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Lattice structures for attractors I
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 |
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.
doi: 10.1137/S0363012900366741. |
[2] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale singular perturbations and homogenization of optimal control problems,, in Geometric Control and Nonsmooth Analysis, (2008), 1.
doi: 10.1142/9789812776075_0001. |
[3] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order hamilton-jacobi equations,, J. Differential Equations, 243 (2007), 349.
doi: 10.1016/j.jde.2007.05.027. |
[4] |
A. Antoulas, Approximation of Large-Scale Dynamical Systems,, SIAM, (2005).
doi: 10.1137/1.9780898718713. |
[5] |
Z. Artstein, On singularly perturbed ordinary differential equations with measure-valued limits,, Math. Bohem., 127 (2002), 139.
|
[6] |
E. Asplund and T. Klüner, Optimal control of open quantum systems applied to the photochemistry of surfaces,, Phys. Rev. Lett., 106 (2011).
doi: 10.1103/PhysRevLett.106.140404. |
[7] |
A. Bensoussan and G. Blankenship, Singular perturbations in stochastic control,, in Singular Perturbations and Asymptotic Analysis in Control Systems (eds. P. V. Kokotovic, (1987), 171.
doi: 10.1007/BFb0007178. |
[8] |
A. Bensoussan, Perturbation Methods in Optimal Control,, Gauthiers-Villars, (1988).
|
[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.
doi: 10.2969/jmsj/04310049. |
[10] |
J.-M. Bismut, Martingales, the malliavin calculus and hypoellipticity under general hörmander's conditions,, Z. Wahrsch. Verw. Gebiete, 56 (1981), 469.
doi: 10.1007/BF00531428. |
[11] |
R. Buckdahn and Y. Hu, Probabilistic approach to homogenizations of systems of quasilinear parabolic PDEs with periodic structures,, Nonlinear Analysis, 32 (1998), 609.
doi: 10.1016/S0362-546X(97)00505-1. |
[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.
doi: 10.1007/s000300050010. |
[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, (2005), 379.
doi: 10.1007/3-540-27909-1_24. |
[14] |
P. Dai Pra, L. Meneghini and W. Runggaldier, Connections between stochastic control and dynamic games,, Mathematics of Control, 9 (1996), 303.
doi: 10.1007/BF01211853. |
[15] |
M. H. Davis and A. R. Norman, Portfolio selection with transaction costs,, Math. Oper. Res., 15 (1990), 676.
doi: 10.1287/moor.15.4.676. |
[16] |
P. Dupuis, K. Spiliopoulos and H. Wang, Importance sampling for multiscale diffusions,, Multiscale Model. Simul., 10 (2012), 1.
doi: 10.1137/110842545. |
[17] |
P. Dupuis and H. Wang, Importance sampling, large deviations, and differential games,, Stochastics and Stochastic Reports, 76 (2004), 481.
doi: 10.1080/10451120410001733845. |
[18] |
L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE,, P. Roy. Soc. Edinb. A, 111 (1989), 359.
doi: 10.1017/S0308210500018631. |
[19] |
W. H. Fleming and W. M. McEneaney, Risk-sensitive control on an infinite time horizon,, SIAM J. Control Optim., 33 (1995), 1881.
doi: 10.1137/S0363012993258720. |
[20] |
W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Springer, (2006).
|
[21] |
V. Gaitsgory, Suboptimization of singularly perturbed control systems,, SIAM J .Control Optim., 30 (1992), 1228.
doi: 10.1137/0330065. |
[22] |
Z. Gajic and M.-T. Lim, Optimal Control of Singularly Perturbed Linear Systems and Applications,, CRC Press, (2001).
doi: 10.1201/9780203907900. |
[23] |
K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their $L^{\infty}$-error bounds,, Int. J. Control, 39 (1984), 1115.
doi: 10.1080/00207178408933239. |
[24] |
G. Grammel, Averaging of singularly perturbed systems,, Nonlinear Analysis, 28 (1997), 1851.
doi: 10.1016/S0362-546X(95)00243-O. |
[25] |
S. Gugercin and A. Antoulas, A survey of model reduction by balanced truncation and some new results,, Int. J. Control, 77 (2004), 748.
doi: 10.1080/00207170410001713448. |
[26] |
C. Hartmann, Balanced model reduction of partially observed Langevin equations: An averaging principle,, Math. Comput. Model. Dyn. Syst., 17 (2011), 463.
doi: 10.1080/13873954.2011.576517. |
[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.
doi: 10.1137/100796844. |
[28] |
C. Hartmann and C. Schütte, Efficient rare event simulation by optimal nonequilibrium forcing,, J. Stat. Mech. Theor. Exp., 2012 (2012).
doi: 10.1088/1742-5468/2012/11/P11004. |
[29] |
C. Hartmann, V. Vulcanov and C. Schütte, Balanced truncation of linear second-order systems: A Hamiltonian approach,, Multiscale Model. Simul., 8 (2010), 1348.
doi: 10.1137/080732717. |
[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.
doi: 10.1002/cpa.3160310406. |
[31] |
N. Ichihara, A stochastic representation for fully nonlinear PDEs and its application to homogenization,, J. Math. Sci. Univ. Tokyo, 12 (2005), 467.
|
[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.
doi: 10.1214/12-AAP901. |
[33] |
Y. Kabanov and S. Pergamenshchikov, Two-scale Stochastic Systems: Asymptotic Analysis and Control,, Springer, (2003).
doi: 10.1007/978-3-662-13242-5. |
[34] |
P. V. Kokotovic, Applications of singular perturbation techniques to control problems,, SIAM Review, 26 (1984), 501.
doi: 10.1137/1026104. |
[35] |
P. Kokotovic, Singular perturbation techniques in control theory,, in Singular Perturbations and Asymptotic Analysis in Control Systems (eds. P. V. Kokotovic, (1987), 1.
doi: 10.1007/BFb0007176. |
[36] |
T. Kurtz and R. H. Stockbridge, Stationary solutions and forward equations for controlled and singular martingale problems,, Electron. J. Probab, 6 (2001).
doi: 10.1214/EJP.v6-90. |
[37] |
H. J. Kushner, Direct averaging and perturbed test function methods for weak convergence,, Lect. Notes Contr. Inf., 81 (1986), 412.
doi: 10.1007/BFb0007118. |
[38] |
H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems,, Birkhäuser, (1990).
doi: 10.1007/978-1-4612-4482-0. |
[39] |
H. J. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time,, Springer, (2001).
doi: 10.1007/978-1-4613-0007-6. |
[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.
doi: 10.4310/CMS.2011.v9.n4.a6. |
[41] |
P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of hamilton-jacobi equations,, Preprint., (). |
[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.
doi: 10.1002/cpa.10101. |
[43] |
P. Malliavin, Stochastic calculus of variations and hypoelliptic operators,, in Proceedings of the International Conference on Stochastic Differential Equations 1976, (1978), 195.
|
[44] |
B. Moore, Principal component analysis in linear system: Controllability, observability and model reduction,, IEEE Trans. Automat. Control, 26 (1981), 17.
doi: 10.1109/TAC.1981.1102568. |
[45] |
E. Nelson, Dynamical Theories of Brownian Motion,, Princeton University Press, (1967).
|
[46] |
G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608.
doi: 10.1137/0520043. |
[47] |
B. K. Øksendal, Stochastic Differential Equations: An Introduction With Applications,, Springer, (2003).
doi: 10.1007/978-3-642-14394-6. |
[48] |
G. Papanicolaou, A. Bensoussan and J. Lions, Asymptotic Analysis for Periodic Structures,, Elsevier, (1978). |
[49] |
A. Papavasiliou, G. A. Pavliotis and A. M. Stuart, Maximum likelihood drift estimation for multiscale diffusions,, Stochastic Process. Appl., 119 (2009), 3173.
doi: 10.1016/j.spa.2009.05.003. |
[50] |
J. H. Park, R. B. Sowers and N. S. Namachchivaya, Dimensional reduction in nonlinear filtering,, Nonlinearity, 23 (2010), 305.
doi: 10.1088/0951-7715/23/2/005. |
[51] |
G. A. Pavliotis and A. M. Stuart, Parameter estimation for multiscale diffusions,, J. Stat. Phys., 127 (2007), 741.
doi: 10.1007/s10955-007-9300-6. |
[52] |
G. Pavliotis and A. Stuart, Multiscale Methods: Averaging and Homogenization,, Springer, (2008).
|
[53] |
H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications,, Stochastic modelling and applied probability, (2009).
doi: 10.1007/978-3-540-89500-8. |
[54] |
M. Robin, Long-term average cost control problems for continuous time Markov processes: A survey,, Acta Appl. Math., 1 (1983), 281.
doi: 10.1007/BF00046603. |
[55] |
C. Schütte, S. Winkelmann and C. Hartmann, Optimal control of molecular dynamics using markov state models,, Math. Program. Ser. B, 134 (2012), 259.
doi: 10.1007/s10107-012-0547-6. |
[56] |
H. Stapelfeldt, Laser aligned molecules: Applications in physics and chemistry,, Physica Scripta, 2004 (2004), 132.
doi: 10.1238/Physica.Topical.110a00132. |
[57] |
A. Steinbrecher, Optimal control of robot guided laser material treatment,, in Progress in Industrial Mathematics at ECMI 2008 (eds. A. D. Fitt, (2010), 505.
doi: 10.1007/978-3-642-12110-4_79. |
[58] |
A. Vigodner, Limits of singularly perturbed control problems with statistical dynamics of fast motions,, SIAM J. Control Optim., 35 (1997), 1.
doi: 10.1137/S0363012994264207. |
[59] |
F. Watbled, On singular perturbations for differential inclusions on the infinite interval,, J. Math. Anal. Appl., 310 (2005), 362.
doi: 10.1016/j.jmaa.2005.01.067. |
[60] |
J. Zabczyk, Exit problem and control theory,, Syst. Control Lett., 6 (1985), 165.
doi: 10.1016/0167-6911(85)90036-2. |
show all references
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.
doi: 10.1137/S0363012900366741. |
[2] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale singular perturbations and homogenization of optimal control problems,, in Geometric Control and Nonsmooth Analysis, (2008), 1.
doi: 10.1142/9789812776075_0001. |
[3] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order hamilton-jacobi equations,, J. Differential Equations, 243 (2007), 349.
doi: 10.1016/j.jde.2007.05.027. |
[4] |
A. Antoulas, Approximation of Large-Scale Dynamical Systems,, SIAM, (2005).
doi: 10.1137/1.9780898718713. |
[5] |
Z. Artstein, On singularly perturbed ordinary differential equations with measure-valued limits,, Math. Bohem., 127 (2002), 139.
|
[6] |
E. Asplund and T. Klüner, Optimal control of open quantum systems applied to the photochemistry of surfaces,, Phys. Rev. Lett., 106 (2011).
doi: 10.1103/PhysRevLett.106.140404. |
[7] |
A. Bensoussan and G. Blankenship, Singular perturbations in stochastic control,, in Singular Perturbations and Asymptotic Analysis in Control Systems (eds. P. V. Kokotovic, (1987), 171.
doi: 10.1007/BFb0007178. |
[8] |
A. Bensoussan, Perturbation Methods in Optimal Control,, Gauthiers-Villars, (1988).
|
[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.
doi: 10.2969/jmsj/04310049. |
[10] |
J.-M. Bismut, Martingales, the malliavin calculus and hypoellipticity under general hörmander's conditions,, Z. Wahrsch. Verw. Gebiete, 56 (1981), 469.
doi: 10.1007/BF00531428. |
[11] |
R. Buckdahn and Y. Hu, Probabilistic approach to homogenizations of systems of quasilinear parabolic PDEs with periodic structures,, Nonlinear Analysis, 32 (1998), 609.
doi: 10.1016/S0362-546X(97)00505-1. |
[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.
doi: 10.1007/s000300050010. |
[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, (2005), 379.
doi: 10.1007/3-540-27909-1_24. |
[14] |
P. Dai Pra, L. Meneghini and W. Runggaldier, Connections between stochastic control and dynamic games,, Mathematics of Control, 9 (1996), 303.
doi: 10.1007/BF01211853. |
[15] |
M. H. Davis and A. R. Norman, Portfolio selection with transaction costs,, Math. Oper. Res., 15 (1990), 676.
doi: 10.1287/moor.15.4.676. |
[16] |
P. Dupuis, K. Spiliopoulos and H. Wang, Importance sampling for multiscale diffusions,, Multiscale Model. Simul., 10 (2012), 1.
doi: 10.1137/110842545. |
[17] |
P. Dupuis and H. Wang, Importance sampling, large deviations, and differential games,, Stochastics and Stochastic Reports, 76 (2004), 481.
doi: 10.1080/10451120410001733845. |
[18] |
L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE,, P. Roy. Soc. Edinb. A, 111 (1989), 359.
doi: 10.1017/S0308210500018631. |
[19] |
W. H. Fleming and W. M. McEneaney, Risk-sensitive control on an infinite time horizon,, SIAM J. Control Optim., 33 (1995), 1881.
doi: 10.1137/S0363012993258720. |
[20] |
W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Springer, (2006).
|
[21] |
V. Gaitsgory, Suboptimization of singularly perturbed control systems,, SIAM J .Control Optim., 30 (1992), 1228.
doi: 10.1137/0330065. |
[22] |
Z. Gajic and M.-T. Lim, Optimal Control of Singularly Perturbed Linear Systems and Applications,, CRC Press, (2001).
doi: 10.1201/9780203907900. |
[23] |
K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their $L^{\infty}$-error bounds,, Int. J. Control, 39 (1984), 1115.
doi: 10.1080/00207178408933239. |
[24] |
G. Grammel, Averaging of singularly perturbed systems,, Nonlinear Analysis, 28 (1997), 1851.
doi: 10.1016/S0362-546X(95)00243-O. |
[25] |
S. Gugercin and A. Antoulas, A survey of model reduction by balanced truncation and some new results,, Int. J. Control, 77 (2004), 748.
doi: 10.1080/00207170410001713448. |
[26] |
C. Hartmann, Balanced model reduction of partially observed Langevin equations: An averaging principle,, Math. Comput. Model. Dyn. Syst., 17 (2011), 463.
doi: 10.1080/13873954.2011.576517. |
[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.
doi: 10.1137/100796844. |
[28] |
C. Hartmann and C. Schütte, Efficient rare event simulation by optimal nonequilibrium forcing,, J. Stat. Mech. Theor. Exp., 2012 (2012).
doi: 10.1088/1742-5468/2012/11/P11004. |
[29] |
C. Hartmann, V. Vulcanov and C. Schütte, Balanced truncation of linear second-order systems: A Hamiltonian approach,, Multiscale Model. Simul., 8 (2010), 1348.
doi: 10.1137/080732717. |
[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.
doi: 10.1002/cpa.3160310406. |
[31] |
N. Ichihara, A stochastic representation for fully nonlinear PDEs and its application to homogenization,, J. Math. Sci. Univ. Tokyo, 12 (2005), 467.
|
[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.
doi: 10.1214/12-AAP901. |
[33] |
Y. Kabanov and S. Pergamenshchikov, Two-scale Stochastic Systems: Asymptotic Analysis and Control,, Springer, (2003).
doi: 10.1007/978-3-662-13242-5. |
[34] |
P. V. Kokotovic, Applications of singular perturbation techniques to control problems,, SIAM Review, 26 (1984), 501.
doi: 10.1137/1026104. |
[35] |
P. Kokotovic, Singular perturbation techniques in control theory,, in Singular Perturbations and Asymptotic Analysis in Control Systems (eds. P. V. Kokotovic, (1987), 1.
doi: 10.1007/BFb0007176. |
[36] |
T. Kurtz and R. H. Stockbridge, Stationary solutions and forward equations for controlled and singular martingale problems,, Electron. J. Probab, 6 (2001).
doi: 10.1214/EJP.v6-90. |
[37] |
H. J. Kushner, Direct averaging and perturbed test function methods for weak convergence,, Lect. Notes Contr. Inf., 81 (1986), 412.
doi: 10.1007/BFb0007118. |
[38] |
H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems,, Birkhäuser, (1990).
doi: 10.1007/978-1-4612-4482-0. |
[39] |
H. J. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time,, Springer, (2001).
doi: 10.1007/978-1-4613-0007-6. |
[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.
doi: 10.4310/CMS.2011.v9.n4.a6. |
[41] |
P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of hamilton-jacobi equations,, Preprint., (). |
[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.
doi: 10.1002/cpa.10101. |
[43] |
P. Malliavin, Stochastic calculus of variations and hypoelliptic operators,, in Proceedings of the International Conference on Stochastic Differential Equations 1976, (1978), 195.
|
[44] |
B. Moore, Principal component analysis in linear system: Controllability, observability and model reduction,, IEEE Trans. Automat. Control, 26 (1981), 17.
doi: 10.1109/TAC.1981.1102568. |
[45] |
E. Nelson, Dynamical Theories of Brownian Motion,, Princeton University Press, (1967).
|
[46] |
G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608.
doi: 10.1137/0520043. |
[47] |
B. K. Øksendal, Stochastic Differential Equations: An Introduction With Applications,, Springer, (2003).
doi: 10.1007/978-3-642-14394-6. |
[48] |
G. Papanicolaou, A. Bensoussan and J. Lions, Asymptotic Analysis for Periodic Structures,, Elsevier, (1978). |
[49] |
A. Papavasiliou, G. A. Pavliotis and A. M. Stuart, Maximum likelihood drift estimation for multiscale diffusions,, Stochastic Process. Appl., 119 (2009), 3173.
doi: 10.1016/j.spa.2009.05.003. |
[50] |
J. H. Park, R. B. Sowers and N. S. Namachchivaya, Dimensional reduction in nonlinear filtering,, Nonlinearity, 23 (2010), 305.
doi: 10.1088/0951-7715/23/2/005. |
[51] |
G. A. Pavliotis and A. M. Stuart, Parameter estimation for multiscale diffusions,, J. Stat. Phys., 127 (2007), 741.
doi: 10.1007/s10955-007-9300-6. |
[52] |
G. Pavliotis and A. Stuart, Multiscale Methods: Averaging and Homogenization,, Springer, (2008).
|
[53] |
H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications,, Stochastic modelling and applied probability, (2009).
doi: 10.1007/978-3-540-89500-8. |
[54] |
M. Robin, Long-term average cost control problems for continuous time Markov processes: A survey,, Acta Appl. Math., 1 (1983), 281.
doi: 10.1007/BF00046603. |
[55] |
C. Schütte, S. Winkelmann and C. Hartmann, Optimal control of molecular dynamics using markov state models,, Math. Program. Ser. B, 134 (2012), 259.
doi: 10.1007/s10107-012-0547-6. |
[56] |
H. Stapelfeldt, Laser aligned molecules: Applications in physics and chemistry,, Physica Scripta, 2004 (2004), 132.
doi: 10.1238/Physica.Topical.110a00132. |
[57] |
A. Steinbrecher, Optimal control of robot guided laser material treatment,, in Progress in Industrial Mathematics at ECMI 2008 (eds. A. D. Fitt, (2010), 505.
doi: 10.1007/978-3-642-12110-4_79. |
[58] |
A. Vigodner, Limits of singularly perturbed control problems with statistical dynamics of fast motions,, SIAM J. Control Optim., 35 (1997), 1.
doi: 10.1137/S0363012994264207. |
[59] |
F. Watbled, On singular perturbations for differential inclusions on the infinite interval,, J. Math. Anal. Appl., 310 (2005), 362.
doi: 10.1016/j.jmaa.2005.01.067. |
[60] |
J. Zabczyk, Exit problem and control theory,, Syst. Control Lett., 6 (1985), 165.
doi: 10.1016/0167-6911(85)90036-2. |
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