The optimality of the velocity-gradient method in the problem of controlling the escape from a potential well

The problem of controlling the escape of a particle from a potential well for a nonlinear system with friction is considered. The velocity-gradient method [Polushin IG, Fradkov AL, Hill, D. Passivity and passivation in non-linear systems. Avtomatika i Telemekhanika 2000;3:3–37] is proved to be optimal in the sense that if it does not guarantee escape from the well, then this is also impossible with any other control law. Nonlinear Duffing and Helmholtz oscillators with one degree of freedom and negative stiffness are considered. For each of them a curve is constructed separating the parameter plane of the problem into two parts: one where escape is feasible and one where it is not. An estimate is obtained for the inclination angle of the tangent to that curve near the origin.