On positive definite solutions to the algebraic Riccati equation
The authors study positive definite solutions of the algebraic Riccati equation (ARE) \(\bar AP+PA-PB\bar BP+\bar CC=0,\) where A, B, C are respectively \(n\times n\), \(n\times m\) and \(p\times n\) real matrices and \(\bar X\) denotes the transpose of X. The discussion is only on the real symmetric solution P. The authors first show that if a positive definite solution exists, the system can be decomposed in such a way as to separate out the unstabilizable, undetectable part from the remainder, which is stabilizable. (These properties are those of the optimal control problem for linear time invariant system associated with the ARE.) Finally, necessary and sufficient conditions are provided for the existence of a positive definite solution.
scientific article; zbMATH DE number 3946358