Asymptotic stability criteria

On state transformation and stability analysis of distributed-parameter systems

“…However T does not have an inverse, and therefore (3) may have a "richer" set of motions than does (2). Thus the conclusion of stability of the equilibrium of (2) can not be carried over to (3) automatically, and would not be meaningful even if it were carried over since the relevant quantity ||T-1(w, m)||« does not exist.…”

“…

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 equation

where 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.

…”

“…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 equation…”

“…In [1] Eq. (1) was replaced by1 u = -vx -e(u -u3 / 3), i) = -ux (2) with the state (u, v) £ (B for all t > 0. A suitable state space (B was assumed to be ® = 02'(0, 1) X W21(0, 1), where the Sobolev spaces and W02t(S2) are as defined in [1], implying the natural norm were then considered [1].…”

“…Such a restriction appears to be too strong, as we shall see by example. Noting that (3) appears to be a better representation of (1) than does (2), we shall perform a stability analysis of (3) and show that the results and analysis for (3) can be easily carried over to (2), even though the converse operation was not possible.…”