scholarly journals Maximum Principle for Forward-Backward Control System Driven by Itô-Lévy Processes under Initial-Terminal Constraints

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Meijuan Liu ◽  
Xiangrong Wang ◽  
Hong Huang
Keyword(s):  
Optimal Control ◽  
Control System ◽  
Control Variable ◽  
Terminal State ◽  
Necessary Condition ◽  
Utility Optimization ◽  
Terminal Constraints ◽  
Stochastic Optimal Control Problem ◽  
Backward Control ◽  
Existence Of Optimal Control

This paper investigates a stochastic optimal control problem where the control system is driven by Itô-Lévy process. We prove the necessary condition about existence of optimal control for stochastic system by using traditional variational technique under the assumption that control domain is convex. We require that forward-backward stochastic differential equations (FBSDE) be fully coupled, and the control variable is allowed to enter both diffusion and jump coefficient. Moreover, we also require that the initial-terminal state be constrained. Finally, as an application to finance, we show an example of recursive consumption utility optimization problem to illustrate the practicability of our result.

scholarly journals A Necessary Condition for Optimal Control of Forward-Backward Stochastic Control System with Lévy Process in Nonconvex Control Domain Case

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Hong Huang ◽  
Xiangrong Wang ◽  
Ying Li
Keyword(s):  
Optimal Control ◽  
Control System ◽  
Lévy Process ◽  
State Constraints ◽  
Terminal State ◽  
Levy Process ◽  
Necessary Condition ◽  
Variational Technique ◽  
The Maximum Principle ◽  
Stochastic Control System

This paper analyzes one kind of optimal control problem which is described by forward-backward stochastic differential equations with Lévy process (FBSDEL). We derive a necessary condition for the existence of the optimal control by means of spike variational technique, while the control domain is not necessarily convex. Simultaneously, we also get the maximum principle for this control system when there are some initial and terminal state constraints. Finally, a financial example is discussed to illustrate the application of our result.

scholarly journals An optimal control problem involving a class of linear time-lag systems

1986 ◽  
Vol 28 (1) ◽  
pp. 93-113 ◽  
Author(s):  
K. L. Teo ◽  
K. H. Wong ◽  
Z. S. Wu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Control Problems ◽  
Linear Time ◽  
Convergence Property ◽  
Time Lag ◽  
Necessary Condition ◽  
Terminal Constraints ◽  
Linear Control Constraints

A class of convex optimal control problems involving linear hereditary systems with linear control constraints and nonlinear terminal constraints is considered. A result on the existence of an optimal control is proved and a necessary condition for optimality is given. An iterative algorithm is presented for solving the optimal control problem under consideration. The convergence property of the algorithm is also investigated. To test the algorithm, an example is solved.

scholarly journals Modeling and Optimal Control on the Spread of Hantavirus Infection

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1192 ◽  
Author(s):  
Fauzi Mohamed Yusof ◽  
Farah Aini Abdullah ◽  
Ahmad Izani Md. Ismail
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Control System ◽  
Maximum Principle ◽  
Optimal Control Theory ◽  
Necessary Condition ◽  
Hantavirus Infection ◽  
Kutta Method ◽  
Rodent Population ◽  
Runge Kutta Method

In this paper, optimal control theory is applied to a system of ordinary differential equations representing a hantavirus infection in rodent and alien populations. The effect of the optimal control in eliminating the rodent population that caused the hantavirus infection is investigated. In addition, Pontryagin’s maximum principle is used to obtain the necessary condition for the controls to be optimal. The Runge–Kutta method is then used to solve the proposed optimal control system. The findings from the optimal control problem suggest that the infection may be eradicated by implementing some controls for a certain period of time. This research concludes that the optimal control mathematical model is an effective method in reducing the number of infectious in a community and environment.

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.

Stochastic optimal control problem of constrained switching system with delay

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 711-720
Author(s):  
Charkaz Aghayeva
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stochastic Optimal Control ◽  
Initial Conditions ◽  
Necessary Condition ◽  
Switching System ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Optimal Control Problem ◽  
Necessary Condition Of Optimality

This paper concerns the stochastic optimal control problem of switching systems with delay. The evolution of the system is governed by the collection of stochastic delay differential equations with initial conditions that depend on its previous state. The restriction on the system is defined by the functional constraint that contains state and time parameters. First, maximum principle for stochastic control problem of delay switching system without constraint is established. Finally, using Ekeland?s variational principle, the necessary condition of optimality for control system with constraint is obtained.

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.

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 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>

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.

scholarly journals Asymptotic expansion of a solution of a singularly perturbed optimal control problem with a convex integral quality index, whose terminal part additively depends on slow and fast variables

2020 ◽  
Vol 55 ◽  
pp. 33-41
Author(s):  
A.R. Danilin ◽  
A.A. Shaburov
Keyword(s):  
Optimal Control ◽  
Control System ◽  
Steady State ◽  
Quality Index ◽  
Control Variable ◽  
Small Mass ◽  
Time Interval ◽  
Terminal Part ◽  
Time Optimal ◽  
Integral Quality

The paper deals with the problem of optimal control with a Boltz-type quality index over a finite time interval for a linear steady-state control system in the class of piecewise continuous controls with smooth control constraints. In particular, we study the problem of controlling the motion of a system of small mass points under the action of a bounded force. The terminal part of the convex integral quality index additively depends on slow and fast variables, and the integral term is a strictly convex function of control variable. If the system is completely controllable, then the Pontryagin maximum principle is a necessary and sufficient condition for optimality. The main difference between this study and previous works is that the equation contains the zero matrix of fast variables and, thus, the results of A.B. Vasilieva on the asymptotic of the fundamental matrix of a control system are not valid. However, the linear steady-state system satisfies the condition of complete controllability. The article shows that problems of optimal control with a convex integral quality index are more regular than time-optimal problems.

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