Monotonicity and convexity properties of matrix Riccati equations (Q2711459)

Given a triple of complex \(n\times n\) matrices \((A,S,Q)\), where \(S\) and \(Q\) are Hermitian, the following equations are considered:NEWLINENEWLINENEWLINE(1) The continuous time algebraic Ricci equation \(A^*X+ XA+ Q- XSX= 0\);NEWLINENEWLINENEWLINE(2) The discrete time algebraic Riccati equation \(A^*X(I+ SX)^{-1}A+ Q- X= 0\);NEWLINENEWLINENEWLINE(3) The initial value problem \(\dot X+ A(t)^* X+ XA(t)+ Q(t)- XS(t)X= 0\), \(X(t_0)= X_0= X^*_0\) for the Riccati differential equation;NEWLINENEWLINENEWLINE(4) The terminal value problem \(A^*(k) X(k+ 1)[I+ S(k)X(k+ 1)]^{-1}Q(k)- X(k)= 0\), \(X(k_0)= X_0= X^*_0\) for the Riccati difference equation.NEWLINENEWLINENEWLINEIt is assumed for the initial value problem that \((A,S,Q)\) is a piecewise continuous function of \(t\) on some interval, and for the terminal value problem it is assumed that \((A,S,Q)\) is a function of the integer valued variable \(k\). Using an approach based on Fréchet derivatives and implicit function theorems, some monotonicity, convexity/concavity, and comparison results are proved for strictly unmixed solutions of the algebraic Riccati equations. It is proved that the solutions of the initial and the terminal value problems are smooth and monotonic functions of the input data and of the initial (or terminal) value; they are also convex or concave functions with respect to certain matrix coefficients.