Riccati matrix differential equations and oscillation theory

Abstract

Chapter 1. A Survey of Classical Oscillation Theory. Starting with the work of Sturm [39], some of the developments in this area are treated. One section deals with the most modern work on the second order equation, and some time is devoted to equations of higher order. It is also shown that all of this theory is a special case of first order matrix oscillation theory. Another section is given over to a discussion of the role of self-adjointness, including a physical interpretation in terms of conservation laws. Chapter 2. Matrix Boundary Value Problems of the First Order, and the Riccati Matrix Differential Equation. We begin our study of the first order boundary value problem: y'=A(x)y My(a)+Ny(b)=0, where M and N are matrices, A is a matrix function and y a vector function. Some standard properties of this system are derived for future use. It is also shown that this problem can be studied with the aid of a Riccati matrix initial value problem (equation (2.20)). In the case where y is two dimensional the Riccati equation has a simple geometrical interpretation. Chapter 3. Properties of a Riccati Matrix Differential Equation. Some properties, most of which are standard, of the matrix Riccati equation are discussed. These include the derivation of the general solution, and the effect on the solution when a parameter, upon which the coefficients depend, is varied. It is also shown that an important class of self-adjoint problems can be made to give rise to Riccati equations whose solutions are hermitean. Finally the effect on the Riccati equation of various transformations is discussed. Chapter 4. Comparison Theorems for Hermitean Riccati Equations. Various monotoneity properties of the Riccati equation's solutions are derived. Theorems 4-4 and 4-7, two comparison theorems for the hermitean Riccati equation, are proved. They are then used to establish comparison theorems for the associated self-adjoint linear problems. Two alternative approaches are used to derived theorems 4-4 and 4-7. Chapter 5. The Unit Circle Theory of Atkinson. Wherein an ingenious transformation developed by Atkinson [1] is used to reduce the self-adjoint matrix boundary value problem to a linear initial value problem, whose solution is a unitary matrix function. The resulting comparison theorems are compared with those of chapter 4. Chapter 6. Linear Equations of the Third Order. In the case of a scalar linear equation of the third order it is shown that, for certain boundary conditions, the Riccati equation degenerates to two coupled scalar equations of the first order. These are used to derive a disconjugacy criterion for the equation. This is the only non- self-adjoint problem considered. During the proof of the disconjugacy criterion a transformation is used which is capable of a geometric interpretation.