Mean-Field Stochastic Linear Quadratic Optimal Control for Jump-Diffusion Systems with Hybrid Disturbances

A mean-field linear quadratic stochastic (MF-SLQ for short) optimal control problem with hybrid disturbances and cross terms in a finite horizon is concerned. The state equation is a systems driven by the Wiener process and the Poisson random martingale measure disturbed by some stochastic perturbat...

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Bibliographic Details
Published inSymmetry (Basel) Vol. 16; no. 6; p. 642
Main Authors Tang, Chao, Li, Xueqin, Wang, Qi
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.06.2024
Subjects
Decoupling method
Differential equations
Disturbances
Equations of state
feedback regulator
Fixed points (mathematics)
hybrid disturbances
Hybrid systems
Martingales
Mathematical functions
mean-field stochastic differential equation
Optimal control
Perturbation
Riccati equation
Riccati equations
SLQ optimal control
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Summary:A mean-field linear quadratic stochastic (MF-SLQ for short) optimal control problem with hybrid disturbances and cross terms in a finite horizon is concerned. The state equation is a systems driven by the Wiener process and the Poisson random martingale measure disturbed by some stochastic perturbations. The cost functional is also disturbed, which means more general cases could be characterized, especially when extra environment perturbations exist. In this paper, the well-posedness result on the jump diffusion systems is obtained by the fixed point theorem and also the solvability of the MF-SLQ problem. Actually, by virtue of adjoint variables, classic variational calculus, and some dual representation, an optimal condition is derived. Throughout our research, in order to connect the optimal control and the state directly, two Riccati differential equations, a BSDE with random jumps and an ordinary equation (ODE for short) on disturbance terms are obtained by a decoupling technique, which provide an optimal feedback regulator. Meanwhile, the relationship between the two Riccati equations and the so-called mean-field stochastic Hamilton system is established. Consequently, the optimal value is characterized by the initial state, disturbances, and original value of the Riccati equations. Finally, an example is provided to illustrate our theoretic results.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym16060642