Accounting for nonlinearities in open-loop protocols for symmetry fault compensation

In this paper, we consider model examples of dynamical systems with only a few degrees of freedom, and with desirable symmetry properties, and explore compensating control strategies for retaining robust symmetric system response even under symmetry-breaking defects. The analysis demonstrates the distinct differences between linear versions of these models, in which fault-compensating strategies are always found, and weakly nonlinear counterparts with varying degrees of asymmetry, for which a multitude of locally optimal solutions may coexist. We further formulate a candidate optimization protocol for fault compensation applied to self-healing systems, which respond to symmetry-breaking defects by a continuous process of fault correction. The analysis shows that such a protocol may exhibit discontinuous changes in the control strategy as the self-healing system successively regains its original symmetry properties. In addition, it is argued that upon return to a symmetric configuration, such a protocol may result in a different control strategy from that applied prior to the occurrence of a fault.