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Li, Yuhang; Han, Yuecai; Gao, Yanwei

Stochastic maximum principle for moving average control system. (English) Zbl 1531.93435

Optim. Control Appl. Methods 45, No. 1, 321-335 (2024).
Summary: In this paper, we consider the stochastic optimal control problem for moving average control system. The corresponding moving average stochastic differential equation is a kind of integral differential equations. We prove the existence and uniqueness of the solution of the moving average stochastic differential equations. We obtain the stochastic maximum principle of the moving average optimal control system by introducing a kind of generalized anticipated backward stochastic differential equations. We prove the existence and uniqueness of the solution of this adjoint equation, which is singular at 0. As an application, the linear quadratic moving average control problem is investigated to illustrate the main results.
© 2023 John Wiley & Sons Ltd.

MSC:

93E20 Optimal stochastic control
49N10 Linear-quadratic optimal control problems
45J05 Integro-ordinary differential equations

Keywords:

integral differential equations; linear quadratic optimal control; maximum principle; moving average control system
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[1] BloomF. Bounds for solutions to a class of integro‐differential equations associated with a theory of rigid nonconducting material dielectrics. SIAM J Math Anal. 1980;11(2):265‐291. · Zbl 0436.45017
[2] HolmåkerK. Global asymptotic stability for a stationary solution of a system of integro‐differential equations describing the formation of liver zones. SIAM J Math Anal. 1993;24(1):116‐128. · Zbl 0767.45005
[3] ForbesLK, CrozierS, DoddrellDM. Calculating current densities and fields produced by shielded magnetic resonance imaging probes. SIAM J Appl Math. 1997;57(2):401‐425. · Zbl 0871.65116
[4] DuH, ZhaoG, ZhaoC. Reproducing kernel method for solving Fredholm integro‐differential equations with weakly singularity. J Comput Appl Math. 2014;255:122‐132. · Zbl 1291.65379
[5] MaoX, RiedleM. Mean square stability of stochastic Volterra integro‐differential equations. Syst Control Lett. 2006;55(6):459‐465. · Zbl 1129.34332
[6] NesterenkoO. Modified projection‐iterative method for weakly nonlinear integrodifferential equations with parameters. J Math Sci. 2014;198(3):328‐336. · Zbl 1301.65134
[7] DzhumabaevDS. On one approach to solve the linear boundary value problems for Fredholm integro‐differential equations. J Comput Appl Math. 2016;294:342‐357. · Zbl 1327.65277
[8] ZhangW, LiangH, GaoJ. Theoretical and numerical analysis of the Euler-Maruyama method for generalized stochastic Volterra integro‐differential equations. J Comput Appl Math. 2020;365:112364. · Zbl 1524.65047
[9] KimJU. Control of a second‐order integro‐differential equation. SIAM J Control Optim. 1993;31(1):101‐110. · Zbl 0781.49005
[10] MashayekhiS, OrdokhaniY, RazzaghiM. Hybrid functions approach for optimal control of systems described by integro‐differential equations. App Math Model. 2013;37(5):3355‐3368. · Zbl 1351.49031
[11] AssanovaAT, BakirovaE, KadirbayevaZM. Numerical solution to a control problem for integro‐differential equations. Comput Math Math Phys. 2020;60:203‐221. · Zbl 1453.65445
[12] WangT. Backward stochastic Volterra integro‐differential equations and applications in optimal control problems. SIAM J Control Optim. 2022;60(4):2393‐2419. · Zbl 1498.93791
[13] El KarouiN, PengS, QuenezMC. Backward stochastic differential equations in finance. Math Fin. 1997;7(1):1‐71. · Zbl 0884.90035
[14] PengS, YangZ. Anticipated backward stochastic differential equations. Ann Probab. 2009;37(3):877‐902. · Zbl 1186.60053
[15] YangZ, ElliottR. Some properties of generalized anticipated backward stochastic differential equations. Electron Commun Probab. 2013;18:1‐10. · Zbl 1329.60204
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