메뉴 건너뛰기




Volumn 1, Issue 2, 2014, Pages 279-306

Optimal control of multiscale systems using reduced-order models

Author keywords

Dynamic programming; Homogenization; Logarithmic transformation; Model order reduction; Singular perturbations; Stochastic control

Indexed keywords


EID: 84979965404     PISSN: None     EISSN: 21582505     Source Type: Journal    
DOI: 10.3934/jcd.2014.1.279     Document Type: Article
Times cited : (24)

References (60)
  • 1
    • 0036338482 scopus 로고    scopus 로고
    • Viscosity solutions methods for singular perturbations in deterministic and stochastic control
    • O. Alvarez and M. Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control, SIAM J. Control Optim., 40 (2002), 1159-1188
    • (2002) SIAM J. Control Optim , vol.40 , pp. 1159-1188
    • Alvarez, O.1    Bardi, M.2
  • 3
    • 36049037276 scopus 로고    scopus 로고
    • Multiscale problems and homogenization for secondorder hamilton-jacobi equations
    • O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for secondorder hamilton-jacobi equations, J. Differential Equations, 243 (2007), 349-387
    • (2007) J. Differential Equations , vol.243 , pp. 349-387
    • Alvarez, O.1    Bardi, M.2    Marchi, C.3
  • 5
    • 9644282182 scopus 로고    scopus 로고
    • On singularly perturbed ordinary differential equations with measure-valued limits
    • Z. Artstein, On singularly perturbed ordinary differential equations with measure-valued limits, Math. Bohem., 127 (2002), 139-152
    • (2002) Math. Bohem , vol.127 , pp. 139-152
    • Artstein, Z.1
  • 6
    • 79960636922 scopus 로고    scopus 로고
    • Optimal control of open quantum systems applied to the photochemistry of surfaces
    • E. Asplund and T. Klüner, Optimal control of open quantum systems applied to the photochemistry of surfaces, Phys. Rev. Lett., 106 (2011), 140404
    • (2011) Phys. Rev. Lett , vol.106
    • Asplund, E.1    Klüner, T.2
  • 7
    • 0001036732 scopus 로고
    • Singular perturbations in stochastic control
    • (eds. P. V. Kokotovic, A. Bensoussan and G. L. Blankenship), vol. 90 of Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg
    • 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
    • (1987) Singular Perturbations and Asymptotic Analysis in Control Systems , pp. 171-260
    • Bensoussan, A.1    Blankenship, G.2
  • 9
    • 84972565055 scopus 로고
    • An ergodic control problem arising from the principal eigenvalue of an elliptic operator
    • 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
    • (1991) J. Math. Soc. Japan , vol.43 , pp. 49-65
    • Bensoussan, A.1    Nagai, H.2
  • 10
    • 34250245856 scopus 로고
    • Martingales, the malliavin calculus and hypoellipticity under general hörmander's conditions
    • J.-M. Bismut, Martingales, the malliavin calculus and hypoellipticity under general hörmander's conditions, Z. Wahrsch. Verw. Gebiete, 56 (1981), 469-505
    • (1981) Z. Wahrsch. Verw. Gebiete , vol.56 , pp. 469-505
    • Bismut, J.-M.1
  • 11
    • 0032061556 scopus 로고    scopus 로고
    • Probabilistic approach to homogenizations of systems of quasilinear parabolic PDEs with periodic structures
    • R. Buckdahn and Y. Hu, Probabilistic approach to homogenizations of systems of quasilinear parabolic PDEs with periodic structures, Nonlinear Analysis, 32 (1998), 609-619
    • (1998) Nonlinear Analysis , vol.32 , pp. 609-619
    • Buckdahn, R.1    Hu, Y.2
  • 12
    • 0041128049 scopus 로고    scopus 로고
    • Probabilistic approach to homogenization of viscosity solutions of parabolic pdes
    • 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
    • (1999) NoDEA Nonlinear Differential Equations Appl , vol.6 , pp. 395-411
    • Buckdahn, R.1    Hu, Y.2    Peng, S.3
  • 13
    • 33646471733 scopus 로고    scopus 로고
    • Benchmark examples for model reduction of linear time invariant dynamical systems, Dimension Reduction of Large-Scale Systems
    • 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
    • (2005) vol. 45 of Lect. Notes Comput. Sci. Eng , pp. 379-392
    • Chahlaoui, Y.1    Van Dooren, P.2
  • 15
    • 0000637746 scopus 로고
    • Portfolio selection with transaction costs
    • M. H. Davis and A. R. Norman, Portfolio selection with transaction costs, Math. Oper. Res., 15 (1990), 676-713
    • (1990) Math. Oper. Res , vol.15 , pp. 676-713
    • Davis, M.H.1    Norman, A.R.2
  • 17
    • 22244431981 scopus 로고    scopus 로고
    • Importance sampling. large deviations. and differential games
    • P. Dupuis and H. Wang, Importance sampling, large deviations, and differential games, Stochastics and Stochastic Reports, 76 (2004), 481-508
    • (2004) Stochastics and Stochastic Reports , vol.76 , pp. 481-508
    • Dupuis, P.1    Wang, H.2
  • 18
    • 84971110219 scopus 로고
    • The perturbed test function method for viscosity solutions of nonlinear PDE
    • L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, P. Roy. Soc. Edinb. A, 111 (1989), 359-375
    • (1989) P. Roy. Soc. Edinb. A , vol.111 , pp. 359-375
    • Evans, L.C.1
  • 19
    • 0001234347 scopus 로고
    • Risk-sensitive control on an infinite time horizon
    • W. H. Fleming and W. M. McEneaney, Risk-sensitive control on an infinite time horizon, SIAM J. Control Optim., 33 (1995), 1881-1915
    • (1995) SIAM J. Control Optim , vol.33 , pp. 1881-1915
    • Fleming, W.H.1    McEneaney, W.M.2
  • 21
    • 0026925030 scopus 로고
    • Suboptimization of singularly perturbed control systems
    • V. Gaitsgory, Suboptimization of singularly perturbed control systems, SIAM J .Control Optim., 30 (1992), 1228-1249
    • (1992) SIAM J .Control Optim , vol.30 , pp. 1228-1249
    • Gaitsgory, V.1
  • 23
    • 0021441691 scopus 로고
    • All optimal Hankel-norm approximations of linear multivariable systems and their L1-error bounds
    • K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L1-error bounds, Int. J. Control, 39 (1984), 1115-1193
    • (1984) Int. J. Control , vol.39 , pp. 1115-1193
    • Glover, K.1
  • 24
    • 0031168753 scopus 로고    scopus 로고
    • Averaging of singularly perturbed systems
    • G. Grammel, Averaging of singularly perturbed systems, Nonlinear Analysis, 28 (1997), 1851-1865
    • (1997) Nonlinear Analysis , vol.28 , pp. 1851-1865
    • Grammel, G.1
  • 25
    • 4544354870 scopus 로고    scopus 로고
    • A survey of model reduction by balanced truncation and some new results
    • S. Gugercin and A. Antoulas, A survey of model reduction by balanced truncation and some new results, Int. J. Control, 77 (2004), 748-766
    • (2004) Int. J. Control , vol.77 , pp. 748-766
    • Gugercin, S.1    Antoulas, A.2
  • 26
    • 80052883821 scopus 로고    scopus 로고
    • Balanced model reduction of partially observed Langevin equations: An averaging principle
    • C. Hartmann, Balanced model reduction of partially observed Langevin equations: An averaging principle, Math. Comput. Model. Dyn. Syst., 17 (2011), 463-490
    • (2011) Math. Comput. Model. Dyn. Syst , vol.17 , pp. 463-490
    • Hartmann, C.1
  • 27
    • 84884835715 scopus 로고    scopus 로고
    • Balanced averaging of bilinear systems with applications to stochastic control
    • 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
    • (2013) J. Control Optim , vol.51 , pp. 2356-2378
    • Hartmann, C.1    Schäfer-Bung, B.2    Zueva, A.3
  • 28
    • 84868569168 scopus 로고    scopus 로고
    • Efficient rare event simulation by optimal nonequilibrium forcing
    • C. Hartmann and C. Schütte, Efficient rare event simulation by optimal nonequilibrium forcing, J. Stat. Mech. Theor. Exp., 2012 (2012), P11004
    • (2012) J. Stat. Mech. Theor. Exp , vol.2012
    • Hartmann, C.1    Schütte, C.2
  • 29
    • 77956763488 scopus 로고    scopus 로고
    • Balanced truncation of linear second-order systems: A Hamiltonian approach
    • C. Hartmann, V. Vulcanov and C. Schütte, Balanced truncation of linear second-order systems: A Hamiltonian approach, Multiscale Model. Simul., 8 (2010), 1348-1367
    • (2010) Multiscale Model. Simul , vol.8 , pp. 1348-1367
    • Hartmann, C.1    Vulcanov, V.2    Schütte, C.3
  • 30
    • 84980156723 scopus 로고
    • A minimum principle for the principal eigenvalue for second-order linear elliptic equations with natural boundary conditions
    • 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
    • (1978) Comm. Pure Appl. Math , vol.31 , pp. 509-519
    • Holland, C.J.1
  • 31
    • 63549106706 scopus 로고    scopus 로고
    • A stochastic representation for fully nonlinear PDEs and its application to homogenization
    • N. Ichihara, A stochastic representation for fully nonlinear PDEs and its application to homogenization, J. Math. Sci. Univ. Tokyo, 12 (2005), 467-492
    • (2005) J. Math. Sci. Univ. Tokyo , vol.12 , pp. 467-492
    • Ichihara, N.1
  • 32
    • 84888334025 scopus 로고    scopus 로고
    • Dimensional reduction in nonlinear filtering: A homogenization approach
    • 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
    • (2013) Ann. Appl. Probab , vol.23 , pp. 2290-2326
    • Imkeller, P.1    Namachchivaya, N.S.2    Perkowski, N.3    Yeong, H.C.4
  • 34
    • 0021501125 scopus 로고
    • Applications of singular perturbation techniques to control problems
    • P. V. Kokotovic, Applications of singular perturbation techniques to control problems, SIAM Review, 26 (1984), 501-550
    • (1984) SIAM Review , vol.26 , pp. 501-550
    • Kokotovic, P.V.1
  • 35
    • 0043268633 scopus 로고
    • Singular perturbation techniques in control theory
    • (eds. P. V. Kokotovic, A. Bensoussan and G. L. Blankenship), vol. 90 of Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg
    • 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
    • (1987) Singular Perturbations and Asymptotic Analysis in Control Systems , pp. 1-55
    • Kokotovic, P.1
  • 36
    • 1942498781 scopus 로고    scopus 로고
    • Stationary solutions and forward equations for controlled and singular martingale problems
    • T. Kurtz and R. H. Stockbridge, Stationary solutions and forward equations for controlled and singular martingale problems, Electron. J. Probab, 6 (2001), 52pp
    • (2001) Electron. J. Probab , vol.6 , pp. 52
    • Kurtz, T.1    Stockbridge, R.H.2
  • 37
    • 85035026377 scopus 로고
    • Direct averaging and perturbed test function methods for weak convergence
    • H. J. Kushner, Direct averaging and perturbed test function methods for weak convergence, Lect. Notes Contr. Inf., 81 (1986), 412-426
    • (1986) Lect. Notes Contr. Inf , vol.81 , pp. 412-426
    • Kushner, H.J.1
  • 40
    • 79961242531 scopus 로고    scopus 로고
    • A structure-preserving numerical discretization of reversible diffusions
    • 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
    • (2011) Commun. Math. Sci , vol.9 , pp. 1051-1072
    • Latorre, J.C.1    Metzner, P.2    Hartmann, C.3    Schütte, C.4
  • 42
    • 0041357028 scopus 로고    scopus 로고
    • Correctors for the homogenization of hamilton-jacobi equations in the stationary ergodic setting
    • 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
    • (2003) Commun. Pure Appl. Math , vol.56 , pp. 1501-1524
    • Lions, P.-L.1    Souganidis, P.E.2
  • 44
    • 0019533482 scopus 로고
    • Principal component analysis in linear system: Controllability. observability and model reduction
    • B. Moore, Principal component analysis in linear system: Controllability, observability and model reduction, IEEE Trans. Automat. Control, 26 (1981), 17-32
    • (1981) IEEE Trans. Automat. Control , vol.26 , pp. 17-32
    • Moore, B.1
  • 46
    • 0000315621 scopus 로고
    • A general convergence result for a functional related to the theory of homogenization
    • G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623
    • (1989) SIAM J. Math. Anal , vol.20 , pp. 608-623
    • Nguetseng, G.1
  • 49
    • 69849095791 scopus 로고    scopus 로고
    • Maximum likelihood drift estimation for multiscale diffusions
    • A. Papavasiliou, G. A. Pavliotis and A. M. Stuart, Maximum likelihood drift estimation for multiscale diffusions, Stochastic Process. Appl., 119 (2009), 3173-3210
    • (2009) Stochastic Process. Appl , vol.119 , pp. 3173-3210
    • Papavasiliou, A.1    Pavliotis, G.A.2    Stuart, A.M.3
  • 50
    • 74849090136 scopus 로고    scopus 로고
    • Dimensional reduction in nonlinear filtering
    • J. H. Park, R. B. Sowers and N. S. Namachchivaya, Dimensional reduction in nonlinear filtering, Nonlinearity, 23 (2010), 305-324
    • (2010) Nonlinearity , vol.23 , pp. 305-324
    • Park, J.H.1    Sowers, R.B.2    Namachchivaya, N.S.3
  • 51
    • 34247359516 scopus 로고    scopus 로고
    • Parameter estimation for multiscale diffusions
    • G. A. Pavliotis and A. M. Stuart, Parameter estimation for multiscale diffusions, J. Stat. Phys., 127 (2007), 741-781
    • (2007) J. Stat. Phys , vol.127 , pp. 741-781
    • Pavliotis, G.A.1    Stuart, A.M.2
  • 54
    • 0042361167 scopus 로고
    • Long-term average cost control problems for continuous time Markov processes: A survey
    • M. Robin, Long-term average cost control problems for continuous time Markov processes: A survey, Acta Appl. Math., 1 (1983), 281-299
    • (1983) Acta Appl. Math , vol.1 , pp. 281-299
    • Robin, M.1
  • 55
    • 84865692627 scopus 로고    scopus 로고
    • Optimal control of molecular dynamics using markov state models
    • C. Schütte, S. Winkelmann and C. Hartmann, Optimal control of molecular dynamics using markov state models, Math. Program. Ser. B, 134 (2012), 259-282
    • (2012) Math. Program. Ser. B , vol.134 , pp. 259-282
    • Schütte, C.1    Winkelmann, S.2    Hartmann, C.3
  • 56
    • 38349172645 scopus 로고    scopus 로고
    • Laser aligned molecules: Applications in physics and chemistry
    • H. Stapelfeldt, Laser aligned molecules: Applications in physics and chemistry, Physica Scripta, 2004 (2004), 132-136
    • (2004) Physica Scripta , vol.2004 , pp. 132-136
    • Stapelfeldt, H.1
  • 57
    • 84919623957 scopus 로고    scopus 로고
    • Optimal control of robot guided laser material treatment
    • (eds. A. D. Fitt, J. Norbury, H. Ockendon and E. Wilson), Springer Berlin Heidelberg, (2010)
    • A. Steinbrecher, Optimal control of robot guided laser material treatment, in Progress in In-dustrial Mathematics at ECMI 2008 (eds. A. D. Fitt, J. Norbury, H. Ockendon and E. Wilson), Springer Berlin Heidelberg, (2010), 505-511
    • (2008) Progress in In-dustrial Mathematics at ECMI , pp. 505-511
    • Steinbrecher, A.1
  • 58
    • 0030826053 scopus 로고    scopus 로고
    • Limits of singularly perturbed control problems with statistical dynamics of fast motions
    • A. Vigodner, Limits of singularly perturbed control problems with statistical dynamics of fast motions, SIAM J. Control Optim., 35 (1997), 1-28
    • (1997) SIAM J. Control Optim , vol.35 , pp. 1-28
    • Vigodner, A.1
  • 59
    • 24044525604 scopus 로고    scopus 로고
    • On singular perturbations for differential inclusions on the infinite interval
    • F. Watbled, On singular perturbations for differential inclusions on the infinite interval, J. Math. Anal. Appl., 310 (2005), 362-378
    • (2005) J. Math. Anal. Appl , vol.310 , pp. 362-378
    • Watbled, F.1
  • 60
    • 0022111472 scopus 로고
    • Exit problem and control theory
    • J. Zabczyk, Exit problem and control theory, Syst. Control Lett., 6 (1985), 165-172
    • (1985) Syst. Control Lett , vol.6 , pp. 165-172
    • Zabczyk, J.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.