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 An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications

2020 ◽  
Vol 28 (1) ◽  
pp. 1-18
Author(s):  
Dahbia Hafayed ◽  
Adel Chala
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Control Problem ◽  
Stochastic Differential Equations ◽  
Doubly Stochastic ◽  
Risk Sensitive ◽  
Performance Functional

AbstractIn this paper, we are concerned with an optimal control problem where the system is driven by a backward doubly stochastic differential equation with risk-sensitive performance functional. We generalized the result of Chala [A. Chala, Pontryagin’s risk-sensitive stochastic maximum principle for backward stochastic differential equations with application, Bull. Braz. Math. Soc. (N. S.) 48 2017, 3, 399–411] to a backward doubly stochastic differential equation by using the same contribution of Djehiche, Tembine and Tempone in [B. Djehiche, H. Tembine and R. Tempone, A stochastic maximum principle for risk-sensitive mean-field type control, IEEE Trans. Automat. Control 60 2015, 10, 2640–2649]. We use the risk-neutral model for which an optimal solution exists as a preliminary step. This is an extension of an initial control system in this type of problem, where an admissible controls set is convex. We establish necessary as well as sufficient optimality conditions for the risk-sensitive performance functional control problem. We illustrate the paper by giving two different examples for a linear quadratic system, and a numerical application as second example.

A general maximum principle for mean-field forward-backward doubly stochastic differential equations with jumps processes

2019 ◽  
Vol 27 (1) ◽  
pp. 9-25 ◽  
Author(s):  
Dahbia Hafayed ◽  
Adel Chala
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Admissible Control ◽  
Mean Field ◽  
Sufficient Optimality Condition ◽  
Doubly Stochastic

Abstract In this paper, we deal with an optimal control, where the system is driven by a mean-field forward-backward doubly stochastic differential equation with jumps diffusion. We assume that the set of admissible control is convex, and we establish a necessary as well as a sufficient optimality condition for such system.

scholarly journals Near Optimality of Linear Delayed Doubly Stochastic Control Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Jie Xu ◽  
Ruiqiang Lin
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Control Problem ◽  
Delay System ◽  
Necessary Condition ◽  
Linear Quadratic ◽  
The Maximum Principle ◽  
Doubly Stochastic ◽  
Convex Control ◽  
Near Optimality

In this paper, we study a kind of near optimal control problem which is described by linear quadratic doubly stochastic differential equations with time delay. We consider the near optimality for the linear delayed doubly stochastic system with convex control domain. We discuss the case that all the time delay variables are different. We give the maximum principle of near optimal control for this kind of time delay system. The necessary condition for the control to be near optimal control is deduced by Ekeland’s variational principle and some estimates on the state and the adjoint processes corresponding to the system.

scholarly journals Study on Indefinite Stochastic Linear Quadratic Optimal Control with Inequality Constraint

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Guiling Li ◽  
Weihai Zhang
Keyword(s):  
Optimal Control ◽  
Dynamic Programming Algorithm ◽  
Inequality Constraint ◽  
Terminal State ◽  
State Feedback Control ◽  
Programming Algorithm ◽  
Necessary Condition ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Lq Optimal Control

This paper studies the indefinite stochastic linear quadratic (LQ) optimal control problem with an inequality constraint for the terminal state. Firstly, we prove a generalized Karush-Kuhn-Tucker (KKT) theorem under hybrid constraints. Secondly, a new type of generalized Riccati equations is obtained, based on which a necessary condition (it is also a sufficient condition under stronger assumptions) for the existence of an optimal linear state feedback control is given by means of KKT theorem. Finally, we design a dynamic programming algorithm to solve the constrained indefinite stochastic LQ issue.

scholarly journals Optimal control for quasilinear retarded parabolic systems

2001 ◽  
Vol 42 (4) ◽  
pp. 532-551 ◽  
Author(s):  
Liping Pan ◽  
Jiongmin Yong
Keyword(s):  
Optimal Control ◽  
Parabolic Equation ◽  
Maximum Principle ◽  
State Constraints ◽  
Quasilinear Parabolic Equation ◽  
Parabolic Systems ◽  
Terminal State ◽  
Spatial Derivative ◽  
The Cost ◽  
Quasilinear Parabolic

AbstractWe study an optimal control problem for a quasilinear parabolic equation which has delays in the highest order spatial derivative terms. The cost functional is Lagrange type and some terminal state constraints are presented. A Pontryagin-type maximum principle is derived.

Time Optimal Swing-Up Control of Single Pendulum1

2000 ◽  
Vol 123 (3) ◽  
pp. 518-527 ◽  
Author(s):  
Yongcai Xu ◽  
Masami Iwase ◽  
Katsuhisa Furuta
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Auxiliary Problem ◽  
Terminal State ◽  
Time Optimal Control ◽  
Necessary Condition ◽  
Optimal Time ◽  
Input Amplitude ◽  
Time Optimal ◽  
Bounded Input

Swing-up of a rotating type pendulum from the pendant to the inverted state is known to be one of most difficult control problems, since the system is nonlinear, underactuated, and has uncontrollable states. This paper studies a time optimal swing-up control of the pendulum using bounded input. Time optimal control of a nonlinear system can be formulated by Pontryagin’s Maximum Principle, which is, however, hard to compute practically. In this paper, a new computational approach is presented to attain a numerical solution of the time optimal swing-up problem. Time optimal control problem is described as minimization of the achievable time to attain the terminal state under the bounded input amplitude, although algorithms to solve this problem are known to be complicated. Therefore, in this paper, it is shown how the optimal time swing-up control is formulated as an auxiliary problem in that the minimal input amplitude is searched so that the terminal state satisfies a specification at a given time. Through the proposed approach, time optimal control can be solved by nonlinear optimization. Its approach is evaluated by numerical simulations of a simplified pendulum model, is checked satisfying the necessary condition of Maximum Principle, and is experimentally verified using the rotating type pendulum.

scholarly journals On Optimal Control Problem for Backward Stochastic Doubly Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Adel Chala
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Diffusion Coefficient ◽  
Maximum Principle ◽  
Optimal Control Problems ◽  
Necessary Conditions ◽  
State Equation ◽  
Control Problems ◽  
Sufficient Condition ◽  
Doubly Stochastic

We are going to study an approach of optimal control problems where the state equation is a backward doubly stochastic differential equation, and the set of strict (classical) controls need not be convex and the diffusion coefficient and the generator coefficient depend on the terms control. The main result is necessary conditions as well as a sufficient condition for optimality in the form of a relaxed maximum principle.

scholarly journals Stochastic maximum principle for systems driven by local martingales with spatial parameters

2021 ◽  
Vol 6 (3) ◽  
pp. 213
Author(s):  
Jian Song ◽  
Meng Wang
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Stochastic Maximum Principle ◽  
Necessary Condition ◽  
Local Martingale ◽  
Linear Quadratic ◽  
Quadratic Problem ◽  
Spatial Parameter ◽  
Stochastic Optimal Control Problem ◽  
Spatial Parameters

<p style='text-indent:20px;'>We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter. Assuming the convexity of the control domain, we obtain the stochastic maximum principle as the necessary condition for an optimal control, and we also prove its sufficiency under proper conditions. The stochastic linear quadratic problem in this setting is also discussed.</p>

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