Motivation

Assume that a device is described by an ordinary differential equation dx/dt = f(x) with an asymptotically stable "nominal" equilibrium x*. Ageing of the device can be described by a slowly time-dependent equation of the form

dx/dt = f(x, λ(εt)),

where f(x, λ₀) = f(x). The function f(x, λ) vanishes on a curve x*(λ), called the "nominal equilibrium curve".

As long as the nominal equilibrium remains asymptotically stable, it is known that the orbits will follow it adiabatically, so that the ageing device will function close to the indented regime. If, however, the nominal equilibrium undergoes a bifurcation (say at λ = 0), several things may happen:

• the state of the system may track a new stable equilibrium curve originating from the bifurcation point (immediate exchange of stabilities);
• the state may jump to another attractor;
• the state may follow the nominal equilibrium, which has become unstable, for some time, before jumping on a stable equilibrium state (delayed exchange of stabilities).

In the first situation, one can detect the fact that the system has reached the stability boundary of the nominal equilibrium, and turn off the device before any damage occurs. In the other situations, however, the jump may have disastrous consequences for the device.

One can remedy this situation by adding a feedback control to the system. A simple affine scalar feedback control would be

dx/dt = f(x, λ(ε t)) + b u(x, λ(εt)),

where b is an imposed vector (the direction in which the system can be steered), and u(x, λ) is the feedback to be designed in such a way as to provoke an immediate exchange of stabilities.