Algebraic Riccati equation

An algebraic Riccati equation is a nonlinear matrix equation that arises in infinite-horizon optimal control problems, in either a continuous-time or a discrete-time setting. In continuous time, the equation seeks a symmetric matrix P satisfying A transpose times P, plus P times A, minus P times B times R inverse times B transpose times P, plus Q, equals zero; the discrete-time version has an analogous form built from the same system matrices. Algebraic Riccati equations are central to control theory because solving them yields the solution to the infinite-horizon, time-invariant linear-quadratic regulator problem and to linear-quadratic-Gaussian control, both of which balance the immediate cost of control action against the cost of letting a dynamic system's state deviate from a target over an unbounded time horizon.