Martin V. Day

On the exponential exit law in the small parameter exit problem

We consider the diffusion dw in a domain D which contains a unique asymptotically stable critical point of the ODE . Using probabilistic estimates we prove the following: 1) The Principle eigenfunction of the differential generator for tghe process x(t converges to a constant as ∊→0, boundedly in D and uniformly on compacts. 2) If τ D is the exit time of x(t) from D, then λτ D converges in distribution to an exponential random variable with mean 1.(λ is the principle eigenvalue). Both of these results were known previosuly in the special case of a gradient flow: . Our arguments apply in the general non-gradient case.

Recent progress on the small parameter exit problem

In this paper we will bring together some recent results of S.-J. Sheu, T.A. Darden and ourselves and develop their application to the small parameter exit problem of A.D. Wentzell and M.I. Freidlin. This problem concerns the asymptotic behavior of the exit distribution from a domain of attraction for an exponentially stable critical point of a dynamical system with an asymptotically small random perturbation. Recent results of Day and Darden on regularity properties of the so-called quasipotential function allow certain improvements and generalizations to be made in the work fo S.-J. Sheu on the asyptotic behavior of the equilibrium density. Applying these results to the exit problem through its connection with the equilibrium density, [2], we obtain a new theorem on the exit problem: Theorem 4 below. This theorem subsumes previous results and generalizes the conclusion of the Matkowksi-Schuss-Kamin approach from smooth to nonsmooth quasipotental functions. In all cases the exit problem is reduced to the...

Robust feedback control of a single server queueing system

Abstract. This paper extends previous work of Ball et al. [BDKY] to control of a model of a simple queueing server. There are n queues of customers to be served by a single server who can service only one queue at a time. Each queue is subject to an unknown arrival rate, called a “disturbance” in accord with standard usage from H∞ theory. An H∞-type performance criterion is formulated. The resulting control problem has several novel features distinguishing it from the standard smooth case already studied in the control literature: the presence of constraining dynamics on the boundary of the state space to ensure the physical property that queue lengths remain nonnegative, and jump discontinuities in any nonconstant state-feedback law caused by the finiteness of the admissible control set (choice of queue to be served). We arrive at the solution to the appropriate Hamilton–Jacobi equation via an analogue of the stable invariant manifold for the associated Hamiltonian flow (as was done by van der Schaft for the smooth case) and relate this solution to the (lower) value of a restricted differential game, similar to that formulated by Soravia for problems without constraining dynamics. An additional example is included which shows that the projection dynamics used to maintain nonnegativity of the state variables must be handled carefully in more general models involving interactions among the different queues. Primary motivation comes from the application to traffic signal control. Other application areas, such as manufacturing systems and computer networks, are mentioned.

Large deviations results for the exit problem with characteristic boundary

Abstract We consider the exit problem for an asymptotically small random perturbation of a stable dynamical system in a region D. We show the standard large deviations results for the exit distribution and mean exit time, as obtained by Wentzell and Freidlin under the assumption of nontangential drift 〈b, n〉

Some regularity results on the Ventcel-Freidlin quasi-potential function

We consider regularity properties of the quasipotential function V defined by A. D. Ventcel and M. I. Freidlin in their work on asymptotically small random perturbations of stable dynamical systems. The regularity properties of V are important for the success of various asymptotic calculations carried out in the literature. Employing classical techniques from the calculus of variations and differential equations, we prove various results about the smoothness of V and its level sets. Among other things, there exists a dense connected open set, containing the stable point for the underlying dynamical system, in which V is continuously differentiable to the same degree as the Lagrangian involved in the defining variational problem.

On the exit law from saddle points

We consider the effects of adding an asymptotically small random (Brownian) perturbation to a planar dynamical system with a saddle point equilibrium. By applying techniques developed for the problem of exit from a stable equilibrium, we obtain a new limit law for the exit time from a neighborhood of the saddle, assuming the initial point is on the stable manifold. The limit law shows that the exit distribution depends on (the logarithm of) the noise parameter in an additive way. This gives a more accurate description of the exit law than the previous (but more general) results of Kifer and Mikami. Generalization to higher dimensions seems likely, although only if the unstable manifold has dimension 1.

Cycling and skewing of exit measures for planar systems

We apply the exit conditioning approach to study two particular phenomena of the characteristic boundary exit problem: cycling of exit measures on periodic boundaries and skewing of exit measures at saddle points on the boundary. Sufficient conditions are given for each phenomenon to occur. Some results about the quasipotential function of Wentzell and Freidlin are established for characteristic boundaries in general.

Boundary local time and small parameter exit problems with characteristic boundaries

The exit problem for an asymptotically small random perturbation of a stable dynamical system

Exponential Leveling for Stochastically Perturbed Dynamical Systems

x(t)

Exit cycling for the Van der Pol oscillator and quasipotential calculations

in a region D is considered. The connection between the distribution of the position of first exit and the equilibrium density of the perturbed system subject to reflection from the boundary of D is developed. Earlier work treated the case in which