Linear-Quadratic Optimal Controls in Finite Horizons

2020 ◽  
pp. 11-59
Author(s):  
Jingrui Sun ◽  
Jiongmin Yong
Keyword(s):  
Optimal Controls ◽  
Linear Quadratic ◽  
Finite Horizons

scholarly journals On linear-quadratic optimal control of implicit difference equations

2018 ◽  
Vol 36 (3) ◽  
pp. 779-833
Author(s):  
Daniel Bankmann ◽  
Matthias Voigt
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Difference Equations ◽  
Matrix Pencil ◽  
Spectral Structure ◽  
Optimal Controls ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Quadratic Optimal Control

Abstract In this work we investigate explicit and implicit difference equations and the corresponding infinite time horizon linear-quadratic optimal control problem. We derive conditions for feasibility of the optimal control problem as well as existence and uniqueness of optimal controls under certain weaker assumptions compared to the standard approaches in the literature which are using algebraic Riccati equations. To this end, we introduce and analyse a discrete-time Lur’e equation and a corresponding Kalman–Yakubovich–Popov (KYP) inequality. We show that solvability of the KYP inequality can be characterized via the spectral structure of a certain palindromic matrix pencil. The deflating subspaces of this pencil are finally used to construct solutions of the Lur’e equation. The results of this work are transferred from the continuous-time case. However, many additional technical difficulties arise in this context.

Linear quadratic control problems of stochastic Volterra integral equations

2018 ◽  
Vol 24 (4) ◽  
pp. 1849-1879 ◽  
Author(s):  
Tianxiao Wang
Keyword(s):  
Integral Equations ◽  
Stochastic Systems ◽  
Open Loop ◽  
Volterra Integral Equations ◽  
Control Problems ◽  
Optimal Controls ◽  
Linear Quadratic Control ◽  
Linear Quadratic ◽  
The Cost ◽  
Quadratic Control Problems

This paper is concerned with linear quadratic control problems of stochastic differential equations (SDEs, in short) and stochastic Volterra integral equations (SVIEs, in short). Notice that for stochastic systems, the control weight in the cost functional is allowed to be indefinite. This feature is demonstrated here only by open-loop optimal controls but not limited to closed-loop optimal controls in the literature. As to linear quadratic problem of SDEs, some examples are given to point out the issues left by existing papers, and new characterizations of optimal controls are obtained in different manners. For the study of SVIEs with deterministic coefficients, a class of stochastic Fredholm−Volterra integral equations is introduced to replace conventional forward-backward SVIEs. Eventually, instead of using convex variation, we use spike variation to obtain some additional optimality conditions of linear quadratic problems for SVIEs.

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