An easy-to-use and effective formula for stability testing of a system with fractional-delay characteristic equation in the general form of $Δ(s)=P_0(s)+\sum_{i=1}^N P_i(s)\exp(-ζ_i s^{β_i}) =0$, where $P_i(s)$ ($i=0,..., N$) are the so-called fractional-order polynomials and $ζ_i$ and $β_i$ are positive real constants, is proposed in this paper. The proposed formula determines the number of unstable roots of the characteristic equation (i.e., those located in the right half-plane of the first Riemann sheet) by applying Rouche's theorem. Numerical simulations are also presented to confirm the efficiency of the proposed formula.
We introduce the delayed Mittag-Leffler type matrix functions, delayed fractional cosine, delayed fractional sine and use the Laplace transform to obtain an analytical solution to the IVP for a Hilfer type fractional linear time-delay system $D_{0,t}^{μ,ν}z\left( t\right) +Az\left( t\right) +Ωz\left( t-h\right) =f\left( t\right) $ of order $1<μ<2$ and type $0\leqν\leq1,$ with nonpermutable matrices $A$ and $Ω$. Moreover, we study Ulam-Hyers stability of the Hilfer type fractional linear time-delay system. Obtained results extend those for Caputo and Riemann-Liouville type fractional linear tim…
In this paper, a fractional Lotka-Volterra mathematical model for a bioreactor is proposed and used to fit the data provided by a bioprocess known as continuous fermentation of Zymomonas mobilis. The model contemplates a time-delay $τ$ due to the dead-time in obtaining the measurement of biomass $x(t)$. A Hopf bifurcation analysis is performed to characterize the inherent self oscillatory experimental bioprocess response. As consequence, stability conditions for the equilibrium point together with conditions for limit cycles using the delay $τ$ as bifurcation parameter are obtained. Under the …
The $μ$-neutral linear fractional multi-delayed differential nonhomogeneous system with noncommutative coefficient matrices is introduced. The novel $μ$-neutral multi-delayed perturbation of Mittag-Leffler type matrix function is proposed. Based on this, an explicit solution to the system is investigated step by step. The existence uniqueness of solutions to $μ$-neutral nonlinear fractional multi-delayed differential system is obtained with regard to the supremum norm. The notion of stability analysis in the sense of solutions to the described system is discussed on the grounds of the fixed po…
In this paper, we consider Caputo type fractional stochastic time-delay system with permutable matrices. We derive stochastic analogue of variation of constants formula via a newly defined delayed Mittag-Leffer type matrix function. Thus, we investigate new results on existence and uniqueness of mild solutions with the help of weighted maximum norm to fractional stochastic time-delay differential equations whose coefficients satisfy standard Lipschitz conditions. The main points in the proof are to apply Ito's isometry and martingale representation theorem, and to show the notion of a coincide…
For the fractional order systems \[D^αx(t)=f(x),\quad 0<α\leq 1,\] one can have a critical value of $α$ viz $α_*$ such that the system is stable for $0<α<α_*$ and unstable for $α_*<α\leq 1$. In general, if such system is stable for some $α_0\in(0,1)$ then it remains stable for all $α<α_0.$ In this paper, we show that there are some delay differential equations \[D^αx(t)=f(x(t),x(t-τ))\] of the fractional order which behave in an exactly opposite way. These systems are unstable for higher values of fractional order and stable for the lower values. The striking observation is the example which i…
Fractional derivative and delay are important tools in modeling memory properties in the natural system. This work deals with the stability analysis of a fractional order delay differential equation \begin{equation*} D^αx(t)=δx(t-τ)-εx(t-τ)^3-px(t)^2+q x(t). \end{equation*} We provide linearization of this system in a neighbourhood of equilibrium points and propose linearized stability conditions. To discuss the stability of equilibrium points, we propose various conditions on the parameters $δ$, $ε$, $p$, $q$ and $τ$. Even though there are five parameters involved in the system, we are able t…
In this article, we give some results for fractional-order delay differential equations. In the first result, we prove the existence and uniqueness of solution by using Bielecki norm effectively. In the second result, we consider a constant delay form of this problem. Then we apply Burton's method to this special form to prove that there is only one solution. Finally, we prove a result regarding the Hyers-Ulam stability of this problem. Moreover, in these results, we omit the conditions for contraction constants seen in many papers.
We study a class of nonlinear fractional differential equations with multiple delays, which is represented by the Voigt creep fractional model of viscoelasticity. We discuss two Voigt models, the first being linear and the second being nonlinear. The linear Voigt model give us the physical interpretation and is associated with important results since the creep function characterizes the viscoelastic behavior of stress and strain. For the nonlinear model of Voigt, our theoretical study and analysis provides existence and stability, where time delays are expressed in terms of Boltzmann's superpo…
Multi-objective LQR with Optimum Weight Selection to Design FOPID Controllers for Delayed Fractional Order Processes
School of Electronics and Computer Science, University of Southampton（南安普顿大学电子与计算机科学学院）；Department of Power Engineering, Jadavpur University（贾达普大学动力工程系）；Department of Earth Science and Engineering, Imperial College London（帝国理工学院地球科学与工程系）；Reactor Control Division, Bhabha Atomic Research Centre（巴哈原子研究中心反应堆控制部）
COVID-19 data released by public health authorities features the presence of notable time-delays, corresponding to the difference between actual time of infection and identification of infection. These delays have several causes, including the natural incubation period of the virus, availability and speed of testing facilities, population demographics, and testing center capacity, among others. Such delays have important ramifications for both the mathematical modeling of COVID-19 contagion and the design and evaluation of intervention strategies. In the present work, we introduce a novel opti…
