We introduce a max-plus analogue of the Petrov-Galerkin finite element method to solve finite horizon deterministic optimal control problems. The method relies on a max-plus variational formulation. We show that the error in the sup-norm can be bounded from the difference between the value function and its projections on max-plus and min-plus semimodules when the max-plus analogue of the stiffness matrix is exactly known. In general, the stiffness matrix must be approximated: this requires approximating the operation of the Lax-Oleinik semigroup on finite elements. We consider two approximations relying on the Hamiltonian. We derive a convergence result, in arbitrary dimension, showing that for a class of problems, the error estimate is of order $\delta+\Delta x(\delta)^{-1}$ or $\sqrt{\delta}+\Delta x(\delta)^{-1}$, depending on the choice of the approximation, where $\delta$ and $\Delta x$ are, respectively, the time and space discretization steps. We compare our method with another max-plus based discretization method previously introduced by Fleming and McEneaney. We give numerical examples in dimensions 1 and 2.

https://hal.inria.fr/inria-00071395/document

The Max-Plus Finite Element Method for Solving Deterministic Optimal Control Problems: Basic Properties and Convergence Analysis

A new fundamental solution for differential Riccati equations arising in control

The H/sub /spl infin// problem for a nonlinear system is considered. The corresponding dynamic programming equation is a fully nonlinear, first-order, steady-state partial differential equation (PDE), possessing a term which is quadratic in the gradient. The solutions are typically nonsmooth, and further, there is nonuniqueness among the class of viscosity solutions. In the case where one tests a feedback control to see if it yields an H/sub /spl infin// controller, the PDE is a Hamilton-Jacobi-Bellman equation. In the case where the "optimal" feedback control is being determined as well, the problem takes the form of a differential game, and the PDE is, in general, an Isaacs equation. The computation of the solution of a nonlinear, steady-state, first-order PDE is typically quite difficult. In this paper, we develop an entirely new class of methods for obtaining the "correct" solution of such PDEs. These methods are based on the linearity of the associated semigroup over the max-plus (or, in some cases, min-plus) algebra. In particular, solution of the PDE is reduced to solution of a max-plus (or min-plus) eigenvector problem for known unique eigenvalue 0 (the max-plus multiplicative identity). It is demonstrated that the eigenvector is unique, and that the power method converges to it. An example is included.

Max-plus eigenvector representations for solution of nonlinear H/sub /spl infin// problems: basic concepts

Error Analysis of a Max-Plus Algorithm for a First-Order HJB Equation

Max-plus summation of Fenchel -transformed semigroups for solution of nonlinear Bellman equations

The H/sub /spl infin// problem for nonlinear systems is considered. The corresponding dynamic programming equation is a fully nonlinear, first-order, partial differential equation. Interestingly, if one switches from the normal definition of addition and multiplication to the max-plus algebra (which is no more complex), the solution operator becomes a linear operator. The solution can be expanded using a max-plus basis. The coefficients in this expansion satisfy a max-plus eigenvector equation for a matrix associated with this solution operator-thus transforming the nonlinear problem into a linear one. In fact there is a parameterized family of matrices for which this holds. Expressions and approximations for the coefficients in these matrices are given.

W.M. McEneaney

Papers

Max-plus eigenvector representations for nonlinear H/sub /spl infin// value functions