Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control

Abstract

10.1 Motivation Optimal control problems for nonlinear partial differential equations are often hard to tackle numerically so that the need for developing novel techniques emerges. One such technique is given by reduced order methods. Recently the application of reduced-order models to optimal control problems for par-tial differential equations has received an increasing amount of attention. The reduced-order approach is based on projecting the dynamical system onto subspaces consisting of basis elements that contain characteristics of the ex-pected solution. This is in contrast to, e.g., finite element techniques, where the elements of the subspaces are uncorrelated to the physical properties of the system that they approximate. The reduced basis method as developed, e.g., in [IR98] is one such reduced-order method with the basis elements cor-responding to the dynamics of expected control regimes. Proper orthogonal decomposition (POD) provides a method for deriving low order models of dynamical systems. It was successfully used in a variety of fields including signal analysis and pattern recognition (see [Fuk90]), fluid dynamics and coherent structures (see [AHLS88, HLB96, NAMTT03, RF94, Sir87]) and more recently in control theory (see [AH01, AFS00, LT01, SK98, TGP99]) and inverse problems (see [BJWW00]). Moreover, in [ABK01] POD was successfully utilized to compute reduced-order controllers. The relation-ship between POD and balancing was considered in [LMG, Row04, WP01]. Error analysis for nonlinear dynamical systems in finite dimensions were car-ried out in [RP02]. In our application we apply POD to derive a Galerkin approximation in the spatial variable, with basis functions corresponding to the solution of the physical system at pre-specified time instances. These are called the snap-262 Michael Hinze and Stefan Volkwein shots. Due to possible linear dependence or almost linear dependence, the snapshots themselves are not appropriate as a basis. Rather a singular value decomposition (SVD) is carried out and the leading generalized eigenfunctions are chosen as a basis, referred to as the POD basis. The paper is organized as follows. In Section 10.2 the POD method and its relation to SVD is described. Furthermore, the snapshot form of POD for abstract parabolic equations is illustrated. Section 10.3 deals with reduced order modeling of nonlinear dynamical systems. Among other things, error estimates for reduced order models of a general equation in fluid mechanics obtained by the snapshot POD method are presented. Section 10.4 deals with suboptimal control strategies based on POD. For optimal open-loop control problems an adaptive optimization algorithm is presented which in every it-eration uses a surrogate model obtained by the POD method instead of the full dynamics. In particular, in Section 10.4.2 first steps towards error estima-tion for optimal control problems are presented whose discretization is based on POD. The practical behavior of the proposed adaptive optimization algo-rithm is illustrated for two applications involving the time-dependent Navier-Stokes system in Section 10.5. For closed-loop control we refer the reader to [Gom02, KV99, KVX04, LV03], for instance. Finally, we draw some conclu-sions and discuss future research perspectives in Section 10.6.