An official website of the United States government
It is already well-understood that many delay differential equations with only a single constant delay exhibit a change in stability according to the value of the delay in relation to a critical delay value. Finding a formula for the critical delay is important to understanding the dynamics of delayed systems and is often simple to obtain when the system only has a single constant delay. However, if we consider a system with multiple constant delays, there is no known way to obtain such a formula that determines for what values of the delays a change in stability occurs. In this paper, we present some single-delay approximations to a multidelay system obtained via a Taylor expansion as well as formulas for their critical delays which are used to approximate where the change in stability occurs in the multidelay system. We determine when our approximations perform well and we give extra analytical and numerical attention to the two-delay and three-delay settings.
more »
« less
Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties
We study a chemostat model with an arbitrary number of competing species, one substrate, and constant dilution rates. We allow delays in the growth rates and additive uncertainties. Using constant inputs of certain species, we derive bounds on the sizes of the delays that ensure asymptotic stability of an equilibrium when the uncertainties are zero, which can allow persistence of multiple species. Under delays and uncertainties, we provide bounds on the delays and on the uncertainties that ensure input-to-state stability with respect to uncertainties
more »
« less
- PAR ID:
- 10066001
- Date Published:
- Journal Name:
- Discrete and continuous dynamical systems. Series B
- Volume:
- 23
- Issue:
- 4
- ISSN:
- 1553-524X
- Page Range / eLocation ID:
- 1851-1872
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We propose a mathematical model to study coupled epidemic and opinion dynamics in a network of communities. Our model captures SIS epidemic dynamics whose evolution is dependent on the opinions of the communities toward the epidemic, and vice versa. In particular, we allow both cooperative and antagonistic interactions, representing similar and opposing perspectives on the severity of the epidemic, respectively. We propose an Opinion-Dependent Reproduction Number to characterize the mutual influence between epidemic spreading and opinion dissemination over the networks. Through stability analysis of the equilibria, we explore the impact of opinions on both epidemic outbreak and eradication, characterized by bounds on the Opinion-Dependent Reproduction Number. We also show how to eradicate epidemics by reshaping the opinions, offering researchers an approach for designing control strategies to reach target audiences to ensure effective epidemic suppression.more » « less
-
Cooperative adaptive cruise control (CACC) is one of the popular connected and automated vehicle (CAV) applications for cooperative driving automation with combined connectivity and automation technologies to improve string stability. This study aimed to derive the string stability conditions of a CACC controller and analyze the impacts of CACC on string stability for both a fleet of homogeneous CAVs and for heterogeneous traffic with human-driven vehicles (HDVs), connected vehicles (CVs) with connectivity technologies only, and autonomous vehicles (AVs) with automation technologies only. We mathematically analyzed the impact of CACC on string stability for both homogeneous and heterogeneous traffic flow. We adopted parameters from literature for HDVs, CVs, and AVs for the heterogeneous traffic case. We found there was a minimum constant time headway required for each parameter design to ensure stability in homogeneous CACC traffic. In addition, the constant time headway and the length of control time interval had positive correlation with stability, but the control parameter had a negative correlation with stability. The numerical analysis also showed that CACC vehicles could maintain string stability better than CVs and AVs under low HDV market penetration rates for the mixed traffic case.more » « less
-
Recently, wireless communication technologies, such as Wireless Local Area Networks (WLANs), have gained increasing popularity in industrial control systems (ICSs) due to their low cost and ease of deployment, but communication delays associated with these technologies make it unsuitable for critical real-time and safety applications. To address concerns on network-induced delays of wireless communication technologies and bring their advantages into modern ICSs, wireless network infrastructure based on the Parallel Redundancy Protocol (PRP) has been proposed. Although application-specific simulations and measurements have been conducted to show that wireless network infrastructure based on PRP can be a viable solution for critical applications with stringent delay performance constraints, little has been done to devise an analytical framework facilitating the adoption of wireless PRP infrastructure in miscellaneous ICSs. Leveraging the deterministic network calculus (DNC) theory, we propose to analytically derive worst-case bounds on network- induced delays for critical ICS applications. We show that the problem of worst-case delay bounding for a wireless PRP network can be solved by performing network-calculus-based analysis on its non-feedforward traffic pattern. Closed-form expressions of worst-case delays are derived, which has not been found previously and allows ICS architects/designers to compute worst- case delay bounds for ICS tasks in their respective application domains of interest. Our analytical results not only provide insights into the impacts of network-induced delays on latency- critical tasks but also allow ICS architects/operators to assess whether proper wireless RPR network infrastructure can be adopted into their systems.more » « less
-
ABSTRACT The time-delay between the arrival of photons of multiple images of time-variable sources can be used to constrain absolute distances in the Universe, and in turn obtain a direct estimate of the Hubble constant and other cosmological parameters. To convert the time-delay into distances, it is well known that the gravitational potential of the main deflector and the contribution of the matter along the line of sight need to be known to a sufficient level of precision. In this paper, we discuss a new astrometric requirement that is becoming important, as time-delay cosmography improves in precision and accuracy with larger samples, and better data and modelling techniques. We derive an analytic expression for the propagation of astrometric uncertainties on the multiple image positions into the inference of the Hubble constant and derive requirements depending on image separation and relative time-delay. We note that this requirement applies equally to the image position measurements and to the accuracy of the model in reproducing them. To illustrate the requirement, we discuss some example lensing configurations and highlight that, especially for time-delays of order 10 d or shorter, the relative astrometric requirement is of order milliarcseconds, setting a tight requirement on both measurements and models. With current optical infrared technology, astrometric uncertainties may be the dominant limitation for strong lensing cosmography in the small image-separation regime when high-precision time-delays become accessible.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3934/dcdsb.2018098
