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.

Modeling and stability analysis of HIV-1 as a time delay fuzzy T–S system via LMIs

“…Time delay models have been the target of different controller design approaches such as the ones presented in [45] [45], [47]- [51] [50].…”

Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations

“…Banks, Banks, and Joyner [9] present a mathematical and statistical framework involving sensitivity for delay systems arising in models for the delayed action of sublethal insecticides in a recent ecological application. None of these presentations give rigorous proofs on the existence of the various Frechet derivatives that define the sensitivity equations, but the authors of [9] cite some initial theoretical foundations presented in [18,19], which rely on an abstract theoretical framework presented in [25]. Finally, the efforts in [23] on the dynamics of behavior change in problem drinkers is a modern application in psychology that motivates the need of sensitivity functions for discrete as well as cumulative delayed effects as represented in longitudinal data for patients undergoing therapy.…”

“…In [25] Banks and Nguyen provide rigorous theoretical sensitivity results for the DDE example for HIV dynamics with measure dependent or distributional parameters given in [10]; however they only present results for the the sensitivity with respect to absolutely continuous probability distributions for the delay. In subsequent efforts [18,19] a rigorous theoretical foundation is developed for sensitivity theory using directional derivatives where the parameter space M is taken as the convex metric space of probability measures (including discrete, continuous or convex combinations thereof) taken with the Prohorov metric topology [8]. Below we give new results for sensitivity with respect to discrete delays.…”

“…Below we give new results for sensitivity with respect to discrete delays. The proofs, while quite tedious, continue with an adaption of the well known ideas for existence and uniqueness of the Frechèt derivative with respect to the delay in nonlinear DDE as employed in [10,18,19,25].…”

“…While we consider here the case of finite dimensional model parameters, similar sensitivity results also hold in a more general case when parameters and delays are distributed, and hence infinite dimensional. Some of these results for infinite dimensional systems are presented in [18,19]. Once established, these results allow us to study traditional and generalized sensitivity functions, where sensitivity is considered with respect to these three quantities (delays, initial conditions, and parameters).…”