Dynamic Programming Principle for One Kind of Stochastic Recursive Optimal Control Problem and Hamilton–Jacobi–Bellman Equation

2008 ◽  
Vol 47 (5) ◽  
pp. 2616-2641 ◽  
Author(s):  
Zhen Wu ◽  
Zhiyong Yu
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Bellman Equation ◽  
Dynamic Programming Principle ◽  
Hamilton Jacobi Bellman Equation ◽  
Hamilton Jacobi Bellman

scholarly journals Dynamic programming principle and associated Hamilton-Jacobi-Bellman equation for stochastic recursive control problem with non-Lipschitz aggregator

2018 ◽  
Vol 24 (1) ◽  
pp. 355-376 ◽  
Author(s):  
Jiangyan Pu ◽  
Qi Zhang
Keyword(s):  
Dynamic Programming ◽  
Control Problem ◽  
Bellman Equation ◽  
Backward Stochastic Differential Equation ◽  
Dynamic Programming Principle ◽  
Unknown Variable ◽  
Hamilton Jacobi Bellman Equation ◽  
Hamilton Jacobi Bellman ◽  
The Stability ◽  
The Value Function

In this work we study the stochastic recursive control problem, in which the aggregator (or generator) of the backward stochastic differential equation describing the running cost is continuous but not necessarily Lipschitz with respect to the first unknown variable and the control, and monotonic with respect to the first unknown variable. The dynamic programming principle and the connection between the value function and the viscosity solution of the associated Hamilton-Jacobi-Bellman equation are established in this setting by the generalized comparison theorem for backward stochastic differential equations and the stability of viscosity solutions. Finally we take the control problem of continuous-time Epstein−Zin utility with non-Lipschitz aggregator as an example to demonstrate the application of our study.

Stochastic Optimal Control

2019 ◽  
pp. 333-350
Author(s):  
Tomas Björk
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stochastic Optimal Control ◽  
Linear Quadratic Regulator ◽  
Linear Quadratic ◽  
Verification Theorem ◽  
Stochastic Optimal Control Problem ◽  
Hamilton Jacobi Bellman

We study a general stochastic optimal control problem within the framework of a controlled SDE. This problem is studied using dynamic programming and we derive the Hamilton–Jacobi–Bellman PDE. By stating and proving a verification theorem we show that solving this PDE is equivalent to solving the control problem. As an example the theory is then applied to the linear quadratic regulator.

scholarly journals Fractional Order Version of the Hamilton–Jacobi–Bellman Equation

2018 ◽  
Vol 14 (1) ◽  
Author(s):  
Abolhassan Razminia ◽  
Mehdi Asadizadehshiraz ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Dynamical Systems ◽  
Optimal Control Problem ◽  
Fractional Order ◽  
Performance Index ◽  
Bellman Equation ◽  
Hjb Equation ◽  
Usual Form ◽  
Hamilton Jacobi Bellman Equation ◽  
Hamilton Jacobi Bellman

We consider an extension of the well-known Hamilton–Jacobi–Bellman (HJB) equation for fractional order dynamical systems in which a generalized performance index is considered for the related optimal control problem. Owing to the nonlocality of the fractional order operators, the classical HJB equation, in the usual form, does not hold true for fractional problems. Effectiveness of the proposed technique is illustrated through a numerical example.

Nonlinear Stochastic Optimal Control of MDOF Partially Observable Linear Systems Excited by Combined Harmonic and Wide-Band Noises

2019 ◽  
Vol 19 (03) ◽  
pp. 1950019 ◽  
Author(s):  
R. C. Hu ◽  
X. F. Wang ◽  
X. D. Gu ◽  
R. H. Huan
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Control Problem ◽  
Linear Systems ◽  
Control Strategy ◽  
Stochastic Optimal Control ◽  
Wide Band ◽  
Dynamic Programming Principle ◽  
Stochastic Dynamic ◽  
Partially Observable

In this paper, nonlinear stochastic optimal control of multi-degree-of-freedom (MDOF) partially observable linear systems subjected to combined harmonic and wide-band random excitations is investigated. Based on the separation principle, the control problem of a partially observable system is converted into a completely observable one. The dynamic programming equation for the completely observable control problem is then set up based on the stochastic averaging method and stochastic dynamic programming principle, from which the nonlinear optimal control law is derived. To illustrate the feasibility and efficiency of the proposed control strategy, the responses of the uncontrolled and optimal controlled systems are respectively obtained by solving the associated Fokker–Planck–Kolmogorov (FPK) equation. Numerical results show the proposed control strategy can dramatically reduce the response of stochastic systems subjected to both harmonic and wide-band random excitations.

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