scholarly journals Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential System and Application

2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Kai Du ◽  
Zhen Wu
Keyword(s):  
Optimal Control ◽  
Differential System ◽  
Optimization Problems ◽  
Stackelberg Game ◽  
Mean Field ◽  
Open Loop ◽  
Stackelberg Equilibrium ◽  
Linear Quadratic ◽  
Stackelberg Differential Game ◽  
Lq Optimal Control

This paper is concerned with a new kind of Stackelberg differential game of mean-field backward stochastic differential equations (MF-BSDEs). By means of four Riccati equations (REs), the follower first solves a backward mean-field stochastic LQ optimal control problem and gets the corresponding open-loop optimal control with the feedback representation. Then the leader turns to solve an optimization problem for a 1×2 mean-field forward-backward stochastic differential system. In virtue of some high-dimensional and complicated REs, we obtain the open-loop Stackelberg equilibrium, and it admits a state feedback representation. Finally, as applications, a class of stochastic pension fund optimization problems which can be viewed as a special case of our formulation is studied and the open-loop Stackelberg strategy is obtained.

On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems

2020 ◽  
Vol 26 ◽  
pp. 41
Author(s):  
Tianxiao Wang
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Time Inconsistency ◽  
Control Problems ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Equilibrium Strategies

This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

scholarly journals Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions

2019 ◽  
Vol 25 ◽  
pp. 17 ◽  
Author(s):  
Qingmeng Wei ◽  
Jiongmin Yong ◽  
Zhiyong Yu
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Control Problems ◽  
Linear Quadratic ◽  
Cost Functional ◽  
The Cost

An optimal control problem is considered for linear stochastic differential equations with quadratic cost functional. The coefficients of the state equation and the weights in the cost functional are bounded operators on the spaces of square integrable random variables. The main motivation of our study is linear quadratic (LQ, for short) optimal control problems for mean-field stochastic differential equations. Open-loop solvability of the problem is characterized as the solvability of a system of linear coupled forward-backward stochastic differential equations (FBSDE, for short) with operator coefficients, together with a convexity condition for the cost functional. Under proper conditions, the well-posedness of such an FBSDE, which leads to the existence of an open-loop optimal control, is established. Finally, as applications of our main results, a general mean-field LQ control problem and a concrete mean-variance portfolio selection problem in the open-loop case are solved.

An LQ Approach to Active Control of Vibrations in Helicopters

1996 ◽  
Vol 118 (3) ◽  
pp. 482-488 ◽  
Author(s):  
Sergio Bittanti ◽  
Fabrizio Lorito ◽  
Silvia Strada
Keyword(s):  
Optimal Control ◽  
Active Control ◽  
Linear Quadratic ◽  
Control Effort ◽  
Vibration Effect ◽  
Suitable Definition ◽  
Lq Optimal Control ◽  
The One ◽  
Definition Of ◽  
And Control

In this paper, Linear Quadratic (LQ) optimal control concepts are applied for the active control of vibrations in helicopters. The study is based on an identified dynamic model of the rotor. The vibration effect is captured by suitably augmenting the state vector of the rotor model. Then, Kalman filtering concepts can be used to obtain a real-time estimate of the vibration, which is then fed back to form a suitable compensation signal. This design rationale is derived here starting from a rigorous problem position in an optimal control context. Among other things, this calls for a suitable definition of the performance index, of nonstandard type. The application of these ideas to a test helicopter, by means of computer simulations, shows good performances both in terms of disturbance rejection effectiveness and control effort limitation. The performance of the obtained controller is compared with the one achievable by the so called Higher Harmonic Control (HHC) approach, well known within the helicopter community.

scholarly journals Backward-forward linear-quadratic mean-field Stackelberg games

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kehan Si ◽  
Zhen Wu
Keyword(s):  
Large Population ◽  
Consistency Condition ◽  
Market Price ◽  
Mean Field ◽  
Mapping Method ◽  
Open Loop ◽  
Stackelberg Games ◽  
Linear Quadratic ◽  
Population System ◽  
The Cost

AbstractThis paper studies a controlled backward-forward linear-quadratic-Gaussian (LQG) large population system in Stackelberg games. The leader agent is of backward state and follower agents are of forward state. The leader agent is dominating as its state enters those of follower agents. On the other hand, the state-average of all follower agents affects the cost functional of the leader agent. In reality, the leader and the followers may represent two typical types of participants involved in market price formation: the supplier and producers. This differs from standard MFG literature and is mainly due to the Stackelberg structure here. By variational analysis, the consistency condition system can be represented by some fully-coupled backward-forward stochastic differential equations (BFSDEs) with high dimensional block structure in an open-loop sense. Next, we discuss the well-posedness of such a BFSDE system by virtue of the contraction mapping method. Consequently, we obtain the decentralized strategies for the leader and follower agents which are proved to satisfy the ε-Nash equilibrium property.

scholarly journals Power Control via Stackelberg Game for Small-Cell Networks

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Yanxiang Jiang ◽  
Hui Ge ◽  
Mehdi Bennis ◽  
Fu-Chun Zheng ◽  
Xiaohu You
Keyword(s):  
Power Control ◽  
Optimization Problems ◽  
Stackelberg Game ◽  
Small Cell ◽  
User Equipment ◽  
Stackelberg Equilibrium ◽  
Small Cell Networks ◽  
Transmit Power ◽  
Control Scheme ◽  
Cell Networks

In this paper, power control in the uplink for two-tier small-cell networks is investigated. We formulate the power control problem as a Stackelberg game, where the macrocell user equipment (MUE) acts as the leader and the small-cell user equipment (SUE) acts as the follower. To reduce the cross-tier and cotier interferences and the power consumption of both the MUE and SUE, we propose optimizing not only the transmit rate but also the transmit power. The corresponding optimization problems are solved through a two-layer iteration. In the inner iteration, the SUE items (SUEs) compete with each other, and their optimal transmit powers are obtained through iterative computations. In the outer iteration, the optimal transmit power of the MUE is obtained in a closed form based on the transmit powers of the SUEs through proper mathematical manipulations. We prove the convergence of the proposed power control scheme, and we also theoretically show the existence and uniqueness of the Stackelberg equilibrium (SE) in the formulated Stackelberg game. The simulation results show that the proposed power control scheme provides considerable improvements, particularly for the MUE.

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