Feedback control theory & Model order reduction for stochastic equations

We analyze structure-preserving model order reduction methods for Ornstein-Uhlenbeck processes and linear SPDEs with multiplicative noise based on balanced truncation with non-zero initial data. We then marry these model order reduction methods with stochastic optimal control theory and prove error bounds for a class of linear quadratic regulator problems. We discuss the application of our approach to enhanced sampling methods from non-equilibrium statistical mechanics.

[1]  A. Menegaki,et al.  SPECTRAL GAP IN O ( n )-MODEL AND CHAIN OF OSCILLATORS USING SCHRÖDINGER OPERATORS , 2019 .

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

[3]  Nadine Gottschalk,et al.  Nonlinear System Theory , 2016 .

[4]  Serkan Gugercin,et al.  Model reduction for systems with inhomogeneous initial conditions , 2016, Syst. Control. Lett..

[5]  Zhang,et al.  Burgers equation with correlated noise: Renormalization-group analysis and applications to directed polymers and interface growth. , 1989, Physical review. A, General physics.

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

[7]  Xun Yu Zhou,et al.  Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls , 2000, IEEE Trans. Autom. Control..

[8]  J. Lebowitz,et al.  FOURIER LAW: A CHALLENGE TO THEORISTS , 2008 .

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

[10]  A. Menegaki,et al.  Spectral Gap in Mean-Field $${\mathcal {O}}(n)$$-Model , 2019 .

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

[12]  Mark R. Opmeer,et al.  Finite-Rank ADI Iteration for Operator Lyapunov Equations , 2013, SIAM J. Control. Optim..

[13]  Tryphon T. Georgiou,et al.  Optimal Steering of a Linear Stochastic System to a Final Probability Distribution—Part III , 2014, IEEE Transactions on Automatic Control.

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

[15]  Athanasios C. Antoulas,et al.  Balanced truncation model reduction for systems with inhomogeneous initial conditions , 2011, Autom..

[16]  Thomas G. Fai,et al.  A reversible mesoscopic model of diffusion in liquids: from giant fluctuations to Fick’s law , 2013, 1312.1894.

[18]  E. Lieb,et al.  Properties of a Harmonic Crystal in a Stationary Nonequilibrium State , 1967 .

[19]  Naji Qatanani,et al.  Balanced model reduction of linear systems with nonzero initial conditions: Singular perturbation approximation , 2019, Appl. Math. Comput..

[20]  J. Zabczyk,et al.  Remarks on the algebraic Riccati equation in Hilbert space , 1975 .

[21]  Grigorios A. Pavliotis,et al.  Optimal control of multiscale systems using reduced-order models , 2014 .

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

[23]  J. Stübinger,et al.  Optimal trading strategies for Lévy-driven Ornstein–Uhlenbeck processes , 2019, Applied Economics.

[24]  R. Bucy.  Nonlinear filtering theory , 1965 .

[25]  Khier Benmahammed,et al.  Model reduction of stochastic systems , 1986, 1986 25th IEEE Conference on Decision and Control.

[26]  M. Freitag,et al.  Balanced model order reduction for linear random dynamical systems driven by Lévy noise , 2018 .

[27]  Tryphon T. Georgiou,et al.  Optimal Steering of a Linear Stochastic System to a Final Probability Distribution, Part II , 2014, IEEE Transactions on Automatic Control.

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

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

[31]  Erkan Nane,et al.  Space-time fractional stochastic partial differential equations with Lévy noise , 2019, 1902.10637.

[32]  D. Kleinman.  On an iterative technique for Riccati equation computations , 1968 .

[33]  Oldrich A. Vasicek.  An equilibrium characterization of the term structure , 1977 .