 49M05, 93A15, 93C10, 93C15. 1. Introduction. Numerical simulation is a fundamental tool in the analysis of dynamical systems, and is required repeatedly in the computational control design, optimization and prediction thereof. Dynamical systems are generally governed by partial differential equations (PDEs), or ordinary differential equations (ODEs), or a combination of both. A high-fidelity approximation of the underlying physical phenomena requires a finely discretized mesh over the interesting spatial domain, leading to complex dynamical systems with a high-dimensional state-space. The simulations of such large-scale systems, however, impose a huge computational burden. This inspires model order reduction (MOR), which aims at constructing simple and reliable surrogate models such that their input-output behaviors approximate that of the original large-scale system accurately. These surrogate models can then be used in engineering studies, which make numerical simulations faster and efficient. In recent decades, numerous theoretical and computational aspects for MOR of linear systems have been developed; e.g., see [1, 6, 14, 41] . These methods have been successfully applied in various fields, e.g., optimal control and PDE constrained optimization and uncertainty quantification; for example, see [16, 28] . In recent years, however, MOR of nonlinear systems has gained significant interest with a goal of extending the input-independent, optimal MOR techniques from linear systems to nonlinear ones. For example, MOR techniques for linear systems such as balanced truncation [1, 35] , or the iterative rational Krylov method (IRKA) [23] , have been extended to a special class of nonlinear systems, so-called bilinear systems, in which nonlinearity arises from the product of the state and input [7, 10, 20, 43] . However,
However, our primary objective is to extend the H 2 -optimal model reduction technique REF to the QBDAE (1).
$\mathcal H_2$-Quasi-Optimal Model Order Reduction for Quadratic-Bilinear Control Systems
Computer Science Mathematics