On existence and uniqueness of solutions of Hamilton-Jacobi equations
Non-Linear Analysis
Numerical methods for stochastic control problems in continuous time
Numerical methods for stochastic control problems in continuous time
${\cal H}_\infty$ Control of Nonlinear Systems: Differential Games and Viscosity Solutions
SIAM Journal on Control and Optimization
Convergence Analysis for a Class of High-Order Semi-Lagrangian Advection Schemes
SIAM Journal on Numerical Analysis
Extending H∞ control to nonlinear systems: control of nonlinear systems to achieve performance objectives
A Max-Plus-Based Algorithm for a Hamilton--Jacobi--Bellman Equation of Nonlinear Filtering
SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
SIAM Journal on Numerical Analysis
Second Order Numerical Methods for First Order Hamilton--Jacobi Equations
SIAM Journal on Numerical Analysis
Max-Plus Eigenvector Methods for Nonlinear H$_\infty$ Problems: Error Analysis
SIAM Journal on Control and Optimization
An efficient algorithm for Hamilton–Jacobi equations in high dimension
Computing and Visualization in Science
A Curse-of-Dimensionality-Free Numerical Method for Solution of Certain HJB PDEs
SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
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In previous work of the first author and others, max-plus methods have been explored for solution of first-order, nonlinear Hamilton-Jacobi-Bellman partial differential equations (HJB PDEs) and corresponding nonlinear control problems. Although max-plus basis expansion and max-plus finite-element methods can provide substantial computational-speed advantages, they still generally suffer from the curse-of-dimensionality. Here we consider HJB PDEs where the Hamiltonian takes the form of a (pointwise) maximum of linear/quadratic forms. The approach to solution will be rather general, but in order to ground the work, we consider only constituent Hamiltonians corresponding to long-run average-cost-per-unit-time optimal control problems for the development. We consider a previously obtained numerical method not subject to the curse-of-dimensionality. The method is based on construction of the dual-space semigroup corresponding to the HJB PDE. This dual-space semigroup is constructed from the dual-space semigroups corresponding to the constituent linear/quadratic Hamiltonians. The dual-space semigroup is particularly useful due to its form as a max-plus integral operator with kernel obtained from the originating semigroup. One considers repeated application of the dual-space semigroup to obtain the solution. Although previous work indicated that the method was not subject to the curse-of-dimensionality, it did not indicate any error bounds or convergence rate. Here we obtain specific error bounds.