Open-loop Nash equilibrium in polynomial differential games via state-dependent Riccati equation

Abstract

This paper studies finite- as well as infinite-time horizon nonzero-sum polynomial differential games. In both cases, we explore the so-called state-dependent Riccati equations to find a set of strategies that guarantee an open loop-Nash equilibrium for this particular class of nonlinear games. We demonstrate that this solution leads the game to an ε - or quasi-equilibrium and provide an upper bound for this ε quantity. The proposed solution is given as a set of N coupled polynomial Riccati-like state-dependent differential equations, where each equation includes a p-linear form tensor representation for its polynomial part.We provide an algorithm for finding the solution of the state-dependent algebraic equation in the infinite-time case based on a Hamiltonian approach. A numerical procedure is detailed to find the solution for this set of strategies. Numerical examples are presented to illustrate the effectiveness of the approach.