Criterion for the convergence of the solution of the Riccati differential equation

The optimal control problem for a linear system with a quadratic cost function leads to the matrix Riccati differential equation. The convergence of the solution of this equation for increasing time interval is investigated as a function of the final state penalty matrix. A necessary and sufficient condition for convergence is derived for stabilizable systems, even if the output in the cost function is not detectable. An algorithm is developed to determine the limiting value of the solution, which is one of the symmetric positive semidefinite solutions of the algebraic Riccati equation. Examples for convergence and nonconvergence are given. A discussion is also included of the convergence properties of the solution of the Riccati differential equation to any real symmetric (not necessarily positive semidefinite) solution of the algebraic Riccati equation.