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Computing the Positive Stabilizing Solution to Algebraic Riccati Equations With an Indefinite Quadratic Term via a Recursive Method

This paper solves a problem that had remained open in the literature for the past 40 years by providing a recursive algorithm for finding the stabilising solution to a class of algebraic Riccati equations. This work is underpinning research elsewhere on new numerical methods to solve Hamilton-Jacobi-Bellman-Isaac equations (http://dx.doi.org/10.1016/j.automatica.2008.11.006), which are fundamental in many optimal nonlinear problems, and were previously numerical intractable (http://dx.doi.org/10.1002/rnc.2814). It also is the foundation of extensions to stochastic systems (http://dx.doi.org/10.1007/s11075-010-9432-7), periodic systems (http://dx.doi.org/10.1109/TAC.2010.2101710), game theory mini-max problems (http://dx.doi.org/10.1016/j.ins.2012.08.012, http://dx.doi.org/10.1002/acs.2348), and finding solutions to Lur'e matrix equations (http://dx.doi.org/10.1137/120861679).