Stability Criteria for Second-Order Dynamical Systems With Time Lag

Stability of second-order dynamical systems with time lag is investigated by using Pontrjagin’s theorems on the zeros of exponential polynomials. The time-lag term may involve the displacement, the velocity, or the acceleration. Systems with negative damping coefficient and/or negative spring constants are also considered. It is shown that a delayed feedback signal of proper strength and proper delay is capable of stabilizing such dynamical systems with negative damping and/or negative spring constants. It is also found that some earlier results given in [2] for the restricted case where the damping coefficient is positive and the spring constant is non-negative are defective. The stability criteria obtained here are expressed in terms of inequalities which impose upper and lower bounds for the system parameters.