Algebraic Ricatti Equation

A type of nonlinear equation that arises in the context of infinite-horizon optimal control problems in continuous or discrete time.

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Continuous time algebraic Riccatti equation

$$A^{T}P+PA-PB{R}^{-1}{B}^{T}P+Q=0$$

Where $P$ is an unknown $n\times n$ Positive-Definite Matrix

Positive-Definite Matrix

A symmetric matrix with all positive eigenvalues. NOTE: All symmetric matrices have only real eigenvalues

For a matrix, $M$, to be positive-semidefinite the following has to hold

Criterion

$x^{T}Mx>0,\forall x\in {\mathbb{R}}^n\backslash \{0\}$ Given that $M$ is a $n\times n$ symmetric real matrix

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Can be written generally as

$$M=\left[\begin{array}{cc}{m}_1& m_2\\ m_2& m_3\end{array}\right]$$Link to original