Decision and Control Lecture Series
Coordinated Science Laboratory
“Controlled Equilibrium Selection in Stochastically
Perturbed Dynamics Under Critical Noise”
Aristotle Arapostathis, Ph.D.
University of Texas at Austin
Wednesday, December 7, 2016
3:00 p.m. to 4:00 p.m.
CSL Auditorium (B02)
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 that 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 that there is a trade-off. We study the optimal stationary distribution of the controlled process as the variance of the noise becomes vanishingly small. It is shown that depending on the relative magnitudes of the noise variance and the `running cost' for control, one can identify three regimes, in each of which the optimal control forces the invariant distribution of the process to concentrate near equilibria that can be characterized according to the regime. We also obtain moment bounds for the optimal stationary distribution. Moreover, we show that in the vicinity of the points of concentration the density of optimal stationary distribution approximates the density of a Gaussian, and we explicitly solve for its covariance matrix.
Prof. Aristotle Arapostathis is a Professor at the Department of Electrical and Computer Engineering at the University of Texas at Austin. His current research interests are in stochastic control, large-scale stochastic networks, and jump diffusion processes. He is an IEEE Fellow and he was an Associate Editor of the IEEE Transactions on Automatic Control and the Journal of Mathematical Systems and Control.