Dynamical systems with multiple equilibria have been of great interest in both physics and engineering. A 'selection principle' to tag one or more equilibria as 'natural' is based on adding a small noise to perturb the system and then checking where its stationary behavior concentrates. Another related concern, particularly in statistical physics, is metastability, where gradient or gradient-like systems exhibit large time quasi-stationary behavior near 'sub-optimal' equilibria, e.g., local minima in the former case. Yet another interesting line of work is noise-induced transitions between equilibria, e.g., in 'stochastic resonance'. A beautiful theoretical basis for all this is provided by the Freidlin-Wentzell theory, by now a cornerstone of applied probability. Our attempt here is to introduce a control which can further modulate the dynamics, giving an extra degree of freedom. For the problem to be interesting, the control has to be 'expensive' in a precise sense, so tha t there is a trade-off. We analyze this situation and interestingly, get three different regimes depending on the scale factor of the noise.
Prof. Aristotle Arapostathis is Texas Atomic Energy Research Foundation Centennial Fellow and Professor of Electrical Engineering at University of Texas, Austin. He obtained his BS from MIT, and MS and PhD degrees from University of California, Berkeley. He has contributed copiously to stability analysis of power systems, adaptive control, hybrid control, and stochastic control. He is an IEEE Fellow.