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

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. Such a method presents advantages in simplicity of the design of equilibrium strategies and yields computationally effective solution algorithms. 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 and give conditions on the existence of such stabilizing solutions for a third order polynomial. A numerical example is presented to illustrate effectiveness of the approach.