Geometric Algorithms For The Coupled Algebraic Riccati Equations

Since the aim of linear-quadratic game is to obtain the Nash equilibrium con-trol strategy restricted by the coupled algebraic Riccati equations (CAREs), it is veryimportant to provide efective solutions for the CAREs. Geometric algorithms areproposed to solve the CAREs numerically, including the Euclidean gradient algorithm(EGA), the Riemannian gradient algorithm (RGA) and the extended Hamiltonian al-gorithm (EHA). The EGA is built on the foundation of the classical Euclidean distance,while the RGA and the EHA are constructed in virtue of the Riemannian structures ofmanifolds consisted by all symmetric-positive defnite matrices. The efciencies of thesealgorithms are compared. Furthermore, two non-zero-sum linear-quadratic games areconsidered to demonstrate the efectiveness of the algorithms. Among all the discussedalgorithms, simulations show that the EHA is the fastest one.