Convergence properties of a class of pointwise control strategies

In recent years, a number of nonlinear power system stabilization problems have been approached using techniques termed "velocity-matching" or "optimal-aiming" strategies. At each instant of time, these strategies require that the velocity of the system be adjusted through instantaneous control action such that the state moves in some "preferred" direction subject to constraints imposed by an admissible input set, the choice of a preferred direction producing the different control strategies. In this paper, we show that for any such control strategy σ and independent of preferred direction choices, there will be a controllable linear time-invariant system which is rendered unstable under σ, even if arbitrarily large inputs and nonlinear dependence of σ on its arguments is permitted.