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Properties of the solutions of rational matrix difference equations. (English) Zbl 1055.39033
Consider the matrix difference equation $$\multline X_t=A^*X_{t+1}A+Q+\Pi_1(X_{t+1})-[S+A^*X_{t+1}B+ \Pi_{12}(X_{t+1})]\times\\ \times[R+B^*X_{t+1}B+\Pi_2(X_{t+1})]^+[S+A^*X_{t+1} B+\Pi_{12}(X_{t+1})]^*\endmultline\tag A$$ and the corresponding algebraic equations $$\multline A^*XA-X+Q+\Pi_1(X)-[S+A^*XB+\Pi_{12}(X)]\times\\ \times [R+B^*XB+\Pi_2(X)]^+[S+A^*XB+\Pi_{12}(X)]^*= 0.\endmultline\tag B$$ Sufficient conditions for the existence and uniqueness of stabilizing solutions for (A) are established. Finally, it is shown that under certain conditions on the coefficients the solutions of (A) converge for any positive value to the stabilizing solutions of (B).

39A20Generalized difference equations
39A70Difference operators
Full Text: DOI
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