Optimal control and stabilization for linear mean-field system with indefinite quadratic cost functional. (English) Zbl 07892491
Summary: In this paper, we consider an optimal control and stabilization problem of linear mean-field (MF) system, where the quadratic cost functional is allowed to be indefinite. Inspired by the equivalent cost functional method, we introduce a subset, which helps us to investigate the convergence property of generalized differential Riccati equations arising in indefinite mean-field linear quadratic (MF-LQ) problems with finite horizon. A coupled generalized algebraic Riccati equation (GARE) is thus obtained. More importantly, the solution pair of corresponding GARE can be decomposed into a solution pair of a coupled standard algebraic Riccati equation (SARE) and a matrix pair in the subset we introduce. Thus, an equivalent relationship is established between GARE and SARE. With this equivalence, we derive necessary and sufficient conditions to stabilize linear MF-system with indefinite weighting matrices in mean-square sense.
© 2023 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
© 2023 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
MSC:
| 93-XX | Systems theory; control |
Keywords:
convergence; generalized algebraic Riccati equation; indefinite weighting matrices; mean-field system; stabilizationReferences:
| [1] | R.Buckdahn, J.Li, and J.Ma, A stochastic maximum principle for general mean‐field systems, Appl. Math. Optim.74 (2016), no. 3, 507-534. · Zbl 1359.93528 |
| [2] | B.Djehiche, H.Tembine, and R.Tempone, A stochastic maximum principle for risk‐sensitive mean‐field type control, IEEE Trans. Autom. Control60 (2015), no. 10, 2640-2649. · Zbl 1360.93772 |
| [3] | G.Wang and Z.Wu, A maximum principle for mean‐field stochastic control system with noisy observation, Automatica137 (2022), 110135. · Zbl 1482.93712 |
| [4] | G.Wang, C.Zhang, and W.Zhang, Stochastic maximum principle for mean‐field type optimal control under partial information, IEEE Trans. Autom. Control59 (2013), no. 2, 522-528. · Zbl 1360.93788 |
| [5] | J.Yong, Linear‐quadratic optimal control problems for mean‐field stochastic differential equations, SIAM J. Control Optim.51 (2013), no. 4, 2809-2838. · Zbl 1275.49060 |
| [6] | J.Huang, X.Li, and J.Yong, A linear‐quadratic optimal control problem for mean‐field stochastic differential equations in infinite horizon, Math. Control Related Fields5 (2015), no. 1, 97-139. · Zbl 1326.49051 |
| [7] | M.Tang and Q.Meng, Linear‐quadratic optimal control problems for mean‐field stochastic differential equations with jumps, Asian J. Control21 (2019), no. 2, 809-823. · Zbl 1423.49034 |
| [8] | R.Elliott, X.Li, and Y.Ni, Discrete time mean‐field stochastic linear‐quadratic optimal control problems, Automatica49 (2013), no. 11, 3222-3233. · Zbl 1358.93189 |
| [9] | Y.Ni, R.Elliott, and X.Li, Discrete‐time mean‐field stochastic linear‐quadratic optimal control problems, ii: Infinite horizon case, Automatica57 (2015), no. 3, 65-77. · Zbl 1330.93244 |
| [10] | S.Chen, X.Li, and X.Zhou, Stochastic linear quadratic regulators with indefinite control weight costs, SIAM J. Control Optim.36 (1998), no. 5, 1685-1702. · Zbl 0916.93084 |
| [11] | M. A.Rami, X.Chen, J. B.Moore, and X.Zhou, Solvability and asymptotic behavior of generalized riccati equations arising in indefinite stochastic lq controls, IEEE Trans. Autom. Control46 (2001), no. 3, 428-440. · Zbl 0992.93097 |
| [12] | H.Wu and X.Zhou, Characterizing all optimal controls for an indefinite stochastic linear quadratic control problem, IEEE Trans. Autom. Control47 (2002), no. 7, 1119-1122. · Zbl 1364.49044 |
| [13] | J.Sun and J.Yong, Stochastic linear quadratic optimal control problems in infinite horizon, Appl. Math. Optim.78 (2018), 145-183. · Zbl 1398.49028 |
| [14] | N.Li, X.Li, and Z.Yu, Indefinite mean‐field type linear‐quadratic stochastic optimal control problems, Automatica122 (2020), 109627. · Zbl 1451.93418 |
| [15] | J.Sun, Mean‐field stochastic linear quadratic optimal control problems: Open‐loop solvabilities, ESAIM: Control, Optim. Calc. Variat.23 (2017), no. 3, 1099-1127. · Zbl 1393.49024 |
| [16] | X.Li, J.Sun, and J.Yong, Mean‐field stochastic linear quadratic optimal control problems: closed‐loop solvability, Probabil. Uncertainty Quant. Risk1 (2016), no. 1, 1-24. · Zbl 1433.49050 |
| [17] | C.Tang, X.Li, and T.Huang, Solvability for indefinite mean‐field stochastic linear quadratic optimal control with random jumps and its applications, Optimal Control Appl. Methods41 (2020), no. 6, 2320-2348. · Zbl 1469.93123 |
| [18] | B.Wang and H.Zhang, Indefinite linear quadratic mean field social control problems with multiplicative noise, IEEE Trans. Autom. Control66 (2020), no. 11, 5221-5236. · Zbl 1536.93968 |
| [19] | Y.Ni, X.Li, and J.Zhang, Indefinite mean‐field stochastic linearquadratic optimal control: from finite horizon to infinite horizon, IEEE Trans. Autom. Control61 (2015), no. 11, 3269-3284. · Zbl 1359.93540 |
| [20] | Y.Ni, J.Zhang, and X.Li, Indefinite mean‐field stochastic linear quadratic optimal control, IEEE Trans. Autom. Control60 (2014), no. 7, 1786-1800. · Zbl 1360.93782 |
| [21] | T.Song and B.Liu, Discrete‐time mean‐field stochastic linear‐quadratic optimal control problem with finite horizon, Asian J. Control23 (2021), no. 2, 979-989. · Zbl 07878864 |
| [22] | H.Zhang, Q.Qi, and M.Fu, Optimal stabilization control for discrete‐time mean‐field stochastic systems, IEEE Trans. Autom. Control64 (2018), no. 3, 1125-1136. · Zbl 1482.93691 |
| [23] | Q.Qi, H.Zhang, and Z.Wu, Stabilization control for linear continuous‐time mean‐field systems, IEEE Trans. Autom. Control64 (2019), no. 8, 3461-3468. · Zbl 1482.93679 |
| [24] | X.Ma, Q.Qi, X.Li, and H.Zhang, Optimal control and stablilization for linear continuous‐time mean‐field systems with delay, IET Control Theory Appl.16 (2022), no. 3, 283-300. · Zbl 07903449 |
| [25] | H.Li, Q.Qi, and H.Zhang, Stabilization control for ito stochastic system with indefinite state and control weight costs, Int. J. Control95 (2022), no. 2, 295-302. · Zbl 1482.93676 |
| [26] | J.Huang and Z.Yu, Solvability of indefinite stochastic riccati equations and linear quadratic optimal control problems, Syst. Control Lett.68 (2014), 68-75. · Zbl 1288.93093 |
| [27] | Z.Yu, Equivalent cost functionals and stochastic linear quadratic optimal control problems, ESAIM: Control, Optim. Calc. Variat.19 (2013), no. 1, 78-90. · Zbl 1258.93129 |
| [28] | S.Boyd, L.El Ghaoui, E.Feron, and V.Balakrishnan, Linear matrix inequality in systems and control theory, 1st ed., SIAM, Philadelphia, 1994. · Zbl 0816.93004 |
| [29] | W.Zhang and B.Chen, On stabilizability and exact observability of stochastic systems with their applications, Automatica40 (2004), no. 1, 87-94. · Zbl 1043.93009 |
| [30] | D.Hinrichsen and A. J.Pritchard, Stochastic h [( {}_{\infty } \]\), SIAM J. Optim. Control36 (1998), no. 5, 1504-1538. · Zbl 0914.93019 |
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