A concise introduction to control theory for stochastic partial differential equations. (English) Zbl 1508.93332
Summary: The aim of this notes is to give a concise introduction to control theory for systems governed by stochastic partial differential equations. We shall mainly focus on controllability and optimal control problems for these systems. For the first one, we present results for the exact controllability of stochastic transport equations, null and approximate controllability of stochastic parabolic equations and lack of exact controllability of stochastic hyperbolic equations. For the second one, we first introduce the stochastic linear quadratic optimal control problems and then the Pontryagin type maximum principle for general optimal control problems. It deserves mentioning that, in order to solve some difficult problems in this field, one has to develop new tools, say, the stochastic transposition method introduced in our previous works.
MSC:
| 93E20 | Optimal stochastic control |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 93B05 | Controllability |
| 93B07 | Observability |
| 49N10 | Linear-quadratic optimal control problems |
Keywords:
stochastic partial differential equation; controllability; observability; optimal control; linear quadratic control problem; Pontryagin type maximum principleReferences:
| [1] | N. Agram; B. Øksendal, Stochastic control of memory mean-field processes, Appl. Math. Optim., 79, 181-204 (2019) · Zbl 1411.60081 · doi:10.1007/s00245-017-9425-1 |
| [2] | A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, Springer, New York, 2013. · Zbl 1287.93002 |
| [3] | J.-M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 14, 419-444 (1976) · Zbl 0331.93086 · doi:10.1137/0314028 |
| [4] | A. M. Bruckner, J. B. Bruckner and B. S. Thomson, Real Analysis, Prentice Hall (Pearson), Upper Saddle River, 1997. · Zbl 0872.26001 |
| [5] | R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications. I. Mean Field FBSDEs, Control, and Games, Probability Theory and Stochastic Modelling, 83. Springer, Cham, 2018. · Zbl 1422.91014 |
| [6] | F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, 264. Springer, London, 2013. · Zbl 1277.49001 |
| [7] | D. A. Dawson, Stochastic evolution equations, Math. Biosci., 15, 287-316 (1972) · Zbl 0251.60040 · doi:10.1016/0025-5564(72)90039-9 |
| [8] | F. Dou; Q. Lü, Partial approximate controllability for linear stochastic control systems, SIAM J. Control Optim., 57, 1209-1229 (2019) · Zbl 1410.93019 · doi:10.1137/18M1164640 |
| [9] | F. Dou; Q. Lü, Time-inconsistent linear quadratic optimal control problems for stochastic evolution equations, SIAM J. Control Optim., 58, 485-509 (2020) · Zbl 1440.93264 · doi:10.1137/19M1250339 |
| [10] | K. Du; Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51, 4343-4362 (2013) · Zbl 1285.49018 · doi:10.1137/120882433 |
| [11] | R. Dumitrescu; B. Øksendal; A. Sulem, Stochastic control for mean-field stochastic partial differential equations with jumps, J. Optim. Theory Appl., 176, 559-584 (2018) · Zbl 1391.60156 · doi:10.1007/s10957-018-1243-3 |
| [12] | G. Fabbri, F. Gozzi and A. Świȩch, Stochastic Optimal Control in Infinite Dimension. Dynamic Programming and HJB Equations, Probability Theory and Stochastic Modelling, 82. Springer, Cham, 2017. · Zbl 1379.93001 |
| [13] | H. O. Fattorini; D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43, 272-292 (1971) · Zbl 0231.93003 · doi:10.1007/BF00250466 |
| [14] | H. Frankowska; Q. Lü, First and second order necessary optimality conditions for controlled stochastic evolution equations with control and state constraints, J. Differential Equations, 268, 2949-3015 (2020) · Zbl 1440.93265 · doi:10.1016/j.jde.2019.09.045 |
| [15] | X. Fu; X. Liu, Controllability and observability of some stochastic complex Ginzburg-Landau equations, SIAM J. Control Optim., 55, 1102-1127 (2017) · Zbl 1361.93010 · doi:10.1137/15M1039961 |
| [16] | X. Fu, Q. Lü and X. Zhang, Carleman Estimates for Second Order Partial Differential Operators and Applications, A Unified Approach, Springer, Cham, 2019. · Zbl 1445.35006 |
| [17] | M. Fuhrman; Y. Hu; G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, Appl. Math. Optim., 68, 181-217 (2013) · Zbl 1282.93274 · doi:10.1007/s00245-013-9203-7 |
| [18] | T. Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Math. J., 89, 129-193 (1983) · Zbl 0531.60095 · doi:10.1017/S0027763000020298 |
| [19] | A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. · Zbl 0862.49004 |
| [20] | P. Gao; M. Chen; Y. Li, Observability estimates and null controllability for forward and backward linear stochastic Kuramoto-Sivashinsky equations, SIAM J. Control Optim., 53, 475-500 (2015) · Zbl 1312.93018 · doi:10.1137/130943820 |
| [21] | P. R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, 1950. · Zbl 0040.16802 |
| [22] | C. Hafizoglu; I. Lasiecka; T. Levajković; H. Mena; A. Tuffaha, The stochastic linear quadratic control problem with singular estimates, SIAM J. Control Optim., 55, 595-626 (2017) · Zbl 1358.49029 · doi:10.1137/16M1056183 |
| [23] | H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach, Second edition, Universitext. Springer, New York, 2010. · Zbl 1198.60005 |
| [24] | R. E. Kalman, On the general theory of control systems, Butterworth, London, 1, 481-492 (1961) |
| [25] | M. V. Klibanov; M. Yamamoto, Exact controllability for the time dependent transport equation, SIAM J. Control Optim., 46, 2071-2195 (2007) · Zbl 1149.93009 · doi:10.1137/060652804 |
| [26] | T. Li, Controllability and Observability for Quasilinear Hyperbolic Systems, American Institute of Mathematical Sciences (AIMS), Springfield, MO; Higher Education Press, Beijing, 2010. · Zbl 1198.93003 |
| [27] | J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome \(1\). Contrôlabilité Exacte, Recherches en Mathématiques Appliquées 8, Masson, Paris, 1988. · Zbl 0653.93002 |
| [28] | J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. · Zbl 0227.35001 |
| [29] | J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. II, Springer-Verlag, New York-Heidelberg, 1972. · Zbl 0223.35039 |
| [30] | X. Liu, Controllability of some coupled stochastic parabolic systems with fractional order spatial differential operators by one control in the drift, SIAM J. Control Optim., 52, 836-860 (2014) · Zbl 1292.93031 · doi:10.1137/130926791 |
| [31] | X. Liu; Y. Yu, Carleman estimates of some stochastic degenerate parabolic equations and application, SIAM J. Control Optim., 57, 3527-3552 (2019) · Zbl 1428.93108 · doi:10.1137/18M1221448 |
| [32] | Q. Lü, Some results on the controllability of forward stochastic parabolic equations with control on the drift, J. Funct. Anal., 260, 832-851 (2011) · Zbl 1213.60105 · doi:10.1016/j.jfa.2010.10.018 |
| [33] | Q. Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems, Inverse Problems, 28 (2012), 045008, 18 pp. · Zbl 1236.35213 |
| [34] | Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM J. Control Optim., 51, 121-144 (2013) · Zbl 1262.93021 · doi:10.1137/110830964 |
| [35] | Q. Lü, Observability estimate and state observation problems for stochastic hyperbolic equations, Inverse Problems, 29 (2013), 095011, 22 pp. · Zbl 1292.60066 |
| [36] | Q. Lü, Exact controllability for stochastic Schrödinger equations, J. Differential Equations, 255, 2484-2504 (2013) · Zbl 1371.93040 · doi:10.1016/j.jde.2013.06.021 |
| [37] | Q. Lü, Exact controllability for stochastic transport equations, SIAM J. Control Optim., 52, 397-419 (2014) · Zbl 1292.93034 · doi:10.1137/130910373 |
| [38] | Q. Lü, Stochastic well-posed systems and well-posedness of some stochastic partial differential equations with boundary control and observation, SIAM J. Control Optim., 53, 3457-3482 (2015) · Zbl 1327.93358 · doi:10.1137/151002605 |
| [39] | Q. Lü, Well-posedness of stochastic Riccati equations and closed-loop solvability for stochastic linear quadratic optimal control problems, J. Differential Equations, 267, 180-227 (2019) · Zbl 1411.93197 · doi:10.1016/j.jde.2019.01.008 |
| [40] | Q. Lü, Stochastic linear quadratic optimal control problems for mean-field stochastic evolution equations, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 127, 28 pp. · Zbl 1467.93332 |
| [41] | Q. Lü, T. Wang and X. Zhang, Characterization of optimal feedback for stochastic linear quadratic control problems, Probab. Uncertain. Quant. Risk., 2 (2017), Paper no. 11, 20 pp. · Zbl 1432.93384 |
| [42] | Q. Lü; J. Yong; X. Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals, J. Eur. Math. Soc., 14, 1795-1823 (2012) · Zbl 1323.60076 · doi:10.4171/JEMS/347 |
| [43] | Q. Lü; J. Yong; X. Zhang, Erratum to “Representation of Itô integrals by Lebesgue/ Bochner integrals”, J. Eur. Math. Soc., 20, 259-260 (2018) · Zbl 1410.60052 · doi:10.4171/JEMS/765 |
| [44] | Q. Lü, H. Zhang and X. Zhang, Second order optimality conditions for optimal control problems of stochastic evolution equations, preprint, arXiv: 1811.07337. |
| [45] | Q. Lü; X. Zhang, Well-posedness of backward stochastic differential equations with general filtration, J. Differential Equations, 254, 3200-3227 (2013) · Zbl 1268.60087 · doi:10.1016/j.jde.2013.01.010 |
| [46] | Q. Lü and X. Zhang, General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, SpringerBriefs in Mathematics. Springer, Cham, 2014. · Zbl 1316.49004 |
| [47] | Q. Lü; X. Zhang, Transposition method for backward stochastic evolution equations revisited, and its application, Math. Control Relat. Fields, 5, 529-555 (2015) · Zbl 1316.93126 · doi:10.3934/mcrf.2015.5.529 |
| [48] | Q. Lü; X. Zhang, Global uniqueness for an inverse stochastic hyperbolic problem with three unknowns, Comm. Pure Appl. Math., 68, 948-963 (2015) · Zbl 1315.60080 · doi:10.1002/cpa.21503 |
| [49] | Q. Lü; X. Zhang, Operator-valued backward stochastic Lyapunov equations in infinite dimensions, and its application, Math. Control Relat. Fields, 8, 337-381 (2018) · Zbl 1405.93233 · doi:10.3934/mcrf.2018014 |
| [50] | Q. Lü and X. Zhang, A mini-course on stochastic control, Control and Inverse Problems for Partial Differential Equations, Ser. Contemp. Appl. Math. CAM, Higher Education Press, Beijing, 22 (2019), 171-254. · Zbl 1425.93303 |
| [51] | Q. Lü and X. Zhang, Mathematical Control Theory for Stochastic Partial Differential Equations, Springer-Verlag, in press. · Zbl 1497.93001 |
| [52] | Q. Lü and X. Zhang, Optimal feedback for stochastic linear quadratic control and backward stochastic Riccati equations in infinite dimensions, preprint, arXiv: 1901.00978. · Zbl 07830345 |
| [53] | Q. Lü and X. Zhang, Exact controllability for a refined stochastic wave equation, preprint, arXiv: 1901.06074. |
| [54] | Q. Lü and X. Zhang, Control theory for stochastic distributed parameter systems, an engineering perspective, in submission. |
| [55] | R. S. Manning; J. H. Maddocks; J. D. Kahn, A continuum rod model of sequence-dependent DNA structure, J. Chem. Phys., 105, 5626-5646 (1996) · doi:10.1063/1.472373 |
| [56] | P.-A. Meyer, Probability and Potentials, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966. · Zbl 0138.10401 |
| [57] | R. M. Murray and et al, Control in an Information Rich World. Report of the Panel on Future Directions in Control, Dynamics, and Systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. · Zbl 1101.93003 |
| [58] | E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, N.J. 1967. · Zbl 0165.58502 |
| [59] | A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. · Zbl 0516.47023 |
| [60] | S.-G. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28, 966-979 (1990) · Zbl 0712.93067 · doi:10.1137/0328054 |
| [61] | S.-G. Peng, Backward stochastic differential equation and exact controllability of stochastic control systems, Progr. Natur. Sci. (English Ed.)., 4, 274-284 (1994) |
| [62] | D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies in Appl. Math., 52, 189-211 (1973) · Zbl 0274.35041 · doi:10.1002/sapm1973523189 |
| [63] | D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open problems, SIAM Rev., 20, 639-739 (1978) · Zbl 0397.93001 · doi:10.1137/1020095 |
| [64] | D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300, 383-431 (1987) · Zbl 0623.93040 · doi:10.2307/2000351 |
| [65] | J. Sun and J. Yong, Stochastic Linear-Quadratic Optimal Control Theory: Open-Loop and Closed-Loop Solutions, SpringerBriefs in Mathematics. Springer, Cham, 2019. |
| [66] | M. Tang; Q. Meng; M. Wang, Forward and backward mean-field stochastic partial differential equation and optimal control, Chin. Ann. Math. Ser. B, 40, 515-540 (2019) · Zbl 1447.60123 · doi:10.1007/s11401-019-0149-1 |
| [67] | S. Tang; X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim., 48, 2191-2216 (2009) · Zbl 1203.93027 · doi:10.1137/050641508 |
| [68] | J. van Neerven, \({\gamma}\)-radonifying operators - A survey, The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, Proc. Centre Math. Appl. Austral. Nat. Univ., Austral. Nat. Univ., Canberra, 44 (2010), 1-61. · Zbl 1236.47018 |
| [69] | B. Wu, Q. Chen and Z. Wang, Carleman estimates for a stochastic degenerate parabolic equation and applications to null controllability and an inverse random source problem, Inverse Problems, 36 (2020), 075014, 38 pp. · Zbl 1443.35207 |
| [70] | D. Yang; J. Zhong, Observability inequality of backward stochastic heat equations for measurable sets and its applications, SIAM J. Control Optim., 54, 1157-1175 (2016) · Zbl 1381.93105 · doi:10.1137/15M1033289 |
| [71] | J. Yong, Time-inconsistent optimal control problems, Proceedings of the International Congress of Mathematicians-Seoul 2014, Seoul, Korea, 4 (2014), 947-969. · Zbl 1531.01037 |
| [72] | J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Applications of Mathematics (New York), 43. Springer-Verlag, New York, 1999. · Zbl 0943.93002 |
| [73] | K. Yosida, Functional Analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995. |
| [74] | G. Yuan, Determination of two kinds of sources simultaneously for a stochastic wave equation, Inverse Problems, 31 (2015), 085003, 13 pp. · Zbl 1327.35450 |
| [75] | G. Yuan, Conditional stability in determination of initial data for stochastic parabolic equations, Inverse Problems, 33 (2017), 035014, 26 pp. · Zbl 1372.35374 |
| [76] | E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems, Springer-Verlag, New York, 1986. · Zbl 0583.47050 |
| [77] | X. Zhang, Carleman and observability estimates for stochastic wave equations, SIAM J. Math. Anal., 40, 851-868 (2008) · Zbl 1159.60335 · doi:10.1137/070685786 |
| [78] | X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, Proceedings of the International Congress of Mathematicians, Hindustan Book Agency, New Delhi, 4, 3008-3034 (2010) · Zbl 1198.60005 · doi:10.1007/978-0-387-89488-1 |
| [79] | X. Zhou, Sufficient conditions of optimality for stochastic systems with controllable diffusions, IEEE Trans. Auto. Control, 41, 1176-1179 (1996) · Zbl 0857.93099 · doi:10.1109/9.533678 |
| [80] | E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Eequations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 3, 527-621 (2006) · Zbl 1193.35234 · doi:10.1016/S1874-5717(07)80010-7 |
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