scholarly journals Nonconvex Duality and Semicontinuous Proximal Solutions of HJB Equation in Optimal Control

2009 ◽  
Vol 43 (2) ◽  
pp. 201-214 ◽  
Author(s):  
Mustapha Serhani ◽  
Nadia Raïssi
Keyword(s):  
Optimal Control ◽  
Hjb Equation ◽  
Nonconvex Duality

scholarly journals A Novel High Dimensional Fitted Scheme for Stochastic Optimal Control Problems

2021 ◽  
Author(s):  
Christelle Dleuna Nyoumbi ◽  
Antoine Tambue
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Optimal Control ◽  
Finite Difference Method ◽  
Stochastic Optimal Control ◽  
Optimal Control Problems ◽  
Hjb Equation ◽  
High Dimensional ◽  
Control Problems ◽  
Volume Method

AbstractStochastic optimal principle leads to the resolution of a partial differential equation (PDE), namely the Hamilton–Jacobi–Bellman (HJB) equation. In general, this equation cannot be solved analytically, thus numerical algorithms are the only tools to provide accurate approximations. The aims of this paper is to introduce a novel fitted finite volume method to solve high dimensional degenerated HJB equation from stochastic optimal control problems in high dimension ($$ n\ge 3$$ n ≥ 3 ). The challenge here is due to the nature of our HJB equation which is a degenerated second-order partial differential equation coupled with an optimization problem. For such problems, standard scheme such as finite difference method losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. We discretize the HJB equation using the fitted finite volume method, well known to tackle degenerated PDEs, while the time discretisation is performed using the Implicit Euler scheme.. We show that matrices resulting from spatial discretization and temporal discretization are M-matrices. Numerical results in finance demonstrating the accuracy of the proposed numerical method comparing to the standard finite difference method are provided.

scholarly journals On Stability of Perturbed Nonlinear Switched Systems with Adaptive Reinforcement Learning

Energies ◽  
2020 ◽  
Vol 13 (19) ◽  
pp. 5069
Author(s):  
Phuong Nam Dao ◽  
Hong Quang Nguyen ◽  
Minh-Duc Ngo ◽  
Seon-Ju Ahn
Keyword(s):  
Optimal Control ◽  
Reinforcement Learning ◽  
Switched Systems ◽  
Tracking Control ◽  
Control Method ◽  
Learning Algorithm ◽  
Hjb Equation ◽  
Control Approach ◽  
Nonlinear Switched Systems ◽  
Hamilton Jacobi Bellman

In this paper, a tracking control approach is developed based on an adaptive reinforcement learning algorithm with a bounded cost function for perturbed nonlinear switched systems, which represent a useful framework for modelling these converters, such as DC–DC converter, multi-level converter, etc. An optimal control method is derived for nominal systems to solve the tracking control problem, which results in solving a Hamilton–Jacobi–Bellman (HJB) equation. It is shown that the optimal controller obtained by solving the HJB equation can stabilize the perturbed nonlinear switched systems. To develop a solution to the translated HJB equation, the proposed neural networks consider the training technique obtaining the minimization of square of Bellman residual error in critic term due to the description of Hamilton function. Theoretical analysis shows that all the closed-loop system signals are uniformly ultimately bounded (UUB) and the proposed controller converges to optimal control law. The simulation results of two situations demonstrate the effectiveness of the proposed controller.

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