# American Institute of Mathematical Sciences

June  2014, 1(2): 279-306. doi: 10.3934/jcd.2014.1.279

## Optimal control of multiscale systems using reduced-order models

 1 Institute of Mathematics, Freie Universität Berlin, 14195 Berlin, Germany, Germany, Germany 2 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Received  June 2014 Revised  September 2014 Published  December 2014

We study optimal control of diffusions with slow and fast variables and address a question raised by practitioners: is it possible to first eliminate the fast variables before solving the optimal control problem and then use the optimal control computed from the reduced-order model to control the original, high-dimensional system? The strategy first reduce, then optimize''---rather than first optimize, then reduce''---is motivated by the fact that solving optimal control problems for high-dimensional multiscale systems is numerically challenging and often computationally prohibitive. We state sufficient and necessary conditions, under which the first reduce, then control'' strategy can be employed and discuss when it should be avoided. We further give numerical examples that illustrate the first reduce, then optmize'' approach and discuss possible pitfalls.
Citation: Carsten Hartmann, Juan C. Latorre, Wei Zhang, Grigorios A. Pavliotis. Optimal control of multiscale systems using reduced-order models. Journal of Computational Dynamics, 2014, 1 (2) : 279-306. doi: 10.3934/jcd.2014.1.279
##### References:
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Bismut, Martingales, the malliavin calculus and hypoellipticity under general hörmander's conditions, Z. Wahrsch. Verw. Gebiete, 56 (1981), 469-505. doi: 10.1007/BF00531428.  Google Scholar [11] R. Buckdahn and Y. Hu, Probabilistic approach to homogenizations of systems of quasilinear parabolic PDEs with periodic structures, Nonlinear Analysis, 32 (1998), 609-619. doi: 10.1016/S0362-546X(97)00505-1.  Google Scholar [12] R. Buckdahn, Y. Hu and S. Peng, Probabilistic approach to homogenization of viscosity solutions of parabolic pdes, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 395-411. doi: 10.1007/s000300050010.  Google Scholar [13] Y. Chahlaoui and P. Van Dooren, Benchmark examples for model reduction of linear time invariant dynamical systems, in Dimension Reduction of Large-Scale Systems, vol. 45 of Lect. Notes Comput. Sci. Eng., (2005), 379-392. doi: 10.1007/3-540-27909-1_24.  Google Scholar [14] P. Dai Pra, L. Meneghini and W. Runggaldier, Connections between stochastic control and dynamic games, Mathematics of Control, Signals and Systems, 9 (1996), 303-326. doi: 10.1007/BF01211853.  Google Scholar [15] M. H. Davis and A. R. Norman, Portfolio selection with transaction costs, Math. Oper. Res., 15 (1990), 676-713. doi: 10.1287/moor.15.4.676.  Google Scholar [16] P. Dupuis, K. Spiliopoulos and H. Wang, Importance sampling for multiscale diffusions, Multiscale Model. Simul., 10 (2012), 1-27. doi: 10.1137/110842545.  Google Scholar [17] P. Dupuis and H. Wang, Importance sampling, large deviations, and differential games, Stochastics and Stochastic Reports, 76 (2004), 481-508. doi: 10.1080/10451120410001733845.  Google Scholar [18] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, P. Roy. Soc. Edinb. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631.  Google Scholar [19] W. H. Fleming and W. M. McEneaney, Risk-sensitive control on an infinite time horizon, SIAM J. 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Simul., 8 (2010), 1348-1367. doi: 10.1137/080732717.  Google Scholar [30] C. J. Holland, A minimum principle for the principal eigenvalue for second-order linear elliptic equations with natural boundary conditions, Comm. Pure Appl. Math., 31 (1978), 509-519. doi: 10.1002/cpa.3160310406.  Google Scholar [31] N. Ichihara, A stochastic representation for fully nonlinear PDEs and its application to homogenization, J. Math. Sci. Univ. Tokyo, 12 (2005), 467-492.  Google Scholar [32] P. Imkeller, N. S. Namachchivaya, N. Perkowski and H. C. Yeong, Dimensional reduction in nonlinear filtering: A homogenization approach, Ann. Appl. Probab., 23 (2013), 2290-2326. doi: 10.1214/12-AAP901.  Google Scholar [33] Y. Kabanov and S. Pergamenshchikov, Two-scale Stochastic Systems: Asymptotic Analysis and Control, Springer, Berlin, Heidelberg, Paris, 2003. doi: 10.1007/978-3-662-13242-5.  Google Scholar [34] P. V. Kokotovic, Applications of singular perturbation techniques to control problems, SIAM Review, 26 (1984), 501-550. doi: 10.1137/1026104.  Google Scholar [35] P. Kokotovic, Singular perturbation techniques in control theory, in Singular Perturbations and Asymptotic Analysis in Control Systems (eds. P. V. Kokotovic, A. Bensoussan and G. L. Blankenship), vol. 90 of Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg, (1987), 1-55. doi: 10.1007/BFb0007176.  Google Scholar [36] T. Kurtz and R. H. Stockbridge, Stationary solutions and forward equations for controlled and singular martingale problems, Electron. J. Probab, 6 (2001), 52pp. doi: 10.1214/EJP.v6-90.  Google Scholar [37] H. J. Kushner, Direct averaging and perturbed test function methods for weak convergence, Lect. Notes Contr. Inf., 81 (1986), 412-426. doi: 10.1007/BFb0007118.  Google Scholar [38] H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston, 1990. doi: 10.1007/978-1-4612-4482-0.  Google Scholar [39] H. J. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, New York, 2001. doi: 10.1007/978-1-4613-0007-6.  Google Scholar [40] J. C. Latorre, P. Metzner, C. Hartmann and C. Schütte, A structure-preserving numerical discretization of reversible diffusions, Commun. Math. Sci., 9 (2011), 1051-1072. doi: 10.4310/CMS.2011.v9.n4.a6.  Google Scholar [41] P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of hamilton-jacobi equations,, Preprint., ().   Google Scholar [42] P.-L. Lions and P. E. Souganidis, Correctors for the homogenization of hamilton-jacobi equations in the stationary ergodic setting, Commun. Pure Appl. Math., 56 (2003), 1501-1524. doi: 10.1002/cpa.10101.  Google Scholar [43] P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, in Proceedings of the International Conference on Stochastic Differential Equations 1976, Wiley, New York, (1978), 195-263.  Google Scholar [44] B. Moore, Principal component analysis in linear system: Controllability, observability and model reduction, IEEE Trans. Automat. Control, 26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568.  Google Scholar [45] E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, 1967.  Google Scholar [46] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar [47] B. K. Øksendal, Stochastic Differential Equations: An Introduction With Applications, Springer, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar [48] G. Papanicolaou, A. Bensoussan and J. Lions, Asymptotic Analysis for Periodic Structures, Elsevier, Burlington, MA, 1978. Google Scholar [49] A. Papavasiliou, G. A. Pavliotis and A. M. Stuart, Maximum likelihood drift estimation for multiscale diffusions, Stochastic Process. Appl., 119 (2009), 3173-3210. doi: 10.1016/j.spa.2009.05.003.  Google Scholar [50] J. H. Park, R. B. Sowers and N. S. Namachchivaya, Dimensional reduction in nonlinear filtering, Nonlinearity, 23 (2010), 305-324. doi: 10.1088/0951-7715/23/2/005.  Google Scholar [51] G. A. Pavliotis and A. M. Stuart, Parameter estimation for multiscale diffusions, J. Stat. Phys., 127 (2007), 741-781. doi: 10.1007/s10955-007-9300-6.  Google Scholar [52] G. Pavliotis and A. Stuart, Multiscale Methods: Averaging and Homogenization, Springer, 2008.  Google Scholar [53] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Stochastic modelling and applied probability, Springer, Berlin, Heidelberg, 2009. doi: 10.1007/978-3-540-89500-8.  Google Scholar [54] M. Robin, Long-term average cost control problems for continuous time Markov processes: A survey, Acta Appl. Math., 1 (1983), 281-299. doi: 10.1007/BF00046603.  Google Scholar [55] C. Schütte, S. Winkelmann and C. Hartmann, Optimal control of molecular dynamics using markov state models, Math. Program. Ser. B, 134 (2012), 259-282. doi: 10.1007/s10107-012-0547-6.  Google Scholar [56] H. Stapelfeldt, Laser aligned molecules: Applications in physics and chemistry, Physica Scripta, 2004 (2004), 132-136. doi: 10.1238/Physica.Topical.110a00132.  Google Scholar [57] A. Steinbrecher, Optimal control of robot guided laser material treatment, in Progress in Industrial Mathematics at ECMI 2008 (eds. A. D. Fitt, J. Norbury, H. Ockendon and E. Wilson), Springer Berlin Heidelberg, (2010), 505-511. doi: 10.1007/978-3-642-12110-4_79.  Google Scholar [58] A. Vigodner, Limits of singularly perturbed control problems with statistical dynamics of fast motions, SIAM J. Control Optim., 35 (1997), 1-28. doi: 10.1137/S0363012994264207.  Google Scholar [59] F. Watbled, On singular perturbations for differential inclusions on the infinite interval, J. Math. Anal. Appl., 310 (2005), 362-378. doi: 10.1016/j.jmaa.2005.01.067.  Google Scholar [60] J. Zabczyk, Exit problem and control theory, Syst. Control Lett., 6 (1985), 165-172. doi: 10.1016/0167-6911(85)90036-2.  Google Scholar

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##### References:
 [1] O. Alvarez and M. Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control, SIAM J. Control Optim., 40 (2002), 1159-1188. doi: 10.1137/S0363012900366741.  Google Scholar [2] O. Alvarez, M. Bardi and C. Marchi, Multiscale singular perturbations and homogenization of optimal control problems, in Geometric Control and Nonsmooth Analysis, vol. 76, World Scientific, Singapore, (2008), 1-27. doi: 10.1142/9789812776075_0001.  Google Scholar [3] O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order hamilton-jacobi equations, J. Differential Equations, 243 (2007), 349-387. doi: 10.1016/j.jde.2007.05.027.  Google Scholar [4] A. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898718713.  Google Scholar [5] Z. Artstein, On singularly perturbed ordinary differential equations with measure-valued limits, Math. Bohem., 127 (2002), 139-152.  Google Scholar [6] E. Asplund and T. Klüner, Optimal control of open quantum systems applied to the photochemistry of surfaces, Phys. Rev. Lett., 106 (2011), 140404. doi: 10.1103/PhysRevLett.106.140404.  Google Scholar [7] A. Bensoussan and G. Blankenship, Singular perturbations in stochastic control, in Singular Perturbations and Asymptotic Analysis in Control Systems (eds. P. V. Kokotovic, A. Bensoussan and G. L. Blankenship), vol. 90 of Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg, (1987), 171-260. doi: 10.1007/BFb0007178.  Google Scholar [8] A. Bensoussan, Perturbation Methods in Optimal Control, Gauthiers-Villars, Chichester, 1988.  Google Scholar [9] A. Bensoussan and H. Nagai, An ergodic control problem arising from the principal eigenvalue of an elliptic operator, J. Math. Soc. Japan, 43 (1991), 49-65. doi: 10.2969/jmsj/04310049.  Google Scholar [10] J.-M. Bismut, Martingales, the malliavin calculus and hypoellipticity under general hörmander's conditions, Z. Wahrsch. Verw. Gebiete, 56 (1981), 469-505. doi: 10.1007/BF00531428.  Google Scholar [11] R. Buckdahn and Y. Hu, Probabilistic approach to homogenizations of systems of quasilinear parabolic PDEs with periodic structures, Nonlinear Analysis, 32 (1998), 609-619. doi: 10.1016/S0362-546X(97)00505-1.  Google Scholar [12] R. Buckdahn, Y. Hu and S. Peng, Probabilistic approach to homogenization of viscosity solutions of parabolic pdes, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 395-411. doi: 10.1007/s000300050010.  Google Scholar [13] Y. Chahlaoui and P. Van Dooren, Benchmark examples for model reduction of linear time invariant dynamical systems, in Dimension Reduction of Large-Scale Systems, vol. 45 of Lect. Notes Comput. Sci. Eng., (2005), 379-392. doi: 10.1007/3-540-27909-1_24.  Google Scholar [14] P. Dai Pra, L. Meneghini and W. Runggaldier, Connections between stochastic control and dynamic games, Mathematics of Control, Signals and Systems, 9 (1996), 303-326. doi: 10.1007/BF01211853.  Google Scholar [15] M. H. Davis and A. R. Norman, Portfolio selection with transaction costs, Math. Oper. Res., 15 (1990), 676-713. doi: 10.1287/moor.15.4.676.  Google Scholar [16] P. Dupuis, K. Spiliopoulos and H. Wang, Importance sampling for multiscale diffusions, Multiscale Model. Simul., 10 (2012), 1-27. doi: 10.1137/110842545.  Google Scholar [17] P. Dupuis and H. Wang, Importance sampling, large deviations, and differential games, Stochastics and Stochastic Reports, 76 (2004), 481-508. doi: 10.1080/10451120410001733845.  Google Scholar [18] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, P. Roy. Soc. Edinb. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631.  Google Scholar [19] W. H. Fleming and W. M. McEneaney, Risk-sensitive control on an infinite time horizon, SIAM J. Control Optim., 33 (1995), 1881-1915. doi: 10.1137/S0363012993258720.  Google Scholar [20] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, 2006.  Google Scholar [21] V. Gaitsgory, Suboptimization of singularly perturbed control systems, SIAM J .Control Optim., 30 (1992), 1228-1249. doi: 10.1137/0330065.  Google Scholar [22] Z. Gajic and M.-T. Lim, Optimal Control of Singularly Perturbed Linear Systems and Applications, CRC Press, New York, 2001. doi: 10.1201/9780203907900.  Google Scholar [23] K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their $L^{\infty}$-error bounds, Int. J. Control, 39 (1984), 1115-1193. doi: 10.1080/00207178408933239.  Google Scholar [24] G. Grammel, Averaging of singularly perturbed systems, Nonlinear Analysis, 28 (1997), 1851-1865. doi: 10.1016/S0362-546X(95)00243-O.  Google Scholar [25] S. Gugercin and A. Antoulas, A survey of model reduction by balanced truncation and some new results, Int. J. Control, 77 (2004), 748-766. doi: 10.1080/00207170410001713448.  Google Scholar [26] C. Hartmann, Balanced model reduction of partially observed Langevin equations: An averaging principle, Math. Comput. Model. Dyn. Syst., 17 (2011), 463-490. doi: 10.1080/13873954.2011.576517.  Google Scholar [27] C. Hartmann, B. Schäfer-Bung and A. Zueva, Balanced averaging of bilinear systems with applications to stochastic control, J. Control Optim., 51 (2013), 2356-2378. doi: 10.1137/100796844.  Google Scholar [28] C. Hartmann and C. Schütte, Efficient rare event simulation by optimal nonequilibrium forcing, J. Stat. Mech. Theor. Exp., 2012 (2012), P11004. doi: 10.1088/1742-5468/2012/11/P11004.  Google Scholar [29] C. Hartmann, V. Vulcanov and C. Schütte, Balanced truncation of linear second-order systems: A Hamiltonian approach, Multiscale Model. Simul., 8 (2010), 1348-1367. doi: 10.1137/080732717.  Google Scholar [30] C. J. Holland, A minimum principle for the principal eigenvalue for second-order linear elliptic equations with natural boundary conditions, Comm. Pure Appl. Math., 31 (1978), 509-519. doi: 10.1002/cpa.3160310406.  Google Scholar [31] N. Ichihara, A stochastic representation for fully nonlinear PDEs and its application to homogenization, J. Math. Sci. Univ. Tokyo, 12 (2005), 467-492.  Google Scholar [32] P. Imkeller, N. S. Namachchivaya, N. Perkowski and H. C. Yeong, Dimensional reduction in nonlinear filtering: A homogenization approach, Ann. Appl. Probab., 23 (2013), 2290-2326. doi: 10.1214/12-AAP901.  Google Scholar [33] Y. Kabanov and S. Pergamenshchikov, Two-scale Stochastic Systems: Asymptotic Analysis and Control, Springer, Berlin, Heidelberg, Paris, 2003. doi: 10.1007/978-3-662-13242-5.  Google Scholar [34] P. V. Kokotovic, Applications of singular perturbation techniques to control problems, SIAM Review, 26 (1984), 501-550. doi: 10.1137/1026104.  Google Scholar [35] P. Kokotovic, Singular perturbation techniques in control theory, in Singular Perturbations and Asymptotic Analysis in Control Systems (eds. P. V. Kokotovic, A. Bensoussan and G. L. Blankenship), vol. 90 of Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg, (1987), 1-55. doi: 10.1007/BFb0007176.  Google Scholar [36] T. Kurtz and R. H. Stockbridge, Stationary solutions and forward equations for controlled and singular martingale problems, Electron. J. Probab, 6 (2001), 52pp. doi: 10.1214/EJP.v6-90.  Google Scholar [37] H. J. Kushner, Direct averaging and perturbed test function methods for weak convergence, Lect. Notes Contr. Inf., 81 (1986), 412-426. doi: 10.1007/BFb0007118.  Google Scholar [38] H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston, 1990. doi: 10.1007/978-1-4612-4482-0.  Google Scholar [39] H. J. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, New York, 2001. doi: 10.1007/978-1-4613-0007-6.  Google Scholar [40] J. C. Latorre, P. Metzner, C. Hartmann and C. Schütte, A structure-preserving numerical discretization of reversible diffusions, Commun. Math. Sci., 9 (2011), 1051-1072. doi: 10.4310/CMS.2011.v9.n4.a6.  Google Scholar [41] P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of hamilton-jacobi equations,, Preprint., ().   Google Scholar [42] P.-L. Lions and P. E. Souganidis, Correctors for the homogenization of hamilton-jacobi equations in the stationary ergodic setting, Commun. Pure Appl. Math., 56 (2003), 1501-1524. doi: 10.1002/cpa.10101.  Google Scholar [43] P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, in Proceedings of the International Conference on Stochastic Differential Equations 1976, Wiley, New York, (1978), 195-263.  Google Scholar [44] B. Moore, Principal component analysis in linear system: Controllability, observability and model reduction, IEEE Trans. Automat. Control, 26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568.  Google Scholar [45] E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, 1967.  Google Scholar [46] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar [47] B. K. Øksendal, Stochastic Differential Equations: An Introduction With Applications, Springer, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar [48] G. Papanicolaou, A. Bensoussan and J. Lions, Asymptotic Analysis for Periodic Structures, Elsevier, Burlington, MA, 1978. Google Scholar [49] A. Papavasiliou, G. A. Pavliotis and A. M. Stuart, Maximum likelihood drift estimation for multiscale diffusions, Stochastic Process. Appl., 119 (2009), 3173-3210. doi: 10.1016/j.spa.2009.05.003.  Google Scholar [50] J. H. Park, R. B. Sowers and N. S. Namachchivaya, Dimensional reduction in nonlinear filtering, Nonlinearity, 23 (2010), 305-324. doi: 10.1088/0951-7715/23/2/005.  Google Scholar [51] G. A. Pavliotis and A. M. Stuart, Parameter estimation for multiscale diffusions, J. Stat. Phys., 127 (2007), 741-781. doi: 10.1007/s10955-007-9300-6.  Google Scholar [52] G. Pavliotis and A. Stuart, Multiscale Methods: Averaging and Homogenization, Springer, 2008.  Google Scholar [53] H. 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