hamilton jacobi bellman
117460677
Optimal Control and Viscosity Solutions of Hamilton Jacobi Bellman Equations
Preface. Basic notations. Outline of the main ideas on a model problem. Continuous viscosity solutions of Hamilton Jacobi equations. Optimal control problems with continuous value functions: unrestricted state space. Optimal control problems with continuous value functions: restricted state space. Discontinuous viscosity solutions and applications. Approximation and perturbation problems. Asymptotic problems. Differential Games. Numerical solution of Dynamic Programming. Nonlinear H infinity control by Pierpaolo Soravia. Bibliography. Index
Martino Bardi Italo Capuzzo-Dolcetta
195886316
Adaptive Deep Learning for High Dimensional Hamilton Jacobi Bellman Equations
Computing optimal feedback controls for nonlinear systems generally requires solving Hamilton Jacobi Bellman (HJB) equations, which, in high dimensions, are notoriously difficult. Existing strategies for high dimensional problems generally rely on specific, restrictive problem structures, or are valid only locally around some nominal trajectory. In this paper, we propose a data driven method to approximate semi global solutions to HJB equations for general high dimensional nonlinear systems and compute optimal feedback controls in real time. To accomplish this, we model solutions to HJB equations with neural networks (NNs) trained on data generated independently of any state space discretization. Training is made more effective and efficient by leveraging the known physics of the problem and using the partially trained NN to aid in adaptive data generation. We demonstrate the effectiveness of our method by learning the approximate solution to the HJB equation corresponding to the stabilization of six dimensional nonlinear rigid body, and controlling the system with the trained NN.
SIAM J. Sci. Comput.
Tenavi Nakamura-Zimmerer Qi Gong Wei Kang
127373032
Mixed Finite Element Approximation of the Hamilton Jacobi Bellman Equation with Cordes Coefficients
A mixed finite element approximation of $H^2$ solutions to the fully nonlinear Hamilton Jacobi Bellman equation, with coefficients that satisfy the Cordes condition, is proposed and analyzed. A p.
SIAM J. Numer. Anal.
Dietmar Gallistl Endre Suli
126089097
Hamilton Jacobi Bellman Quasi Variational Inequality arising in an environmental problem and its numerical discretization
Abstract A Hamilton Jacobi Bellman Quasi Variational Inequality (HJBQVI) for a river environmental restoration problem with wise use of sediment is formulated and its mathematical properties are analyzed. A finite difference scheme with a penalization technique is then established for solving the HJBQVI. The scheme is free from any iterative solvers and is unconditionally stable and convergent in the viscosity sense under certain conditions. A demonstrative application example of the HJBQVI is finally presented.
Comput. Math. Appl.
Hidekazu Yoshioka Yuta Yaegashi Yumi Yoshioka Kunihiko Hamagami
213672963
Tensor Decompositions for High dimensional Hamilton Jacobi Bellman Equations
A tensor decomposition approach for the solution of high dimensional, fully nonlinear Hamilton Jacobi Bellman equations arising in optimal feedback control of nonlinear dynamics is presented. The method combines a tensor train approximation for the value function together with a Newton like iterative method for the solution of the resulting nonlinear system. The tensor approximation leads to a polynomial scaling with respect to the dimension, partially circumventing the curse of dimensionality. A convergence analysis for the linear quadratic case is presented. For nonlinear dynamics, the effectiveness of the high dimensional control synthesis method is assessed in the optimal feedback stabilization of the Allen Cahn and Fokker Planck equations with a hundred of variables.
Sergey V Dolgov Dante Kalise Karl Kunisch
30296567
Polynomial Approximation of High Dimensional Hamilton Jacobi Bellman Equations and Applications to Feedback Control of Semilinear Parabolic PDEs
A procedure for the numerical approximation of high dimensional Hamilton Jacobi Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the successive Galerkin approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen.
SIAM J. Sci. Comput.
Dante Kalise Karl Kunisch
37004030
Remarks on Schauder Estimates and Existence of Classical Solutions for a Class of Uniformly Parabolic Hamilton Jacobi Bellman Integro PDEs
We prove Schauder estimates and obtain existence of classical solutions of Dirichlet initial boundary value problems for a class of uniformly parabolic non local Hamilton Jacobi Bellman equations.
Chenchen Mou
125923113
Solving a class of fractional optimal control problems by the Hamilton Jacobi Bellman equation
The performance index of both the state and control variables with a constrained dynamic optimization problem of a fractional order system with fixed final Time have been considered here. This paper presents a general formulation and solution scheme of a class of fractional optimal control problems. The method is based upon finding the numerical solution of the Hamilton Jacobi Bellman equation, corresponding to this problem, by the Legendre Gauss collocation method. The main reason for using this technique is its efficiency and simple application. Also, in this work, we use the fractional derivative in the Riemann Liouville sense and explain our method for a fractional derivative of order of 0 a 1 Numerical examples are provided to show the effectiveness of the formulation and solution scheme.
Seyed Ali Rakhshan Sohrab Effati Ali Vahidian Kamyad
119320278
Convergence of Implicit Schemes for Hamilton Jacobi Bellman Quasi Variational Inequalities
In [P. Azimzadeh and P. A. Forsyth, SIAM J. Numer. Anal. 54 (2016) pp. 1341 1364] we outlined the theory and implementation of computational methods for implicit schemes for Hamilton Jacobi Bellman quasi variational inequalities (HJBQVIs) No convergence proofs were given therein. This work closes the gap by giving rigorous proofs of convergence. We do so by introducing the notion of nonlocal consistency and appealing to a Barles Souganidis type analysis. Our results rely only on a well known comparison principle and are independent of the specific form of the intervention operator.
SIAM J. Control. Optim.
Parsiad Azimzadeh Erhan Bayraktar George Labahn
119136616
Viscosity Solutions of Stochastic Hamilton Jacobi Bellman Equations
In this paper we study the fully nonlinear stochastic Hamilton Jacobi Bellman (HJB) equation for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and we prove that the value function of the optimal stochastic control problem is the maximal viscosity solution of the associated stochastic HJB equation. For the superparabolic cases when the diffusion coefficients are deterministic functions of time, states and controls, the uniqueness is addressed as well.
SIAM J. Control. Optim.
Jinniao Qiu