Matrix Riccati differential equations on time scales (Q2782939)

The author considers the Riccati matrix dynamic equation NEWLINE\[NEWLINER^\Delta(t)=A(t)-R(\sigma(t))R(t) \tag{1}NEWLINE\]NEWLINE on an arbitrary time scale \({\mathbb T}\) (a nonempty closed subset of \(\mathbb{R}\)), which is associated to the second-order linear dynamic equation \(X^{\Delta\Delta}-A(t)X(t)=0\) via the Riccati substitution \(R(t)=X^\Delta(t)X^{-1}(t)\). This includes as special cases the differential (\(\mathbb{T}=\mathbb{R}\)) and difference (\(\mathbb{T}=\mathbb{Z}\)) equations. A quasi-linearization technique is applied to (1) to obtain an auxiliary Lyapunov-type linear dynamic equation, which is solved by the variation of parameters formula. In the last section, the author then obtains iterative approximations to the auxiliary Lyapunov-type equation. Furthermore, it is shown that the solutions to these approximate equations form a monotone decreasing sequence, whose lower bound is the solution \(R(t)\) to the Riccati equation (1). NEWLINENEWLINENEWLINEThe paper will be useful for researchers interested in linear dynamic equations on time scales (measure chains) and/or in Riccati matrix equations.