# American Institute of Mathematical Sciences

## Journals

DCDS-B

We consider an Internet congestion control system which is presented as a group of differential equations with time delay, modeling the random early detection (RED) algorithm. Although this model achieves success in many aspects, some basic problems are not clear. We provide the result on the existence of the equilibrium and the positivity and boundedness of the solution. Also, we implement the model by route switch mechanism, based on the minimum delay principle, to model the dynamic routing. For the simple network topology, we show that the Filippov solution exists under some restrictions on parameters. For the case with a single user group and two alternative links, we prove that the discontinuous boundary, or equivalently the sliding region, always exists and is locally attractive. This result implies that for some cases this type of routing may deviate from the purpose of the original design.

keywords: Internet congestion control dynamic routing switch system Filippov solution sliding motion
DCDS-B
A proportionally-fair controller with time delay is considered to control Internet congestion. The time delay is chosen to be a controllable parameter. To represent the relation between the delay and congestion analytically, the method of multiple scales is employed to obtain the periodic solution arising from the Hopf bifurcation in the congestion control model. A new control method is proposed by perturbing the delay periodically. The strength of the perturbation is predicted analytically in order that the oscillation may disappear gradually. It implies that the proved control scheme may decrease the possibility of the congestion derived from the oscillation. The proposed control scheme is verified by the numerical simulation.
keywords: delayed differential equation time-varying delay. Internet congestion control method of multiple scales Hopf bifurcation
DCDS
Considered herein is the blow-up mechanism to the periodic modified Camassa-Holm equation with varying linear dispersion. We first consider the case when linear dispersion is absent and derive a finite-time blow-up result. The key feature is the ratio between solution and its gradient. Using the continuity of the solutions and the right transformation, we then obtain this blow-up criterion to the case with negative linear dispersion and determine that the finite time blow-up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild-oscillation region. Finally, we demonstrate that when the linear dispersion is non-negative, formation of singularity can be induced by an initial datum with a sufficiently steep profile.
keywords: Periodic modified Camassa-Holm equation varying linear dispersion. integrable equation blow up sign-changing momentum
DCDS
We derive conditions on the initial data, including cases where the initial momentum density is not of one sign, that produce blow-up of the induced solution to the periodic modified Camassa-Holm equation with cubic nonlinearity. The blow-up conditions and the blow-up rate are formulated in terms of the initial momentum density and the average initial energy.
keywords: Modified Camassa-Holm equation peakon. integrable equation blow up
DCDS
This paper is concerned with the existence of multi-dimensional non-isothermal subsonic phase transitions in a steady supersonic flow with the van der Waals type state function. Due to the subsonic property, the Lax entropy inequality [15] is no longer valid for subsonic phase transitions. Hence, physical admissible planar waves are chosen by the viscosity capillarity criterion [24]. Based on the uniform stability result in [28], we perform the iteration scheme [20] and establish the existence.
keywords: Supersonic flows Euler equations. subsonic phase transitions
JIMO

The classical multifacility Weber problem (MFWP) is one of the most important models in facility location. This paper considers more general and practical case of MFWP called constrained multifacility Weber problem (CMFWP), in which the gauge is used to measure distances and locational constraints are imposed to facilities. In particular, we develop a variational inequality approach for solving it. The CMFWP is reformulated into a linear variational inequality, whose special structures lead to new projection-type methods. Global convergence of the projection-type methods is proved under mild assumptions. Some preliminary numerical results are reported which verify the effectiveness of proposed methods.

keywords: Facility location multifacility Weber problem gauge locational constraints variational inequality approach
NACO
Iterative Water-filling Algorithm (IWFA) is a well-known distributed multi-carrier power control method for multi-user communication. It was empirically observed (and conjectured) to be convergent under all channel conditions. In this paper, we present an example showing that IWFA can oscillate, therefore disproving the conjecture.
keywords: multi-user communication system. Iterative water-filling algorithm
PROC
Macrophage derived foam cells are a major constituent of the fatty deposits characterizing the disease atherosclerosis. Foam cells are formed when certain immune cells (macrophages) take on oxidized low density lipoproteins through failed phagocytosis. High density lipoproteins (HDL) are known to have a number of anti-atherogenic effects. One of these stems from their ability to remove excess cellular cholesterol for processing in the liver---a process called reverse cholesterol transport (RCT). HDL perform macrophage RCT by binding to forming foam cells and removing excess lipids by efflux transporters.
We propose a model of foam cell formation accounting for macrophage RCT. This model is presented as a system of non-linear ordinary differential equations. Motivated by experimental observations regarding time scales for oxidation of lipids and MRCT, we impose a quasi-steady state assumption and analyze the resulting systems of equations. We focus on the existence and stability of equilibrium solutions as determined by the governing parameters with the results interpreted in terms of their potential bio-medical implications.
keywords: stability analysis steady state. Atherosclerosis modeling
JIMO

In this paper, we present an optimal feedback control model to deal with the problem of energy efficiency management. Especially, an emission permits trading scheme is considered in our model, in which the decision maker can trade the emission permits flexibly. We make use of the optimal control theory to derive a Hamilton-Jacobi-Bellman (HJB) equation satisfied by the value function, and then propose an upwind finite difference method to solve it. The stability of this method is demonstrated and the accuracy, as well as the usefulness, is shown by the numerical examples. The optimal management strategies, which maximize the discounted stream of the net revenue, together with the value functions, are obtained. The effects of the emission permits price and other parameters in the established model on the results have been also examined. We find that the influences of emission permits price on net revenue for the economic agents with different initial quotas are quite different. All the results demonstrate that the emission permits trading scheme plays an important role in the energy efficiency management.

keywords: Energy efficiency management emission permits trading upwind finite difference method optimal management strategies
JIMO

It becomes increasingly important to manage water and improve the efficiency of irrigation under higher temperatures and irregular precipitation patterns. The choice of investment in water saving technologies and its timing play key roles in improving efficiency of water use. In this paper, we use a real option approach to establish a model to handle future uncertainties about the water price. In addition, to match the practical situation, the expiration of the real option is considered to be finite in our model, such that it is difficult to solve the model. Therefore, we reformulate the problem into a linear parabolic variational inequality (Ⅵ) and develop a power penalty method to solve it numerically. Thus, a nonlinear partial differential equation (PDE) is obtained, which is shown to be uniquely solvable and the solution of the nonlinear PDE converges to that of the Ⅵ at the rate of $O(λ^{-\frac{k}{2}})$ with $λ$ being the penalty number. Furthermore, a so-called fitted finite volume method is proposed to solve the nonlinear PDE. Finally, several numerical experiments are performed. It is shown that the subjective discount rate will affect the investment boundary mostly, and the flexibility to suspend operation will enlarge the investment region.

keywords: Water management real options fitted finite volume method power penalty method optimal investment decision