# Algebraic Riccati equation

An algebraic Riccati equation is a type of nonlinear equation that arises in the context of infinite-horizon optimal control problems in continuous time or discrete time. In such a problem, one cares about the value of some variable of interest arbitrarily far into the future, and one must optimally choose a value of a controlled variable right now, knowing that one will also behave optimally at all times in the future. The optimal current values of the problem's control variables at any time can be found using the solution of the Riccati equation and the current observations on evolving state variables. With multiple state variables and multiple control variables, the Riccati equation will be a matrix equation.

A typical algebraic Riccati equation is similar to one of the following:

the continuous time algebraic Riccati equation (CARE):

$A^T X + X A - X B R^{-1} B^T X + Q = 0 \,$

or the discrete time algebraic Riccati equation (DARE):

$X = A^T X A -(A^T X B)(R + B^T X B)^{-1}(B^T X A) + Q.\,$

X is the unknown n by n symmetric matrix and A, B, Q, R are known real coefficient matrices.

The name Riccati is given to the CARE equation by analogy to the Riccati differential equation: the unknown elements of X appear linearly and in quadratic terms (but no higher-order terms). The DARE arises in place of the CARE when studying discrete time systems; it is not obviously related to the differential equation studied by Riccati.

The algebraic Riccati equation determines the solution of the infinite-horizon time-invariant Linear-Quadratic Regulator problem (LQR) as well as that of the infinite horizon time-invariant Linear-Quadratic-Gaussian control problem (LQG). These are two of the most fundamental problems in control theory.

A solution to the algebraic Riccati equation can be obtained by matrix factorizations or by iterating on the Riccati equation. One type of iteration can be obtained in the discrete time case by using the dynamic Riccati equation that arises in the finite-horizon problem: in the latter type of problem each iteration of the value of the matrix is relevant for optimal choice at each period that is a finite distance in time from a final time period, and if it is iterated infinitely far back in time it converges to the specific matrix that is relevant for optimal choice an infinite length of time prior to a final period—that is, for when there is an infinite horizon.