Stabilization of the equilibria of dynamical systems

Based on Rumyantsev's method, a procedure is developed for stabilizing stable and unstable equilibria of dynamical systems by continuous and modulus-constrained control actions. It is shown that, when modulus constraints are imposed on the controls, and when the quadratic-form coefficients are reduced in modulus, the optimal stabilization, in terms of this method, approximates to time-optimal stabilization. A solution is obtained of the problem of stabilizing unstable equilibrium positions at which the potential energy of the system has neither a maximum nor a minimum (or, in particular, at which the potential energy is identically equal to zero).