Time delay systems with distribution dependent dynamics

Abstract

General delay dynamical systems in which uncertainty is present in the form of probability measure dependent dynamics are considered. Several motivating examples arising in biology are discussed. A functional analytic framework for investigating well-posedness (existence, uniqueness and continuous dependence of solutions), inverse problems, sensitivity analysis and approximations of the measures for computational purposes is surveyed.

Sensitivity Functions and Their Uses in Inverse Problems

“…In recent years however, due to an increasing interest in incorporating uncertainty into models and in ascertaining the sensitivity of parameter estimates with respect to data measurements, the uses of sensitivity have broadened significantly [7,26]. In one direction, sensitivity of systems with probability measures embedded in the dynamics (problems involving aggregate dynamics) have become important in applications in biology, electromagnetics, and hysteretic and polymeric materials (see [7,8] and the references therein). On the other hand, investigators' attention has also recently turned to the sensitivity of the solutions to inverse problems with respect to data, in a quest for optimal selection of data measurements in experimental design.…”

“…North Carolina State University,Center for Research in Scientific Computation,Raleigh,NC,27695-8205 8. PERFORMING ORGANIZATION REPORT NUMBER…”

Inverse Problems, Control and Modeling in the Presence of Uncertainty

“…As part of our ongoing study of estimation of function space parameters in complex nonlinear systems, we have also developed a sensitivity theory [13,14,27] for systems depending explicitly on functions and probability measures.…”

“…From the sensitivity analysis theory for dynamical systems (numerous references are given in [13,14,27]), we have that s = (so,,..., so,) is an N x p vector function that satisfies the ODE system…”

A brief review of some approaches to hysteresis in viscoelastic polymers

“…For the latter there is a growing body of literature [11,12,13,45,54,68] on the Preisach and related theories for hysteretic control input such as arises in smart material systems [27,39,64]. Applications in which delays and hysteresis play a basic role in the underlying dynamics include sustained efforts in biology with early efforts employing delay systems [1,2,16,17,28,29,30,31,32,42,51,53,55,62] and more recent investigations involving hysteretic probabilistic structures [5,7] as well as classical materials and electromagnetics research (see [8] and the references therein). These applications drove a substantial amount of mathematical and computational research on hysteretic systems in the last half of the 20th century, e.g., see [26,40,41,46,47] among the many books, research monographs, and research articles written.…”