Max-plus eigenvector representations for nonlinear H∞ value functions

The H_∞ 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.