# Balanced Averaging of Bilinear Systems with Applications to Stochastic Control

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## References

*A new model reduction scheme for k-power bilinear systems*, in Proceedings of the 32nd IEEE Conference on Decision and Control, IEEE, New York, 1993, pp. 22--27.

*Approximation of Large-Scale Dynamical Systems*, Adv. Des. Control, SIAM, Philadelphia, 2005.

*Invariant measures of differential inclusions applied to singular perturbations*, Weizman Institute Research Report, 1998.

*A projection method for model reduction of bilinear dynamical systems*, Linear Algebra Appl., 415 (2006), pp. 406--425.

*Hypercontractivité des semi-groupes de diffusion*, C.R. Acad. Sci. Paris Sér. I Math., 299 (1984), pp. 775--778.

*Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems*, Numer. Linear Algebra Appl., 15 (2008), pp. 755--777.

*Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems*, SIAM J. Control Optim., 49 (2011), pp. 686--711.

*Nonnegative Matrices in the Mathematical Sciences*, Academic Press, New York, 1979.

*Interpolation-based $\mathcal{H}_{2}$-model reduction of bilinear control system*, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 859--885.

*Krylov subspace methods for model order reduction of bilinear control systems*, Systems Control Lett., 59 (2011), pp. 443--450.

*Asymptotic Methods in the Theory of Nonlinear Oscillations*, Gordon and Breach, New York, 1961.

*Krylov subspace methods for model order reduction of bilinear control systems*, Systems Control Lett., 59 (2010), pp. 443--450.

*Empirical balanced truncation for nonlinear systems*, J. Nonlinear Sci., 14 (2004), pp. 405--414.

*Nonlinear systems---algebraic Gramians and model reduction*, COMPEL, 24 (2005), pp. 202--219.

*Krylov subspace from bilinear representations of nonlinear systems*, COMPEL, 26 (2007), pp. 11--26.

*Realization and structure theory of bilinear dynamical systems*, SIAM J. Control, 12 (1974), pp. 517--535.

*Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations*, Numer. Linear Algebra Appl., 15 (2008), pp. 853--871.

*Rational Matrix Equations in Stochastic Control*, Lecture Notes in Control and Inform. Sci. 297, Springer, Berlin, 2004.

*On the structure of energy functions of singularly perturbed bilinear systems*, J. Robust Nonlinear Control, 15 (2005), pp. 601--618.

*A Tichonov-type theorem for singularly perturbed differential inclusion*, Nonlinear Anal., 26 (1996), pp. 1547--1554.

*A note on projection techniques for model order reduction of bilinear systems*, in International Conference on Numerical Analysis and Applied Mathematics, AIP Conf. Proc. 936, 2007, pp. 208--211.

*Singular perturbational model reduction of balanced systems*, IEEE Trans. Automat. Control, AC-27 (1982), pp. 466--468.

*Interpolation Methods for the Model Reduction of Bilinear Systems*, Ph.D. Thesis, Virginia Tech, Blacksburg, VA, 2012.

*Adhesion forces between individual ligand-receptor pairs*, Science, 264 (1994), pp. 415--417.

*Random Perturbations of Dynamical Systems*, Springer, New York, 1998.

*Singular value analysis and balanced realizations for nonlinear systems*, in Model Order Reduction: Theory, Research Aspects and Applications, W.H.A. Schilders, H.A. Vorst, J. van der Rommes, eds., Springer, Berlin, 2008, pp. 251--272.

*Balanced realization and model order reduction for nonlinear systems based on singular value analysis*, SIAM J. Control Optim., 48 (2010), pp. 4591--4623.

*Suboptimization of singularly perturbed control systems*, SIAM J. Control Optim., 30 (1992), pp. 1228--1249.

*Singular perturbation analysis of system order reduction via system balancing*, Proc. Amer. Control Conf., 4 (2000), pp. 2420--2424.

*All optimal Hankel-norm approximations of linear multivariable systems and their $L^\infty$-error bounds*, Internat. J. Control, 39 (1984), pp. 1115--1193.

*Averaging of singularly perturbed systems*, Nonlinear Anal., 28 (1997), pp. 1851--1865.

*Energy functions and algebraic Gramians for bilinear systems*, in Preprints of the 4th IFAC Nonlinear Control Systems Design Symposium, Enschede, Netherlands, 1998, Pergamon, Oxford, 1998, pp. 103--108.

*Observability functions for linear and nonlinear systems*, Systems Control Lett., 38 (1999), pp. 99--113.

*On the nonuniqueness of singular value functions and balanced nonlinear realizations*, Systems Control Lett., 44 (2001), pp. 219--232.

*Balanced truncation of linear second-order systems: A Hamiltonian approach*, Multiscale Model. Simul., 8 (2010), pp. 1348--1367.

*Balanced model reduction of partially-observed Langevin equations: An averaging principle*, Math. Comput. Modelling Dynam., 17 (2011), pp. 463--490.

*Turbulence, Coherent Structures, Dynamical Systems and Symmetry*, Cambridge University Press, Cambridge, UK, 1996.

*Realization theory of bilinear systems*, in Geometric Methods in System Theory, D.Q. Mayne and R.W. Brocken, eds., Reidel, Dordrecht, 1973, pp. 81--130.

*Folding-unfolding transitions in single Titin molecules characterized with laser tweezers*, Science, 276 (1997), pp. 1112--1116.

*A limit theorem for the solution of differential equations with random right-hand sides*, Theory Probab. Appl., 11 (1966), pp. 390--406.

*Systems Biology in Practice: Concepts, Implementation and Application*, Wiley-VCH, Weinheim, Germany, 2005.

*Applications of singular perturbation techniques to control problems*, SIAM Rev., 26 (1984), pp. 501--550.

*Numerical Methods for Stochastic Control Problems in Continuous Time*, Springer, New York, 2001.

*A subspace approach to balanced truncation for model reduction of nonlinear control systems*, Internat. J. Robust Nonlinear Control, 12 (2002), pp. 519--535.

*A model-order reduction method based on Krylov subspaces for MIMO bilinear dynamical systems*, J. Appl. Math. Comput., 25 (2007), pp. 293--304.

*Order reduction of bilinear MIMO dynamical systems using new block Krylov subspaces*, Comput. Math. Appl., 58 (2009), pp. 1093--1102.

*Singular perturbation approximation of balanced systems*, Internat. J. Control, 50 (1989), pp. 1379--1405.

*Principles and applications of control in quantum systems*, Internat. J. Robust Nonlinear Control, 15 (2005), pp. 647--667.

*Energy landscapes of receptor-ligand bonds explored with dynamic force spectroscopy*, Nature, 397 (1999), pp. 50--53.

*Principal component analysis in linear systems: Controllability, observability, and model reduction*, IEEE Trans. Automat. Control, AC-26 (1981), pp. 17--32.

*Stochastic Differential Equations: An Introduction with Applications*. Springer, Berlin, 2003.

*Some probabilistic problems and methods in singular perturbations*, Rocky Mountain J. Math., 6 (1976), pp. 653--674.

*Multiscale Methods: Averaging and Homogenization*, Springer, New York, 2008.

*Projection-based approaches for model reduction of weakly nonlinear, time-varying systems*, IEEE Trans. Comp. Aided Design, 22 (2003), pp. 171--187.

*Domain Decomposition Methods for Partial Differential Equations*, Clarendon Press, Oxford, 1999.

*Reversible unfolding of individual Titin Immunoglobulin domains by AFM*, Science, 276 (1997), pp. 1109--1112.

*The Fokker-Planck Equation: Methods of Solutions and Applications*, Springer, Berlin, 1996.

*Model reduction for fluids using balanced proper orthogonal decomposition*, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), pp. 997--1013.

*Applications of the singular perturbation approximation of balanced systems*, in Proceedings of the Third IEEE Conference on Control Applications, Piscataway, NJ, 1994, pp. 1823--1828.

*Dimension reduction by balanced truncation: Application to light-induced control of open quantum systems*, J. Chem. Phys., 135 (2011), 014112.

*Balancing for nonlinear systems*, Systems Control Lett., 21 (1993), pp. 143--153.

*Positive operators and an inertia theorem*, Numer. Math., 7 (1965), pp. 11--17.

*A new bound of the $L^{2}[0,T]$-induced norm and applications to model reduction*, in Proceedings of the 2002 American Control Conference, IEEE, Piscataway, NJ, 2002, pp. 1180--1185.

*Limits of singularly perturbed control problems with statistical dynamics of fast motions*, SIAM J. Control Optim., 35 (1997), pp. 1--28.

*Topics in Optimal Transportation*, Graduate Studies in Mathematics 58, AMS, Providence, RI, 2003.

*Iterative solution of the Lyapunov matrix equation*, Appl. Math. Lett., 1 (1988), pp. 87--90.

*Model reduction of bilinear systems based on Laguerre series expansion*, J. Franklin Inst., 349 (2012), pp. 1231--1246.

*On singular perturbations for differential inclusions on the infinite time interval*, J. Math. Anal. Appl., 310 (2005), pp. 362--378.

*On $H_{2}$ model reduction of bilinear systems*, Automatica J. IFAC, 38 (2002), pp. 205--216.

*Robust and Optimal Control*, Prentice Hall, NJ, 1998.

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**Submitted**: 1 June 2010

**Accepted**: 8 April 2013

**Published online**: 4 June 2013

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