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.

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

the continuous time algebraic Riccati equation (CARE):

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

or the discrete time algebraic Riccati equation (DARE):

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

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.