Finite-time stability results for fractional damped dynamical systems with time delays

This paper is explored with the stability procedure for linear nonautonomous multiterm fractional damped systems involving time delay. Finite-time stability (FTS) criteria have been developed based on the extended form of Gronwall inequality. Also, the result is deduced to a linear autonomous case. Two examples of applications of stability analysis in numerical formulation are described showing the expertise of theoretical prediction.

Existence of solutions for the semilinear abstract Cauchy problem of fractional order

In this paper we establish the existence of solutions for the nonlinear abstract Cauchy problem of order α ∈ (1, 2), where the fractional derivative is considered in the sense of Caputo. The autonomous and nonautonomous cases are studied. We assume the existence of an α-resolvent family for the homogeneous linear problem. By using this α-resolvent family and appropriate conditions on the forcing function, we study the existence of classical solutions of the nonhomogeneus semilinear problem. The non-autonomous problem is discussed as a perturbation of the autonomous case. We establish a variation of the constants formula for the nonautonomous and nonhomogeneous equation.

Mathematical modeling and assessment of barrier measures and temperature on the transmission and persistence of Novel coronavirus disease 2019 (COVID-19)

In December 2019, human cases of novel coronavirus infection were detected in Wuhan, China which have been named as COVID-19 by the World Health Organization (WHO). Since COVID-19 was first detected in China, the virus has reached more than 120 countries and was declared a global pandemic on March 11, 2020 by the WHO. In this paper, we have highlighted the influence of temperature on the spread of COVID-19. For this, the dynamic transmission of COVID-19 is modeled taking into account the influence of the temperature on the persistence of coronavirus in the environment. We also took into account the impact of proportion of people who respect the barrier measures published by the WHO on the scale of the COVID-19 pandemic. Taking into account the influence of the temperature on the persistence of the virus in the environment, the dynamic transmission has been described by a system of ordinary differential equations (ODEs). First, we analyzed the solutions of system in the case where the impact of the temperature on the virus is neglected. Second, we carried out the mathematical analysis of the solutions of the system in the non-autonomous case. Mathematical analyzes have enabled us to establish that the temperature and proportion of persons who respect the barrier rules can affect the spread of COVID-19. Some numerical simulations have been proposed to illustrate the behavior of the pandemic in some countries.

The feasibility of energy autonomy for municipalities: local energy system optimisation and upscaling with cluster and regression analyses

The sheer number of alternative technologies and measures make the optimal planning of energy system transformations highly complex, requiring decision support from mathematical optimisation models. Due to the high computational expenses of these models, only individual case studies are usually examined. In this article, the approach from the author’s PhD thesis to transfer the optimisation results from individual case studies to many other energy systems is summarised. In the first step, a typology of the energy systems to be investigated was created. Based on this typology, representative energy systems were selected and analysed in an energy system optimisation model. In the third step, the results of the representative case studies were transferred to all other energy systems. This approach was applied to a case study for determining the minimum costs of energy system transformation for all 11,131 German municipalities from 2015 to 2035 in the completely energy autonomous case. While a technical potential to achieve energy autonomy is present in 56% of the German municipalities, energy autonomy shows only low economic potential under current energy-political conditions. However, energy system costs in the autonomous case can be greatly reduced by the installation and operation of base-load technologies like deep-geothermal plants combined with district heating networks. The developed approach can be applied to any type of energy system and should help decision makers, policy makers and researchers to estimate optimal results for a variety of energy systems using practical computational expenses.

Nonlinear Compartment Models with Time-Dependent Parameters

A nonlinear compartment model generates a semi-process on a simplex and may have an arbitrarily complex dynamical behaviour in the interior of the simplex. Nonetheless, in applications nonlinear compartment models often have a unique asymptotically stable equilibrium attracting all interior points. Further, the convergence to this equilibrium is often wave-like and related to slow dynamics near a second hyperbolic equilibrium on the boundary. We discuss a generic two-parameter bifurcation of this equilibrium at a corner of the simplex, which leads to such dynamics, and explain the wave-like convergence as an artifact of a non-smooth nearby system in C0-topology, where the second equilibrium on the boundary attracts an open interior set of the simplex. As such nearby idealized systems have two disjoint basins of attraction, they are able to show rate-induced tipping in the non-autonomous case of time-dependent parameters, and induce phenomena in the original systems like, e.g., avoiding a wave by quickly varying parameters. Thus, this article reports a quite unexpected path, how rate-induced tipping can occur in nonlinear compartment models.

The Value of Autonomous Vehicles for Last-Mile Deliveries in Urban Environments

We demonstrate that autonomous-assisted delivery can yield significant improvements relative to today’s system in which a delivery person must park the vehicle before delivering packages. We model an autonomous vehicle that can drop off the delivery person at selected points in the city where the delivery person makes deliveries to the final addresses on foot. Then, the vehicle picks up the delivery person and travels to the next reloading point. In this way, the delivery person would never need to look for parking or walk back to a parking place. Based on the number of customers, driving speed of the vehicle, walking speed of the delivery person, and the time for loading packages, we characterize the optimal solution to the autonomous case on a solid rectangular grid of customers, showing the optimal solution can be found in polynomial time. To benchmark the completion time of the autonomous case, we introduce a traditional model for package delivery services that includes the time to search for parking. If the time to find parking is ignored, we show the introduction of an autonomous vehicle reduces the completion time of delivery to all customers by 0%–33%. When nonzero times to find parking are considered, the delivery person saves 30%–77% with higher values achieved for longer parking times, smaller capacities, and lower fixed time for loading packages. This paper was accepted by Vishal Gaur, operations management.

Existence of positive periodic solutions for a class of in-host MERS-CoV infection model with periodic coefficients

<abstract><p>In this paper, a dynamic model of Middle East Respiratory Syndrome Coronavirus (MERS-CoV) with periodic coefficients is proposed and studied. By using the continuation theorem of the coincidence degree theory, we obtain some sufficient conditions for the existence of positive periodic solutions of the model. The periodic model degenerates to an autonomous case, and our conditions can be degenerated to the basic reproductive number $ R_0 &gt; 1 $. Finally, we give some numerical simulations to illustrate our main theoretical results.</p></abstract>

On the motions of a near-autonomous Hamiltonian system in the cases of two zero frequencies

We consider the motion of a near-autonomous, time-periodic two-degree-of- freedom Hamiltonian system in the vicinity of trivial equilibrium. It is assumed that the system depends on three parameters, one of which is small, and when it is zero, the system is autonomous. Suppose that in the autonomous case for a set of two other parameters, both frequencies of small linear oscillations of the system in the vicinity of the equilibrium are equal to zero, and the rank of the coefficient matrix of the linearized equations of perturbed motion is three, two, or one. We study the structure of the regions of stability and instability of the trivial equilibrium of the system in the vicinity of the resonant point of a three-dimensional parameter space, as well as the existence, number and stability (in a linear approximation) of periodic motions of the system that are analytic in integer or fractional powers of the small parameter. As an application, periodic motions of a dynamically symmetric satellite (solid) with respect to the center of mass are obtained in the vicinity of its stationary rotation (cylindrical precession) in a weakly elliptical orbit in the case of two zero frequencies under study, and their instability is proved.

Stochastic dynamic for an extensible beam equation with localized nonlinear damping and linear memory

In this paper, we concerned to prove the existence of a random attractor for the stochastic dynamical system generated by the extensible beam equation with localized non-linear damping and linear memory defined on bounded domain. First we investigate the existence and uniqueness of solutions, bounded absorbing set, then the asymptotic compactness. Longtime behavior of solutions is analyzed. In particular, in the non-autonomous case, the existence of a random attractor attractors for solutions is achieved.