scholarly journals Time-inconsistent stochastic optimal control problems and backward stochastic Volterra integral equations

2021 ◽  
Author(s):  
Jiongmin Yong ◽  
Hanxiao Wang
Keyword(s):  
Optimal Control ◽  
Mesh Size ◽  
Equilibrium Strategy ◽  
Time Interval ◽  
Well Posedness ◽  
Cost Functional ◽  
New Class ◽  
Time Inconsistent ◽  
Hamilton Jacobi Bellman ◽  
The Cost

An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situation with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate equilibrium strategies is constructed associated with partitions of the time intervals. By sending the mesh size of the time interval partition to zero, an equilibrium Hamilton--Jacobi--Bellman (HJB, for short) equation is derived, through which the equilibrium valued function and an equilibrium strategy are obtained. Under certain conditions, a verification theorem is proved and the well-posedness of the equilibrium HJB is established. As a sort of Feynman-Kac formula for the equilibrium HJB equation, a new class of BSVIEs (containing the diagonal values $Z(r,r)$ of $Z(\cd\,,\cd)$) is naturally introduced and the well-posedness of such kind of equations is briefly discussed.

scholarly journals Equilibrium strategies for time-inconsistent stochastic switching systems

2019 ◽  
Vol 25 ◽  
pp. 64 ◽  
Author(s):  
Hongwei Mei ◽  
Jiongmin Yong
Keyword(s):  
Optimal Control ◽  
Regime Switching ◽  
Hjb Equation ◽  
Equilibrium Strategy ◽  
Cost Functional ◽  
State Dependent ◽  
Global Optimal ◽  
Time Inconsistent ◽  
The Cost ◽  
Exponential Discounting

An optimal control problem is considered for a stochastic differential equation containing a state-dependent regime switching, with a recursive cost functional. Due to the non-exponential discounting in the cost functional, the problem is time-inconsistent in general. Therefore, instead of finding a global optimal control (which is not possible), we look for a time-consistent (approximately) locally optimal equilibrium strategy. Such a strategy can be represented through the solution to a system of partial differential equations, called an equilibrium Hamilton–Jacob–Bellman (HJB) equation which is constructed via a sequence of multi-person differential games. A verification theorem is proved and, under proper conditions, the well-posedness of the equilibrium HJB equation is established as well.

scholarly journals Fractional-Order Modelling and Optimal Control of Cholera Transmission

2021 ◽  
Vol 5 (4) ◽  
pp. 261
Author(s):  
Silvério Rosa ◽  
Delfim F. M. Torres
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Sensitivity Analysis ◽  
Fractional Order ◽  
Control Measures ◽  
Time Interval ◽  
Variable Order ◽  
Effectiveness Analysis ◽  
The Cost ◽  
Derivative Order

A Caputo-type fractional-order mathematical model for “metapopulation cholera transmission” was recently proposed in [Chaos Solitons Fractals 117 (2018), 37–49]. A sensitivity analysis of that model is done here to show the accuracy relevance of parameter estimation. Then, a fractional optimal control (FOC) problem is formulated and numerically solved. A cost-effectiveness analysis is performed to assess the relevance of studied control measures. Moreover, such analysis allows us to assess the cost and effectiveness of the control measures during intervention. We conclude that the FOC system is more effective only in part of the time interval. For this reason, we propose a system where the derivative order varies along the time interval, being fractional or classical when more advantageous. Such variable-order fractional model, that we call a FractInt system, shows to be the most effective in the control of the disease.

scholarly journals On a system of Hamilton-Jacobi-Bellman inequalities associated to a minimax problem with additive final cost

2003 ◽  
Vol 2003 (72) ◽  
pp. 4517-4538 ◽  
Author(s):  
Silvia C. Di Marco ◽  
Roberto L. V. González
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Existence And Uniqueness ◽  
Original Problem ◽  
Auxiliary Problem ◽  
Minimax Problem ◽  
Dynamical Programming ◽  
Minimax Optimal Control ◽  
Hamilton Jacobi Bellman ◽  
The Cost

We study a minimax optimal control problem with finite horizon and additive final cost. After introducing an auxiliary problem, we analyze the dynamical programming principle (DPP) and we present a Hamilton-Jacobi-Bellman (HJB) system. We prove the existence and uniqueness of a viscosity solution for this system. This solution is the cost function of the auxiliary problem and it is possible to get the solution of the original problem in terms of this solution.

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.

BILINEAR OPTIMAL CONTROL FOR A WAVE EQUATION

1999 ◽  
Vol 09 (01) ◽  
pp. 45-68 ◽  
Author(s):  
MIN LIANG
Keyword(s):  
Optimal Control ◽  
Wave Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Optimality System ◽  
Cost Functional ◽  
Quadratic Cost ◽  
Uniqueness Of The Solution ◽  
Initial Boundary Value ◽  
The Cost

We consider the problem of optimal control of a wave equation. A bilinear control is used to bring the state solutions close to a desired profile under a quadratic cost of control. We establish the existence of solutions of the underlying initial boundary-value problem and of an optimal control that minimizes the cost functional. We derive an optimality system by formally differentiating the cost functional with respect to the control and evaluating the result at an optimal control. We establish existence and uniqueness of the solution of the optimality system and thus determine the unique optimal control in terms of the solution of the optimality system.

Optimal Pole Assignment for Discrete Linear Regulator With Deterministic Disturbances

1988 ◽  
Vol 110 (4) ◽  
pp. 433-436
Author(s):  
Tsu-Tian Lee ◽  
Shiow-Harn Lee
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Pole Assignment ◽  
External Input ◽  
Circular Region ◽  
Cost Functional ◽  
Regulator Problem ◽  
Input Disturbance ◽  
Linear Regulator ◽  
The Cost

This paper presents the solution of the linear discrete optimal control to achieve poles assigned in a circular region and, meanwhile, accommodate deterministic disturbances. It is shown that by suitable manipulations, the problem can be reduced to a standard discrete quadratic regulator problem that simultaneously ensures: 1) closed-loop-poles all inside a circle centered at (β, 0) with a radius α, where |β| + α ≤ 1, 2) minimizes the cost functional, and 3) accommodates external input disturbance.

scholarly journals Optimal control of an SIR epidemic through finite-time non-pharmaceutical intervention

2020 ◽  
Author(s):  
David I. Ketcheson
Keyword(s):  
Optimal Control ◽  
Finite Time ◽  
Bellman Equation ◽  
Time Interval ◽  
Finite Time Interval ◽  
Sir Epidemic ◽  
Hamilton Jacobi Bellman Equation ◽  
Long Time ◽  
Hamilton Jacobi Bellman ◽  
Number Of Individuals

AbstractWe consider the problem of controlling an SIR-model epidemic by temporarily reducing the rate of contact within a population. The control takes the form of a multiplicative reduction in the contact rate of infectious individuals. The control is allowed to be applied only over a finite time interval, while the objective is to minimize the total number of individuals infected in the long-time limit, subject to some cost function for the control. We first consider the no-cost scenario and analytically determine the optimal control and solution. We then study solutions when a cost of intervention is included, as well as a cost associated with overwhelming the available medical resources. Examples are studied through the numerical solution of the associated Hamilton-Jacobi-Bellman equation. Finally, we provide some examples related directly to the current pandemic.AMS subject classification. 92D30, 34H05, 49N90, 92-10, 49L12

scholarly journals Solution to an Optimal Control Problem via Canonical Dual Method

2009 ◽  
Vol 2009 ◽  
pp. 1-5 ◽  
Author(s):  
Jinghao Zhu ◽  
Jiani Zhou
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Analytic Solution ◽  
Dual Method ◽  
Cost Functional ◽  
Nonconvex Problem ◽  
Pontryagin Principle ◽  
Dual Variables ◽  
The Cost

The analytic solution to an optimal control problem is investigated using the canonical dual method. By means of the Pontryagin principle and a transformation of the cost functional, the optimal control of a nonconvex problem is obtained. It turns out that the optimal control can be expressed by the costate via canonical dual variables. Some examples are illustrated.

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