Stability of the Nonwandering Set in the Region of Attraction Boundary under Perturbations with Application to Vulnerability Assessment

SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3390-3430, December 2023.
Abstract. For many engineered systems it is important to assess vulnerability to potential disturbances in order to ensure reliable operation. Whether the system will recover from a particular finite-time disturbance to a desired stable equilibrium point depends on uncertain and time-varying system parameter values. Therefore, it is valuable to determine, for specific fixed disturbances, the margins for safe operation: the smallest change in parameter values that would cause the system to become vulnerable to the disturbance. The natural setting for this problem is a parameter-dependent vector field with a family of stable equilibria and a parameter-dependent initial condition representing the disturbance. The system recovers for a particular parameter value if its initial condition lies within the region of attraction of the desired stable equilibrium point. Prior work has developed algorithms for numerically computing the margins for safe operation. However, the theoretical guarantees provided for these methods require a very restrictive assumption: that the nonwandering set in the region of attraction boundary is stable under perturbations to the vector field. This assumption is generally intractable to verify, so feasibility of the above algorithms cannot be determined in advance, and even when these algorithms do converge their convergence to the correct values cannot be guaranteed. Thus, this assumption limits the effective application of these algorithms in practice. This work relaxes this restrictive assumption while still obtaining similar results under weaker assumptions, thereby guaranteeing effectiveness of these algorithms. For the setting under consideration, it is shown for vector fields on compact Riemannian manifolds that the restrictive assumption follows immediately and does not need to be independently verified. A motivating example shows that this is not the case for vector fields on Euclidean space, but in this setting it is shown that the restrictive assumption can still be relaxed provided there exist a neighborhood of infinity with suitable properties and some additional generic assumptions. These results are then used to provide theoretical guarantees for the numerical algorithms discussed above under far weaker assumptions.