Abstract
Along the ideas of Curtain and Glover (in: Bart, Gohberg, Kaashoek (eds) Operator theory and systems, Birkhäuser, Boston, 1986), we extend the balanced truncation method for (infinite-dimensional) linear systems to arbitrary-dimensional bilinear and stochastic systems. In particular, we apply Hilbert space techniques used in many-body quantum mechanics to establish new fully explicit error bounds for the truncated system and prove convergence results. The functional analytic setting allows us to obtain mixed Hardy space error bounds for both finite-and infinite-dimensional systems, and it is then applied to the model reduction of stochastic evolution equations driven by Wiener noise.
Highlights
Model reduction of bilinear systems has become a major field of research, partly triggered by applications in optimal control and the advancement of iterative numerical methods for solving large-scale matrix equations
High-dimensional bilinear systems often appear in connection with semi-discretized controlled partial differential equations or stochastic differential equations with multiplicative noise
The purpose of this paper is to extend balanced truncation to bilinear and stochastic evolution equations, to establish convergence results and prove explicit truncation error bounds for the bilinear and stochastic systems
Summary
Model reduction of bilinear systems has become a major field of research, partly triggered by applications in optimal control and the advancement of iterative numerical methods for solving large-scale matrix equations. High-dimensional bilinear systems often appear in connection with semi-discretized controlled partial differential equations or stochastic (partial) differential equations with multiplicative noise. A popular class of model reduction methods that is well established in the field of linear systems theory is based on first transforming the system to a form in which highly controllable states are highly observable and vice versa (“balancing”) and eliminating the least controllable and observable states.
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