scholarly journals Study on Stochastic Linear Quadratic Optimal Control with Quadratic and Mixed Terminal State Constraints

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Yang Hongli
Keyword(s):  
State Constraints ◽  
Terminal State ◽  
Equality Constraints ◽  
Necessary Condition ◽  
Mathematical Programming Problem ◽  
Lagrangian Multiplier ◽  
Lq Problem ◽  
Indefinite Stochastic Lq Control ◽  
Lq Control ◽  
Stochastic Lq Control

This paper studies the indefinite stochastic LQ control problem with quadratic and mixed terminal state equality constraints, which can be transformed into a mathematical programming problem. By means of the Lagrangian multiplier theorem and Riesz representation theorem, the main result given in this paper is the necessary condition for indefinite stochastic LQ control with quadratic and mixed terminal equality constraints. The result shows that the different terminal state constraints will cause the endpoint condition of the differential Riccati equation to be changed. It coincides with the indefinite stochastic LQ problem with linear terminal state constraint, so the result given in this paper can be viewed as the extension of the indefinite stochastic LQ problem with the linear terminal state equality constraint. In order to guarantee the existence and the uniqueness of the linear feedback control, a sufficient condition is also presented in the paper. A numerical example is presented at the end of the paper.

scholarly journals Indefinite LQ Optimal Control with Terminal State Constraint for Discrete-Time Uncertain Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Yuefen Chen ◽  
Minghai Yang
Keyword(s):  
Optimal Control ◽  
Discrete Time ◽  
State Constraint ◽  
Terminal State ◽  
Necessary Condition ◽  
Lagrangian Multiplier ◽  
Lq Optimal Control ◽  
System States ◽  
Lq Problem ◽  
Terminal State Constraint

Uncertainty theory is a branch of mathematics for modeling human uncertainty based on the normality, duality, subadditivity, and product axioms. This paper studies a discrete-time LQ optimal control with terminal state constraint, whereas the weighting matrices in the cost function are indefinite and the system states are disturbed by uncertain noises. We first transform the uncertain LQ problem into an equivalent deterministic LQ problem. Then, the main result given in this paper is the necessary condition for the constrained indefinite LQ optimal control problem by means of the Lagrangian multiplier method. Moreover, in order to guarantee the well-posedness of the indefinite LQ problem and the existence of an optimal control, a sufficient condition is presented in the paper. Finally, a numerical example is presented at the end of the paper.

Linear Quadratic Optimal Control Via Fourier-Based State Parameterization

1991 ◽  
Vol 113 (2) ◽  
pp. 206-215 ◽  
Author(s):  
V. Yen ◽  
M. Nagurka
Keyword(s):  
Optimal Control ◽  
Quadratic Programming ◽  
Programming Problem ◽  
Quadratic Programming Problem ◽  
Necessary Condition ◽  
Mathematical Programming Problem ◽  
Linear Quadratic ◽  
Linear Algebraic Equations ◽  
Linearly Constrained ◽  
Lq Problem

A method for determining the optimal control of unconstrained and linearly constrained linear dynamic systems with quadratic performance indices is presented. The method is based on a modified Fourier series approximation of each state variable that converts the linear quadratic (LQ) problem into a mathematical programming problem. In particular, it is shown that an unconstrained LQ problem can be cast as an unconstrained quadratic programming problem where the necessary condition of optimality is derived as a system of linear algebraic equations. Furthermore, it is shown that a linearly constrained LQ problem can be converted into a general quadratic programming problem. Simulation studies for constrained LQ systems, including a bang-bang control problem, demonstrate that the approach is accurate. The results also indicate that in solving high order unconstrained LQ problems the approach is computationally more efficient and robust than standard methods.

scholarly journals A Maximum Principle for Controlled Time-Symmetric Forward-Backward Doubly Stochastic Differential Equation with Initial-Terminal Sate Constraints

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Shaolin Ji ◽  
Qingmeng Wei ◽  
Xiumin Zhang
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Maximum Principle ◽  
Stochastic Differential Equation ◽  
State Constraints ◽  
Terminal State ◽  
Necessary Condition ◽  
Variation Principle ◽  
Linear Quadratic ◽  
Doubly Stochastic

We study the optimal control problem of a controlled time-symmetric forward-backward doubly stochastic differential equation with initial-terminal state constraints. Applying the terminal perturbation method and Ekeland’s variation principle, a necessary condition of the stochastic optimal control, that is, stochastic maximum principle, is derived. Applications to backward doubly stochastic linear-quadratic control models are investigated.

scholarly journals Discrete-Time Indefinite Stochastic LQ Control via SDP and LMI Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Shaowei Zhou ◽  
Weihai Zhang
Keyword(s):  
Discrete Time ◽  
Time Horizon ◽  
Algebraic Riccati Equation ◽  
Mean Square ◽  
Infinite Time ◽  
Lq Problem ◽  
The Cost ◽  
And Control ◽  
Indefinite Stochastic Lq Control ◽  
Lq Control

This paper studies a discrete-time stochastic LQ problem over an infinite time horizon with state-and control-dependent noises, whereas the weighting matrices in the cost function are allowed to be indefinite. We mainly use semidefinite programming (SDP) and its duality to treat corresponding problems. Several relations among stability, SDP complementary duality, the existence of the solution to stochastic algebraic Riccati equation (SARE), and the optimality of LQ problem are established. We can test mean square stabilizability and solve SARE via SDP by LMIs method.

scholarly journals Necessary conditions for a mathematical programming problem with set and cone constraints

1988 ◽  
Vol 29 (3) ◽  
pp. 310-321
Author(s):  
Youji Nagahisa
Keyword(s):  
Mathematical Programming ◽  
Programming Problem ◽  
State Constraints ◽  
Necessary Conditions ◽  
Open Mapping ◽  
Necessary Condition ◽  
Mathematical Programming Problem ◽  
Open Mapping Theorem ◽  
Cone Constraints ◽  
Inequality State

AbstractThis paper is devoted to the derivation of a necessary condition of F. John type which must be satisfied by a solution of a mathematical programming problem with set and cone constraints. The necessary condition is applied to an optimisation problem defined on functional spaces with inequality state constraints. Furthermore a pseudo open mapping theorem is developed in the course of proving the main theorem.

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