From Max-plus Algebra to Non-linear Perron-frobenius Theory

The max-plus (or tropical) algebra is obtained by replacing the addition by the maximisation (or the minimisation) and the multiplication by the addition. It arises in the dynamic programming approach to deterministic optimal control. In particular , the evolution semigroup of a first order Hamilton-Jacobi equation is linear in the max-plus sense if the Hamiltonian is convex in the adjoint variable. This observation has been made by several authors, including Maslov. It has motivated the development of a max-plus analogue of the theory of linear operators or linear semi-groups, including spectral theory. The max-plus eigenvalue gives the optimal mean payoff per time unit and the eigenvectors can be used to parametrise the stationary optimal strategies. Stochastic control or zero-sum game problems yield more general dynamic programming operators, which are no longer max-plus linear, but preserve the order and have some homogeneity or nonexpansiveness properties. Such operators belong to non-linear Perron-Frobenius theory, which allows one to include in a common perspective the positive linear operators and the max-plus linear ones. The goal of this talk is to present this perspective, which has inspired some results of non-linear spectral theory, has suggested algorithms, and has been helpful in applications. The max-plus spectral theorem allows one to represent any eigenvector as a linear combination of extremal eigenvectors. In the finite dimensional case, the extremal eigenvectors correspond to " recurrence classes " ; in the case of a noncompact state space, extremal eigenvectors correspond either to recurrence classes or to the limits of geodesics leading off to infinity (Busemann points) [1],[2]. This is reminiscent of the Martin representation of harmonic functions, and this is related to the representation of weak-KAM solutions of Fathi in terms of the Aubry set, although the setting is different.