We construct a family of globally defined dynamical systems for a nonlinear programming problem, such that (a) the equilibrium points are the unknown (and sought) critical points of the problem, (b) for every initial condition, the solution of the corresponding initial value problem converges to the set of critical points, (c) every strict local minimum is locally asymptotically stable, (d) the feasible set is a positively invariant set, and (e) the dynamical system is given explicitly and does not involve the unknown critical points of the problem. No convexity assumption is employed. The construction of the family of dynamical systems is based on an extension of the control Lyapunov function methodology, which employs extensions of LaSalle's theorem and are of independent interest. Examples illustrate the obtained results.