On state transformation and stability analysis of distributed-parameter systems

Abstract

Some time ago Slemrod [1] presented an extension of LaSalle's invariance principle [2,3,4] to distributed-parameter systems. As an example he considered van der Pol's nonlinear partial differential equationwhere u = u(x, t), u = du/dt, ux = du/dx, e > 0, and the boundary conditions are m(0, t) = u{ 1, t) -0, t > 0.In order to study the stability properties of a formal partial differential equation by topological (Liapunov) methods, it is necessary to replace the formal equation by an evolution equation which describes the time evolution of a quantity called the state in a metric space called the state space for all time t > 0. In