Combining Legendre/Fenchel transformed operators on max-plus spaces for nonlinear control solution

Max-plus methods have been explored for solution of first-order, nonlinear Hamilton-Jacobi-Bellan partial differential equations (HJB PDEs) and corresponding nonlinear control problems. Specific applications to date have been to H_∞ control and estimation. These methods exploit the max-plus linearity of the associated semigroups. In particular, although the problems are nonlinear, the semigroups are linear in the max-plus sense. These methods have been used successfully to compute solutions. Although they may be faster than more traditional methods, they still generally suffer from the curse of dimensionality - only the coefficient in the exponential rate of computational growth is reduced. There are natural spaces over the max-plus algebra in which to express solutions of HJB PDEs. The natural analog to the Laplace transform in ordinary spaces is the Legendre/Fenchel transform over max-plus spaces, the range space being referred to as the dual space. One can transform the semigroup operators into operators on the dual space. There are natural operations on the transformed operators which may be used to construct solutions of nonlinear control problems. In this paper, a method for exploiting operations in the Legendre/Fenchel transform space is used to develop a method for certain problems where the computational growth in the most time-consuming portion of the computations can be hugely reduced. One has computational growth which is linear in a certain measure of problem complexity for which linear/quadratic problems have the minimal complexity. Although the curse of dimensionality is unavoidable, the computational cost reductions are very high for some classes of problems.