Second-order self-adjoint differential equations using a proportional-derivative controller

Abstract

Using a differential operator modeled after a proportional-derivative controller (PD controller), linear second-order differential equations are shown to be formally self ad-joint with respect to a certain inner product and the associated self-adjoint boundary conditions. Defining a Wronskian, we establish a Lagrange identity and Abel’s formula. Several reduction-of-order theorems are given. Solutions of the second-order self-adjoint equation are then shown to be related to corresponding solutions of a first-order Riccati equation and a related quadratic functional and a Picone identity. A comprehensive roundabout theorem relating key equivalences among all these results is given, followed by a Lyapunov inequality.