Infinite-dimensional bilinear and stochastic balanced truncation with explicit error bounds

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.

[1]  A. Isidori,et al.  Realization and Structure Theory of Bilinear Dynamical Systems , 1974 .

[2]  Ruth F. Curtain,et al.  Balanced realisations for infinite dimensional systems , 1985 .

[3]  Timo Reis,et al.  Balancing Transformations for Infinite-Dimensional Systems with Nuclear Hankel Operator , 2014 .

[4]  Peter Benner,et al.  Interpolation-Based H2-Model Reduction of Bilinear Control Systems pdfsubject , 2011 .

[5]  P. Benner,et al.  Balanced Truncation for Stochastic Linear Systems with Guaranteed Error Bound , 2014 .

[6]  Mark R. Opmeer,et al.  Model Reduction by Balanced Truncation for Systems with Nuclear Hankel Operators , 2014, SIAM J. Control. Optim..

[7]  J. Weidmann.  Lineare Operatoren in Hilberträumen , 2000 .

[8]  Jiongmin Yong,et al.  Optimal Control Theory for Infinite Dimensional Systems , 1994 .

[9]  Gianluigi Rozza,et al.  Model Reduction of Parametrized Systems , 2017 .

[10]  Martin Redmann.  Type II Singular Perturbation Approximation for Linear Systems with Lévy Noise , 2018, SIAM J. Control. Optim..

[11]  Martin Redmann.  Type II Balanced Truncation for Deterministic Bilinear Control Systems , 2018, SIAM J. Control. Optim..

[12]  Martin Redmann.  Energy estimates and model order reduction for stochastic bilinear systems , 2020, Int. J. Control.

[13]  Carsten Hartmann,et al.  Dimension reduction by balanced truncation: application to light-induced control of open quantum systems. , 2011, The Journal of chemical physics.

[14]  Tobias Breiten,et al.  Krylov subspace methods for model order reduction of bilinear control systems , 2010, Syst. Control. Lett..

[15]  Peter Benner,et al.  Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems , 2011, SIAM J. Control. Optim..

[16]  H. Lebesgue.  Sur l'intégration des fonctions discontinues , 1910 .

[17]  Peter Benner,et al.  Model reduction for stochastic systems , 2015 .

[18]  R. Curtain,et al.  Realisation and approximation of linear infinite-dimensional systems with error bounds , 1988 .

[19]  Carsten Hartmann,et al.  Balanced Averaging of Bilinear Systems with Applications to Stochastic Control , 2013, SIAM J. Control. Optim..

[20]  J Lam,et al.  ON H2 MODEL REDUCTION OF BILINEAR SYSTEM , 2002 .

[21]  The Itô Integral with respect to an Infinite Dimensional Lévy Process: A Series Approach , 2013, 1907.01450.

[22]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[23]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[24]  Jerzy Zabczyk,et al.  Stochastic Partial Differential Equations with Lévy Noise: References , 2007 .

[25]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .