Asymptotic Integration and Stability

The following sections are included:

• Dedication
• Preface
• Contents

When asked about the significance of the non-integer order for a differential operator by anyone without special interest in the mathematics of differential equations, the fractionist¹ can provide the inquirer with a simple analogy, described in the following…

Let us introduce the initial value problem

Several unrelated results, presented in the following, enrich the beautiful panorama of fractional differential equations of order less than 1 that has been given a modest sketch in the previous chapter.

We shall focus here on establishing fractional variants for two major issues in the classical asymptotic integration paradigm: the Trench asymptotic development [Trench (1963)] for the solutions of a differential equation as t→+∞ and the set of hypotheses needed for the absence of non-trivial L^p–solutions, which is the heart of the limit-circle/limit-point classification [Weyl (1910)] of differential operators due to H. Weyl.

The retarded integral inequalities of Henry-Gronwall type were applied to certain properties of solutions to fractional differential equations with delay [Ye and Gao (2011)]…

The existence of positive solution for fractional differential equations with delay is an interesting issue both from theoretical and applied viewpoints (see for example refs. [Babakhani and Enteghami (2009); Daftardar-Gejji and Babakhani (2004); Babakhani and Daftardar-Gejji (2003, 2006); Sign and Chuanzhi (2006); Daftardar-Gejji (2005); Ye et al. (2007); Belarbi et al. (2006); Zhang (2008)] and references therein)…

We notice that the discrete fractional calculus is a new concept suggested in [Miller and Ross (1989)]. Since that pioneering work several results were reported on this subject (see for example [Atici and Eloe (2007, 2009a,b); Abdeljawad and Baleanu (2011a); Abdeljawad (2011); Jarad et al. (2012); Baleanu et al. (2006); Abdeljawad et al. (2013); Abdeljawad and Baleanu (2011b)] and the references therein). This new concept describes better in some cases the dynamics of complex or hypercomplex systems. Some of the recent reported applications can be see in [Wu and Baleanu (2014a); Wu et al. (2014); Wu and Baleanu (2014b)]. One of the key concept in this new field is the stability of fractional-order dynamic systems which was already investigated in [Chen et al. (1995); Lazarevic (2006); Deng et al. (2007); Merrikh-Bayat and Karimi-Ghartemani (2009); Zhang (2008); Momani and Hadid (2004); Li et al. (2010); Jarad et al. (2012)] and the references therein, while the stability of discrete dynamic systems was a the subject of many books and articles such as [Agarwal (2000)] and the references therein.

The dynamics of time-delay system is an important subject from both theoretical applied point of views [Richard (2003)]. We recall that the time delay appears naturally in various technical systems, e.g., electric, chemical, pneumatic, and hydraulic networks, long transmission lines and many others. Various results were reported on this matter, with particular emphasis on the application of Lyapunov's second method (see for example references [Chen et al. (1995); Lee and Dianat (1981)] and the references therein)…

As it well known, the Razumikhin stability theory is extensively utilized in the literature to prove the stability of time-delay systems [Hale and Verduyn Lunel (1993); Kequin et al. (2003)], mainly because the building of Lyapunov-Krasovskii functional seems more difficult than that of Lyapunov-Razumikhin function…

The following sections are included:

• Preliminaries
• The Problem
• A Controllability Result

The controllability of the deterministic and stochastic dynamical control systems in infinite dimensional spaces is an well-developed field. We recall that the theory of controllability for nonlinear fractional dynamical systems started to be developed intensively during the last few years (see for example [Sukavanam and Kumar (2011); Sukavanam and Tomar (2007); Kerboua et al. (2013)] and the references therein). The exact controllability for semilinear fractional order system, such that the nonlinear term is independent of the control function, was reported in [Bashirov and Mahmudov (1999); Debbouche and Baleanu (2011, 2012); Sakthivel et al. (2012a)]. In these works it was proved the exact controllability by assuming that the controllability operator has an induced inverse on a quotient space. If the semigroup associated with the system is compact, then the controllability operator is also compact,as a result the induced inverse does not exist mainly because the state space is infinite dimensional [Yan (2012a)]. Hence, the concept of exact controllability is too strong and implicitly has limited applicability. as it is known the approximate controllability is a weaker concept than the complete one and it is completely adequate in applications for these control systems…

The following sections are included:

• Bibliography
• Index