Conjugate points, triangular matrices, and Riccati equations

Abstract:

Let $A$ be a real continuous $n \times n$ matrix on an interval $\Gamma$, and let the $n$-vector $x$ be a solution of the differential equation $x’ = Ax$ on $\Gamma$. If $[\alpha ,\beta ] \in \Gamma ,\beta$ is called a conjugate point of $\alpha$ if the equation has a nontrivial solution vector $x = ({x_1}, \ldots ,{x_n})$ such that ${x_1}(\alpha ) = \ldots = {x_k}(\alpha ) = {x_{k + 1}}(\beta ) = \ldots = {x_n}(\beta ) = 0$ for some $k \in [1,n - 1]$. It is shown that the absence on $({t_1},{t_2})$ of a point conjugate to ${t_1}$ with respect to the equation $x’ = Ax$ is equivalent to the existence on $({t_1},{t_2})$ of a continuous matrix solution $L$ of the nonlinear differential equation $L’ = {[L{A^ \ast }{L^{ - 1}}]_{{\tau _0}}}L$ with the initial condition $L({t_1}) = I$, where ${[B]_{{\tau _0}}}$ denotes the matrix obtained from the $n \times n$ matrix $B$ by replacing the elements on and above the main diagonal by zeros. This nonlinear equation—which may be regarded as a generalization of the Riccati equation, to which it reduces for $n = 2$—can be used to derive criteria for the presence or absence of conjugate points on a given interval.