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.

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

the continuous time algebraic Riccati equation (CARE):

$${\displaystyle A^{\top }P+PA-PBR^{-1}B^{\top }P+Q=0}$$

or the discrete time algebraic Riccati equation (DARE):

$${\displaystyle P=A^{\top }PA-(A^{\top }PB)(R+B^{\top }PB)^{-1}(B^{\top }PA)+Q.}$$

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

Though generally this equation can have many solutions, it is usually specified that we want to obtain the unique stabilizing solution, if such a solution exists.