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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 161246, 25 pages
http://dx.doi.org/10.1155/2011/161246
Research Article

Positivity and Stability of the Solutions of Caputo Fractional Linear Time-Invariant Systems of Any Order with Internal Point Delays

Institute for Research and Development of Processes, Faculty of Science and Technology, University of Basque Country, Campus of Leioa, Aptdo. 544, 48080 Bilbao, Spain

Received 21 September 2010; Revised 15 November 2010; Accepted 11 January 2011

Academic Editor: Marcia Federson

Copyright © 2011 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is devoted to the investigation of the nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic systems involving delayed dynamics with point delays. The obtained results are independent of the sizes of the delays.

1. Introduction

The theory of fractional calculus is basically concerned with the calculus of integrals and derivatives of any arbitrary real or complex orders. In this sense, it may be considered as a generalization of classical calculus which is included in the theory as a particular case. The former ideas have been stated about three hundred years ago, but the main mathematical developments and applications of fractional calculus have been of increasing interest from the seventies. There is a good compendium of the state of the art of the subject and the main related existing mathematical results with examples and case studies in [1]. There are a lot of results concerning the exact and approximate solutions of fractional differential equations of Riemann-Liouville and Caputo types, [14], fractional derivatives involving products of polynomials, [5, 6], fractional derivatives and fractional powers of operators, [79], boundary value problems concerning fractional calculus (see, e.g., [1, 10]), and so forth. There is also an increasing interest in the recent mathematical literature in the characterization of dynamic fractional differential systems oriented towards several fields of science like physics, chemistry or control theory because it is a powerful tool for later applications in all fields requiring support via ordinary, partial derivatives, and functional differential equations. Perhaps the reason of interest of fractional calculus is that the numerical value of the fraction parameter allows a closer characterization of eventual uncertainties present in the dynamic model compared to the alternative use of structured uncertainties. We can find, in particular, a lot of literature concerned with the development of Lagrangian and Hamiltonian formulations where the motion integrals are calculated though fractional calculus and also in related investigations concerning dynamic and damped and diffusive systems [1117] as well as the characterization of impulsive responses or its use in applied optics related, for instance, to the formalism of fractional derivative Fourier plane filters (see, e.g., [1618]) and Finance [19]. Fractional calculus is also of interest in control theory concerning, for instance, heat transfer, lossless transmission lines, the use of discretizing devices supported by fractional calculus, and so forth (see, e.g., [2022]). In particular, there are several recent applications of fractional calculus in the fields of filter design, circuit theory and robotics, [21, 22], and signal processing, [17]. Fortunately, there is an increasing mathematical literature, currently available on fractional differ-integral calculus, which can formally support successfully the investigations in other related disciplines.

This paper is concerned with the investigation of the solutions of time-invariant fractional differential dynamic systems, [23, 24], involving point delays which leads to a formalism of a class of functional differential equations, [2531]. Functional equations involving point delays are a crucial mathematical tool to investigate real process where delays appear in a natural way like, for instance, transportation problems, war and peace problems, or biological and medical processes. The main interest of this paper is concerned with the positivity and stability of solutions independent of the sizes of the internal delays and also with obtaining results being independent of the eventual mutual coincidence of some values of delays, [3133]. It has to be pointed out that the positivity of the solutions is a crucial property in investigating some dynamic systems like biological systems or epidemic models, [32, 33], where positivity is an essential requirement since negative solutions have nonsense at any time instant. It is also a relevant property concerning the existence and characterization of oscillatory solutions of differential equations, [34]. Most of the results are centred in characterizations via Caputo fractional differentiation although some extensions are presented concerned with the classical Riemann-Liouville differ integration. It is proved that the existence of nonnegative solutions independent of the sizes of the delays and the stability properties of linear time-invariant fractional dynamic differential systems subject to point delays may be characterized with sets of precise mathematical results.

1.1. Notation

𝐙, 𝐑, and 𝐂 are the sets of integer, real, and complex numbers, 𝐙+ and 𝐑+ are the positive integer and real numbers, and 𝐙0+=𝐙+{0},𝐑0+=𝐑+{0},𝐂+𝐂={𝑧𝐂Re𝑧>0},0+={𝑧𝐂Re𝑧0},𝑛={1,2,,𝑛}.(1.1)

The following notation is used to characterize different levels of positivity of matrices: 𝐑𝑛×𝑚0+={𝑀=(𝑀𝑖𝑗)𝐑𝑛×𝑚𝑀𝑖𝑗0;(𝑖,𝑗)𝑛×𝑚} is the set of all 𝑛×𝑚 real matrices of nonnegative entries. If 𝑀𝐑𝑛×𝑚 then 𝑀0 is used as a simpler notation for 𝑀𝐑𝑛×𝑚0+.

𝐑+𝑛×𝑚={0𝑀=(𝑀𝑖𝑗)𝐑𝑛×𝑚𝑀𝑖𝑗0;(𝑖,𝑗)𝑛×𝑚} is the set of all nonzero 𝑛×𝑚 real matrices of nonnegative entries (i.e., at least one of their entries is positive). If 𝑀𝐑𝑛×𝑚 then 𝑀>0 is used as a simpler notation for 𝑀𝐑+𝑛×𝑚.

𝐑𝑛×𝑚++={𝑀=(𝑀𝑖𝑗)𝐑𝑛×𝑚𝑀𝑖𝑗>0;(𝑖,𝑗)𝑛×𝑚} is the set of all 𝑛×𝑚 real matrices of positive entries. If 𝑀𝐑𝑛×𝑚 then 𝑀0 is used as a simpler notation for 𝑀𝐑𝑛×𝑚++. The superscript 𝑇 denotes the transpose, 𝑀𝑇𝑖 and 𝑀𝑗 are, respectively, the 𝑖th row and the 𝑗th column of the matrix 𝑀.

A close notation to characterize the positivity of vectors is the following: 𝐑𝑛0+={𝑣=(𝑣1,𝑣2,,𝑣𝑛)𝑇𝐑𝑛𝑣𝑖0;𝑖𝑛} is the set of all 𝑛 real vectors of nonnegative components. If 𝑣𝐑𝑛 then 𝑣0 is used as a simpler notation for 𝑣𝐑𝑛0+. 𝐑𝑛+={0𝑣=(𝑣1,𝑣2,,𝑣𝑛)𝑇𝐑𝑛𝑣𝑖0;𝑖𝑛} is the set of all 𝑛 real nonzero vectors of nonnegative components (i.e., at least one component is positive). If 𝑣𝐑𝑛 then 𝑣>0 is used as a simpler notation for 𝑣𝐑𝑛+.

𝐑𝑛++={𝑣=(𝑣1,𝑣2,,𝑣𝑛)𝑇𝐑𝑛𝑣𝑖>0;𝑖𝑛} is the set of all 𝑛 real vectors of positive components. If 𝑣𝐑𝑛 then 𝑣0 is used as a simpler notation for 𝑣𝐑𝑛++.

𝑀=(𝑀𝑖𝑗)𝐑𝑛×𝑛 is a Metzler matrix if 𝑀𝑖𝑗0; for all (𝑖,𝑗𝑖)𝑛×𝑛. 𝑀𝐑𝑛×𝑛 is the set of Metzler matrices of order 𝑛.

The maximum real eigenvalue, if any, of a real matrix 𝑀, is denoted by 𝜆max(𝑀). Multiple subscripts of vector, matrices, and vector and matrix functions are separated by commas only in the case that, otherwise, some confusion could arise as, for instance, when some of the subscripts is an expression involving several indices.

2. Some Background on Fractional Differential Systems

Assume that 𝑓[𝑎,𝑏]𝐂𝑛 for some real interval [𝑎,𝑏]𝐑 satisfies𝑓𝐶𝑘2((𝑎,𝑏),𝐑𝑛) and, furthermore, 𝑑𝑘1𝑓(𝜏)/𝑑𝜏𝑘1 exists everywhere in [𝑎,𝑏] for 𝑘=[Re𝛼]+1 for some 𝛼𝐂0+. Then, the Riemann-Liouville left-sided fractional derivative RL𝐷𝛼𝑎+𝑓 of order 𝛼𝐂0+ of the vector function 𝑓 in [𝑎,𝑏] is pointwise defined in terms of the Riemann-Liouville integral asRL𝐷𝛼𝑎+𝑓(1𝑡)=𝑑Γ(𝑘𝛼)𝑘𝑑𝑡𝑘𝑡𝑎𝑓(𝜏)(𝑡𝜏)𝛼+1𝑘[]𝑑𝜏,𝑡𝑎,𝑏,(2.1) where the integer 𝑘 is given by 𝑘=[Re𝛼]+1 and Γ𝐂𝐙0𝐂, where 𝐙0={𝑛𝐙𝑛0}, is the Γ-function defined by Γ(𝑧)=0𝜏𝑧1𝑒𝜏𝑑𝜏; 𝑧𝐂𝐙0. If 𝑓𝐶𝑘1((𝑎,𝑏),𝐑𝑛) and, furthermore, 𝑓(𝑘)(𝜏)𝑑𝑘𝑓(𝜏)/𝑑𝜏𝑘 exists everywhere in [𝑎,𝑏], then the Caputo left-sided fractional derivative 𝐶𝐷𝛼𝑎+𝑓 of order 𝛼𝐂0+ of the vector function 𝑓 in [𝑎,𝑏] is pointwise defined in terms of the Riemann-Liouville integral as𝐶𝐷𝛼𝑎+𝑓1(𝑡)=Γ(𝑘𝛼)𝑡𝑎𝑓(𝑘)(𝜏)(𝑡𝜏)𝛼+1𝑘[]𝑑𝜏,𝑡𝑎,𝑏,(2.2) where 𝑘=[Re𝛼]+1 if 𝛼𝐙0+ and 𝑘=𝛼 if 𝛼𝐙0+. The following relationship between both fractional derivatives holds provided that they exist (i.e., if 𝑓[𝑎,𝑏]𝐂𝑛 possesses Caputo left-sided fractional derivative in [𝑎,𝑏]), [1]𝐶𝐷𝛼𝑎+𝑓(𝑡)=RL𝐷𝛼𝑎+𝑓(𝜏)𝑘1𝑗=0𝑓(𝑗)(𝑎)(𝜏𝑎)𝑗[]𝑗!(𝑡),𝑡𝑎,𝑏.(2.3) Since Re𝛼𝑘, the above formula relating both fractional derivatives proves the existence of the Caputo left-sided fractional derivative in [𝑎,𝑏] if the Riemann-Liouville one exists in [𝑎,𝑏].

3. Solution of a Fractional Differential Dynamic System of Any Order 𝛼 with Internal Point Delays

Consider the linear and time-invariant differential functional Caputo fractional differential system of order 𝛼:𝐶𝐷𝛼0+𝑥1(𝑡)=Γ(𝑘𝛼)𝑡𝑎𝑓(𝑘)(𝜏)(𝑡𝜏)𝛼+1𝑘𝑑𝜏=𝑝𝑖=0𝐴𝑖𝑥𝑡𝑖+𝐵𝑢(𝑡)(3.1) with 𝑘1<𝛼(𝐑+)𝑘; 𝑘1, 𝑘𝐙0+, 0=0<1<2<<𝑝=< being distinct constant delays, 𝐴0,𝐴𝑖𝐑𝑛×𝑛(𝑖𝑝={1,2,,𝑝}), are the matrices of dynamics for each delay 𝑖,𝑖𝑝{0}, 𝐵𝐑𝑛×𝑚 is the control matrix. The initial condition is given by 𝑘  𝑛-real vector functions 𝜑𝑗[,0]𝐑𝑛, with 𝑗𝑘1{0}, which are absolutely continuous except eventually in a set of zero measure of [,0]𝐑 of bounded discontinuities with 𝜑𝑗(0)=𝑥𝑗(0)=𝑥(𝑗)(0)=𝑥𝑗0. The function vector 𝑢𝐑0+𝐑𝑚 is any given bounded piecewise continuous control function. The following result is concerned with the unique solution on 𝐑0+ of the above differential fractional system (3.1). The proof follows directly from a parallel existing result from the background literature on fractional differential systems by grouping all the additive forcing terms of (3.1) in a unique one (see, e.g., [1, (1.8.17), (3.1.34)–(3.1.49)], with 𝑓(𝑡)𝑝𝑖=1𝐴𝑖𝑥(𝑡𝑖)+𝐵𝑢(𝑡)).

Theorem 3.1. The linear and time-invariant differential functional fractional differential system (3.1) of any order 𝛼𝐂0+ has a unique solution on 𝐑0+ for each given set of initial functions 𝜑𝑗[,0]𝐑𝑛, 𝑗𝑘1{0} being absolutely continuous except eventually in a set of zero measures of [,0]𝐑 of bounded discontinuities with 𝜑𝑗(0)=𝑥𝑗(0)=𝑥(𝑗)(0)=𝑥𝑗0; 𝑗𝑘1{0} and each given control 𝑢𝐑0+𝐑𝑚 being a bounded piecewise continuous control function. Such a solution is given by 𝑥𝛼(𝑡)=𝑘1𝑗=0Φ𝛼𝑗0(𝑡)𝑥𝑗0+𝑝𝑖=1𝑖0Φ𝛼(𝑡𝜏)𝐴𝑖𝜑𝑗𝜏𝑖+𝑑𝜏𝑝𝑖=1𝑡𝑖Φ𝛼(𝑡𝜏)𝐴𝑖𝑥𝛼𝜏𝑖𝑑𝜏+𝑡0Φ𝛼(𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏,𝑡𝐑0+(3.2) with 𝑘=[Re𝛼]+1 if 𝛼𝐙+ and 𝑘=𝛼 if 𝛼𝐙+, Φ𝛼𝑗0(𝑡)=𝑡𝑗𝐸𝛼,𝑗+1𝐴0𝑡𝛼,Φ𝛼(𝑡)=𝑡𝛼1𝐸𝛼,𝛼𝐴0𝑡𝛼,𝐸(3.3)𝛼,𝑗𝐴0𝑡𝛼==0𝐴0𝑡𝛼Γ(𝛼+𝑗),𝑗𝑘1{0,𝛼}(3.4) for 𝑡0 and Φ𝛼0(𝑡)=Φ𝛼(𝑡)=0 for 𝑡<0, where 𝐸𝛼,𝑗(𝐴0𝑡𝛼) are the Mittag-Leffler functions.

Now consider that the right-hand side of (3.1) is the evaluation of a Riemann-Liouville fractional differential system of the same order 𝛼 as follows: RL𝐷𝛼0+𝑥(𝑡)=𝑝𝑖=0𝐴𝑖𝑥𝑡𝑖+𝐵𝑢(𝑡)(3.5) under the same functions of initial conditions as those of (3.1). Through the formula (2.3) relating Caputo and Riemann-Liouville left-sided fractional derivatives of the same order 𝛼, one gets𝐶𝐷𝛼0+𝑥(𝑡)=𝑝𝑖=0𝐴𝑖𝑥𝑡𝑖+𝐵𝑢(𝑡)RL𝐷𝛼0+𝑘1𝑗=0𝑥𝑗0𝜏𝑗𝑗!(𝑡).(3.6) Since the Caputo left-sided fractional derivative and the Riemann-Liouville fractional integral of order 𝐂𝛼+={𝐙+{𝑧𝐂+Re𝑧𝐙+}} are inverse operators (what is not the case if 𝐂𝛼+), (see [1, Lemma 2.21(a)]), one gets from (3.6), (2.3), and (3.2) if 𝐂𝛼 the subsequent result for the fractional differential system (3.5) on 𝐑0+.

Corollary 3.2. If (3.5) of any order 𝐂𝛼+ is replaced with (3.1) under the same initial conditions then its unique solution on 𝐑0+ is given by 𝑥𝛼(𝑡)=𝑘1𝑗=0Φ𝛼𝑗0𝑡(𝑡)𝑗𝐼𝑗!𝑛𝑥𝑗0+𝑝𝑖=1𝑖0Φ𝛼(𝑡𝜏)𝐴𝑖𝜑𝑗𝜏𝑖+𝑑𝜏𝑝𝑖=1𝑡𝑖Φ𝛼(𝑡𝜏)𝐴𝑖𝑥𝛼𝜏𝑖𝑑𝜏+𝑡0Φ𝛼(𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏,𝛼𝐙0+;𝑡𝐑0+=𝑘1𝑗=0𝐸𝛼,𝑗+11(𝑡)𝐼𝑗!𝑛𝑡𝑗𝑥𝑗0+𝑝𝑖=1𝑖0Φ𝛼(𝑡𝜏)𝐴𝑖𝜑𝑗𝜏𝑖+𝑑𝜏𝑝𝑖=1𝑡𝑖Φ𝛼(𝑡𝜏)𝐴𝑖𝑥𝛼𝜏𝑖𝑑𝜏+𝑡0Φ𝛼(𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏,𝛼𝐙0+,𝑡𝐑0+(3.7) with 𝑘=[Re𝛼]+1 if 𝛼𝐙+ and 𝑘=𝛼 if 𝛼𝐙+.

Another mild evolution operator can be considered to construct the unique solution of (3.1) by considering the control effort as the unique forcing term of (3.1) and the functions of initial conditions as forcing terms. See the corresponding expressions obtainable from [1, (1.8.17), (3.1.34)–(3.1.49)], with the identity (𝑓(𝑡)𝐵𝑢(𝑡)) and the evolution operator defined in [2, 3] for the standard (nonfractional differential system), that is, 𝛼=1 in (3.1). Thus, another equivalent expression for the unique solution of the Caputo fractional differential system of order 𝛼 is given in the subsequent result.

Theorem 3.3. The solution of (3.1) given in Theorem 3.1 is equivalently rewritten as follows: 𝑥𝛼(𝑡)=𝑘1𝑗=0Ψ𝛼𝑗0(𝑡)𝑥𝑗0+𝑝𝑖=1𝑖0Ψ𝛼𝑗0(𝑡𝜏)𝜑𝑗𝜏𝑖+𝑑𝜏𝑡0Ψ𝛼(𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏(3.8) for 𝑡𝐑0+, any 𝛼𝐂+ with 𝑘=[Re𝛼]+1 if 𝛼𝐙+ and 𝑘=𝛼 if 𝛼𝐙+; Ψ𝛼𝑗0(𝑡)=𝑡𝑗𝐸𝛼,𝑗+1𝐴0𝑡𝛼+𝑝𝑖=1𝑡0𝜏𝛼1𝐸𝛼𝛼𝐴0𝜏𝛼𝐴𝑖Ψ𝛼𝑗0𝑡𝜏𝑖Ψ𝑑𝜏,𝛼(𝑡)=𝑡𝛼1𝐸𝛼𝛼𝐴0𝑡𝛼+𝑝𝑖=1𝑡0𝜏𝛼1𝐸𝛼𝛼𝐴0𝜏𝛼𝐴𝑖Ψ𝛼𝑡𝜏𝑖𝑑𝜏(3.9) for 𝑡0 and Ψ𝛼𝑗0(𝑡)=Ψ𝛼(𝑡)=0, 𝑗𝑘1{0} for 𝑡[,0).

Also, the solution to the Riemann-Liouville fractional differential system (3.5) under the same initial conditions as those of (3.4) is given in the next result for 𝑘=[Re𝛼]+1 if 𝛼𝐙+ based on (3.6).

Corollary 3.4. If (3.5) being of order 𝐂𝛼+ is replaced with (3.1) under the same initial conditions then its unique solution on 𝐑0+ is given by 𝑥𝛼(𝑡)=𝑘1𝑗=0Ψ𝛼𝑗0𝑡(𝑡)𝑗𝐼𝑗!𝑛𝑥𝑗0+𝑝𝑖=1𝑖0Ψ𝛼𝑗0(𝑡𝜏)𝜑𝑗𝜏𝑖+𝑑𝜏𝑡0Ψ𝛼(𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏(3.10) with 𝑘=[Re𝛼]+1 if 𝛼𝐙+ and 𝑘=𝛼 if 𝛼𝐙+ which is identical to that given in Corollary 3.2.

Particular cases of interest of the solution of (3.1) given in Theorem 3.3 are(1)𝛼=𝑘 which yields the solution:𝑥𝑘(𝑡)=𝑘1𝑗=0Ψ𝑘𝑗0(𝑡)𝑥𝑗0+𝑝𝑖=1𝑖0Ψ𝑘𝑗0(𝑡𝜏)𝜑𝑗𝜏𝑖+𝑑𝜏𝑡0Ψ𝑘(𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏,(3.11)(2)a further particular case 𝛼=𝑘=1 which yields the solution: 𝑥1(𝑡)=Ψ1(𝑡)𝑥𝑗0+𝑝𝑖=1𝑖0Ψ1(𝑡𝜏)𝜑0𝜏𝑖𝑑𝜏+𝑡0Ψ1(𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏(3.12) since Ψ100(𝑡)=Ψ1(𝑡), 𝑡𝐑0+ which is the unique solution of (𝐷𝑥)(𝑡)=𝑝𝑖=0𝐴𝑖(𝑡𝑖)+𝐵𝑢(𝑡) under any almost everywhere absolutely continuous function (except eventually in some subset of zero measure of [,0] of bounded discontinuities) of initial conditions  𝜑𝜑0[,0]𝐑𝑛. Use for this case, the less involved notations (Ψ𝑥)(𝑡)=(Ψ100𝑥)(𝑡)=(Ψ1𝑥)(𝑡) for the smooth evolution operator from 𝐑0+×𝐑𝑛 to 𝐑𝑛, and Φ(𝑡)=Φ100(𝑡)=Φ1(𝑡)=𝑒𝐴0𝑡, 𝑡𝐑+ for the exponential matrix function 𝑒𝐴0𝑡 from 𝐑0+ to 𝐑𝑛×𝑛, which defines a 𝐶0-semigroup (𝑒𝐴0𝑡,𝑡𝐑0+) of infinitesimal generator 𝐴0 from 𝐑0+ to 𝐿(𝐑𝑛). Then, the unique solution 𝑥(𝑡)𝑥1(𝑡),𝑡𝐑+ for the given function of initial conditions is𝑥(𝑡)=Φ(𝑡)𝑥0+𝑝𝑖=1𝑖0Φ(𝑡𝜏)𝐴𝑖𝜑𝜏𝑖+𝑝𝑖=1𝑡𝑖Φ(𝑡𝜏)𝐴𝑖𝑥𝜏𝑖𝑑𝜏+𝑡0Φ(𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏=Ψ(𝑡)𝑥0+𝑝𝑖=1𝑖0Ψ(𝑡𝜏)𝜑𝜏𝑖𝑑𝜏+𝑡0Ψ(𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏,𝑡𝐑0+(3.13) and 𝑥(𝑡)=𝜑(𝑡) for 𝑡[,0], where Φ(𝑡)=𝑒𝐴0𝑡 satisfies ̇Φ(𝑡)=𝐴0Φ(𝑡)𝑡𝐑, and ̇Ψ(𝑡)=𝑝𝑖=0𝐴𝑖Ψ(𝑡𝑖) with Ψ(0)=Φ(0)=𝐼𝑛 (the 𝑛-identity matrix) and Ψ(𝑡)=0, 𝑡[,0) which has a unique solution Ψ(𝑡)=𝑒𝐴0𝑡(𝐼𝑛+𝑝𝑖=1𝑡𝑖𝑒𝐴0𝜏𝐴𝑖Ψ(𝜏𝑖)𝑑𝜏) for 𝑡𝐑0+, [2, 3, 25, 26].

A problem of interest when considering a set of 𝑝 delays in [0,] is the case of potentially repeated delays, then subject to 0=012𝑝<, with 𝑞 of them {𝑗𝑝,𝑗𝑞{0}} being distinct, each being repeated 1𝜈𝑗𝑝(𝑗𝑞{0}) times so that0=0=0𝑝<1𝑝<<𝑞𝑝<,𝑝𝑗=0𝜈𝑗𝑝=𝑝+1,𝑗𝑝=𝑘+𝑖,𝑘=𝑗1=0𝜈,𝑖𝜈𝑗,𝑗𝑞{0}.(3.14)

Thus, the following result holds from Theorem 3.3 by grouping the terms of the delayed dynamics corresponding to the same potentially repeated delays.

Theorem 3.5. The Caputo solutions to the subsequent Caputo and Riemann-Liouville fractional differential systems of order 𝛼 with 𝑝0 (potentially repeated) delays and 0𝑞𝑝 distinct delays: 𝐶𝐷𝛼0+𝑥(𝑡)=𝑝𝑖=0𝐴𝑖𝑡𝑖+𝐵𝑢(𝑡),RL𝐷𝛼0+𝑥(𝑡)=𝑝𝑖=0𝐴𝑖𝑡𝑖+𝐵𝑢(𝑡)(3.15) on 𝐑0+ for the given set of initial conditions on [,0) are given by 𝑥𝛼(𝑡)=𝑘1𝑗=0Ψ𝛼𝑗0(𝑡)𝑥𝑗0+𝑞𝑖=1𝑖𝑝0Ψ𝛼𝑗0(𝑡𝜏)𝜑𝑗𝜏𝑖𝑝+𝑑𝜏𝑡0Ψ𝛼(𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏(3.16) for any 𝛼𝐂+ with 𝑘=[Re𝛼]+1 if 𝛼𝐙0+ and 𝑘=𝛼 if 𝛼𝐙0+, and, respectively by 𝑥𝛼(𝑡)=𝑘1𝑗=0Ψ𝛼𝑗0𝑡(𝑡)𝑗𝑥𝑗!𝑗0+𝑞𝑖=1𝑖𝑝0Ψ𝛼𝑗0(𝑡𝜏)𝜑𝑗𝜏𝑖𝑝+𝑑𝜏𝑡0Ψ𝛼(𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏(3.17) for any 𝐂𝛼+ with 𝑘=[Re𝛼]+1 if 𝛼𝐙+ and 𝑘=𝛼 if 𝛼𝐙+, where Ψ𝛼𝑗0(𝑡)=𝐸𝛼,𝑗+1𝜈01𝑖=0𝐴𝑖𝑡𝛼+𝑞𝑖=1𝑡0𝜏𝛼1𝐸𝛼,𝛼𝜈01𝑖=0𝐴𝑖𝜏𝛼𝜈𝑖=1𝐴(𝑖1𝑗=0𝜈𝑗+)Ψ𝛼𝑗0𝑡𝜏𝑖𝑑𝜏,(3.18)𝑡0 and Ψ𝛼𝑗0(𝑡)=Ψ𝛼(𝑡)=0, 𝑗𝑘1{0} for 𝑡[,0).

4. Nonnegativity of the Solutions

The positivity of the solutions of (3.1) independent of the values of the delays is now investigated under initial conditions 𝜑𝑗[,0]𝐑𝑛0+, 𝑗𝑘1{0}.

Theorem 4.1. The Caputo fractional differential system (3.1) under the delay constraint 0=0<1<2<<𝑝=< for any given absolutely continuous functions of initial conditions 𝜑𝑗[,0]𝐑𝑛0+, 𝑗𝑘1{0} and any piecewise continuous vector function 𝑢𝐑0+𝐑𝑛0+ if 𝑘=[𝛼]+1 if 𝛼𝐙+ and 𝑘=𝛼𝐙+,forall𝑡𝐑0+; for all 𝛼𝐑+ has following properties:
(i)Φ𝛼𝑗0(𝑡) is nonsingular; forall𝑗𝑘1{0} and Φ𝛼(𝑡)0; for all 𝑡𝐑0+ (if 𝐵𝐑+𝑛×𝑚 then Φ𝛼(𝑡)>0; for all 𝑡𝐑0+),(ii)(1)𝐴0𝑀𝐑𝑛×𝑛Φ(𝑡)Φ100(𝑡)>0; for all 𝑡𝐑0+,(2)𝐴0𝑀𝐑𝑛×𝑛Φ𝛼𝑗0(𝑡)>0;forall𝑗𝑘1{0}; for all 𝑡[0,𝑡𝑗) for some sufficiently small 𝑡𝑗𝐑+ with Φ𝛼00(𝑡)>0, for all 𝑡𝐑0+(i.e.,𝑡0=). This property holds for all 𝑡𝐑0+  (i.e., 𝑡𝑗=; for all 𝑗𝑘1{0}) if, in addition, either 𝐴00 or if 𝐴0 is nilpotent or if 0<𝛼𝑘=1. Furthermore, there are at least n entries (one per row) of Φ𝛼𝑗0(𝑡) being positive; forall𝑡𝐑0+; (iii)Any solution (3.2) to any Caputo fractional differential system (3.1) is nonnegative independent of the delays; that is, 𝑥𝛼(𝑡)𝐑𝑛0+; for all 𝑡[,𝑡)𝐑0+ for some 𝑡𝐑0+, for any set of delays satisfying 0=0<1<2<<𝑝< and any absolutely continuous functions of initial conditions 𝜑𝑗[,0]𝐑𝑛0+, for all 𝑗𝑘1{0} and any piecewise continuous control 𝑢𝐑0+𝐑𝑚0+, if and only if 𝐴0𝑀𝐑𝑛×𝑛 for 𝑡𝐑0+ being sufficiently small. Furthermore, 𝑥𝛼(𝑡)𝐑𝑛0+; for all 𝑡[,0)𝐑0+ if, in addition, either 𝐴00 or if 𝐴0 is nilpotent or if 0<𝛼𝑘=1,𝐴𝑖𝐑𝑛×𝑛0+(𝑖𝑝) and 𝐵𝐑𝑛×𝑚0+.

Proof. It is now proven that Φ𝛼𝑗0(𝑡)0; for all 𝑡𝐑0+𝐴0𝑀𝐑𝑛×𝑛; for all 𝛼𝐑+ for any 𝑗𝑘1{0}. First, note the following. If 𝛼=𝑘=1 then Φ𝛼00(𝑡)=𝐸1,1(𝐴0𝑡)=Φ(𝑡)==0𝐴0𝑡/!=𝑒𝐴0𝑡0 if 𝐴0𝑀𝐑𝑛×𝑛 from the above part of the proof and also 𝐴0𝑀𝐑𝑛×𝑛Φ(𝑡)0; for all 𝑡𝐑0+. This follows by contradiction. Assume that Φ𝑖𝑚(𝑡)<0 for some 𝑡𝐑+. Consider the positive differential system ̇𝑥(𝑡)=𝐴0𝑥(𝑡), 𝑥(0)=𝑒𝑗, 𝐴0𝑀𝐑𝑛×𝑛 so that 𝑥𝑖(𝑡)=|Φ𝑖𝑚(𝑡)|<0 which contradicts the system being positive. Thus, 𝐴0𝑀𝐑𝑛×𝑛Φ(𝑡)0; for all 𝑡𝐑0+. Furthermore, since Φ(𝑡) is a fundamental matrix of solutions of the differential system, it is non-singular for all finite time and the above result is weakened as follows:
𝐴0𝑀𝐑𝑛×𝑛(Φ(𝑡)=Φ𝛼00(𝑡)=𝑒𝐴0𝑡>0Φ(𝑡) is non-singular; for all 𝑡𝐑0+). Since Φ(𝑡) is nonsingular; for all 𝑡𝐑0+ at least 𝑛 of its entries (one per-row) is positive. Property (i) has been proven. Now, one gets from (3.3)-(3.4): Φ𝛼𝑗0(𝑡)=𝐸𝛼,𝑗+1𝐴0𝑡𝛼==0𝐴0𝑡𝛼=Γ(𝛼+𝑗+1)=0𝐴0𝑡𝑡!(𝛼1)!(𝛼+𝑗)Γ(𝛼+𝑗),𝑗𝑘1{0}.(4.1) Let 𝑒𝑖 the 𝑖th unit Euclidean vector of 𝐑𝑛 whose 𝑖th component is 1. Then, one obtains for all 𝑗𝑘1{0}, irrespective of the value of 𝛼𝐑+ and 𝑘𝐙+ being 𝑘=[𝛼]+1 if 𝛼𝐙+ and 𝑘=𝛼𝐙+, provided that 𝐴0𝑀𝐑𝑛×𝑛: Φ𝛼𝑗0𝑖𝑚(𝑡)=𝑒𝑇𝑖Φ𝛼𝑗0(𝑡)𝑒𝑚=𝑒𝑇𝑖=0𝐴0𝑡𝑡!(𝛼1)!𝑒Γ(𝛼+𝑗+1)𝑚=𝑒𝑇𝑖=0𝐴0𝑡𝑡!(𝛼1)!𝑒(𝛼+𝑗)Γ(𝛼+𝑗)𝑚𝑒𝑇𝑖𝑒𝐴0𝑡𝑒𝑚min0𝑡(𝛼1)!Γ(𝛼+𝑘)=𝑒𝑇𝑖=0𝐴0𝑡𝑡!(𝛼1)!𝑒(𝛼+𝑗)Γ(𝛼+𝑗)𝑚(4.2)=𝑒𝑇𝑖𝑁=0𝐴0𝑡𝑒!𝑚min0𝑁𝑡(𝛼1)!(𝛼+𝑗)Γ(𝛼+𝑘)=𝑒𝑇𝑖𝑒𝐴0𝑡𝑒𝑚min0𝑁𝑡(𝛼1)Γ(+1)(𝛼+𝑗)Γ(𝛼+𝑘)𝑒𝑇𝑖𝑒𝐴0𝑡𝑒𝑚min0𝑁𝑡(𝛼1)Γ(+1)(𝛼+𝑗)Γ(𝛼+𝑘)(4.3) for all (𝑖,𝑚)𝑛×𝑛, for all 𝑡𝐑0+ since 𝐴0𝑀𝐑𝑛×𝑛Φ(𝑡)0; for all 𝑡𝐑0+, for some 𝑁()𝐙0+ and 𝑁 is finite if and only if 𝐴0 is nilpotent (of degree 𝑁). Equation (4.3) implies that 𝑒𝑇𝑖(𝑒𝐴0𝑡)𝑒𝑚0 and then (Φ𝛼𝑗0𝑖𝑚(𝑡))0, for all (𝑖,𝑚)𝑛×𝑛 in the following cases:
(a) 𝑁, 𝑡[0,𝑡), since 𝐑0+ and some sufficiently small 𝑡𝐑+, since 𝐴0𝑀𝐑𝑛×𝑛𝐼+𝐴0𝑡>0; for all 𝑡𝐑0+ and Φ𝛼𝑗0(𝑡)=𝑁=0(𝐴0𝑡/!)(𝑡(𝛼1)!/(𝛼+𝑗)Γ(𝛼+𝑗))=𝐼+𝐴0𝑡+𝑜(𝑡)>0 for some sufficiently small 𝑡𝐑0+ for any 𝛼𝐑+.
(b) 𝑁 and 𝐴00 since 𝑡(𝛼1)!/(𝛼+𝑗)Γ(𝛼+𝑗)0; forall𝐙0+, for all 𝑗𝑘1{0} for any 𝛼𝐑+. It follows from inspection of (4.2) since 𝑒𝑇𝑖(𝑒𝐴0𝑡)𝑒𝑚0 for all (𝑖,𝑚)𝑛×𝑛, since 𝐴0𝑀𝐑𝑛×𝑛. This implies (Φ𝛼𝑗0𝑖𝑚(𝑡))0; for all 𝑡𝐑0+.
(c) 𝑁<min0𝑁(𝑡(𝛼1)/Γ(𝛼+𝑘))>0; for all 𝑗𝑘1{0} for any 𝛼𝐑+ so that 𝑒𝑇𝑖(𝑒𝐴0𝑡)𝑒𝑚0, for all (𝑖,𝑚)𝑛×𝑛, since 𝐴0𝑀𝐑𝑛×𝑛, irrespectively of 𝐴00 or not, what follows from (4.3). This implies (Φ𝛼𝑗0𝑖𝑚(𝑡))0; for all 𝑡𝐑0+.
(d) 𝑁, 0<𝛼𝑘=1. Then, 𝑗=0 so that 𝑡(𝛼1)!=𝑡Γ(𝛼+𝑗+1)(𝛼1)!=𝑡Γ(𝛼+1)(𝛼1)!=𝑡𝛼Γ(𝛼)(𝛼1)(1)!=𝛼Γ(𝛼)Γ()𝛼𝑡(1𝛼)1Γ(𝛼)𝛼𝑡(1𝛼),𝐙0+,𝑡𝐑0+(4.4) since 0<𝛼1 implies Γ(𝛼)=0𝜏𝛼1𝑒𝜏𝑑𝜏Γ()=0𝜏1𝑒𝜏𝑑𝜏,𝐙0+.(4.5) As a result, Φ𝛼00(𝑡) from (4.2); for all 𝑡𝐑0+. Also, direct calculations with (3.3)-(3.4) lead to Φ𝛼(𝑡)=𝑡𝛼1𝐸𝛼𝛼𝐴0𝑡𝛼==0𝑡𝛼1𝐴0𝑡𝛼=Γ((+1)𝛼)=0𝐴0𝑡𝑡!(𝛼1)(+1)!Γ((+1)𝛼)(4.6) and similar developments to the above ones yield (Φ𝛼(𝑡))𝑖𝑚0; for all (𝑖,𝑚)𝑛×𝑛, for all 𝑡𝐑0+ under the same conditions as above in the cases (a) to (d) for Φ𝛼00(𝑡). On the other hand, one gets from (3.2)–(3.4) for the unforced system with point initial conditions at 𝑡=0: 𝑥𝛼(𝑡)=𝑘1𝑗=0Φ𝛼𝑗0(𝑡)𝑥𝑗0=Φ𝛼00(𝑡),,Φ𝛼,𝑘1,0𝑥(𝑡)𝑇00,,𝑥𝑇𝑘1,0𝑇(4.7) which leads to 𝑥𝛼(𝑡)=Φ𝛼𝑖0(𝑡)𝑥𝑖0 by taking point initial conditions 𝑥𝑖00,𝑥𝑗0=0,(𝑖𝑗),  𝑗𝑘1{0} so that Φ𝛼𝑖0(𝑡) is nonsingular for all 𝑡𝐑0+ since otherwise the solution is not unique for each given set of initial conditions since any trajectory solution subject to some set of initial conditions 𝑥𝑖00, 𝑥𝑗0=0, would have infinitely many initial conditions, subject to identical constraint, so that such a trajectory is not unique which is a contradiction. Since this reasoning may be made for any 𝑗𝑘1{0}, Φ𝛼𝑗0(𝑡) is nonsingular for all 𝑗𝑘1{0}, all and, in addition, Φ𝛼𝑗0(𝑡)>0; forall𝑗𝑘1{0}, for all 𝑡𝐑0+ if either 𝐴00 or if 𝐴0 is nilpotent or if 0<𝛼𝑘=1 or without these restricting condition within some first interval [0,𝑡). The following properties have been proven:
(a) 𝐴0𝑀𝐑𝑛×𝑛Φ𝛼00(𝑡)>0; for all 𝑡𝐑0+,
(b) 𝐴0𝑀𝐑𝑛×𝑛(Φ𝛼𝑗0(𝑡)>0detΦ𝛼𝑗0(𝑡)0Φ𝛼(𝑡)0; for all 𝑗𝑘1{0}, 𝑘=[𝛼]+1 if 𝛼𝐙+and 𝑘=𝛼𝐙+, for all 𝑡𝐑0+).
It remains to prove Φ𝛼𝑗0(𝑡)>0; for all 𝑡𝐑0+𝐴0𝑀𝐑𝑛×𝑛; for all 𝑗𝑘1, some 𝑡𝐑0+. This is equivalent to its contrapositive logic proposition. Proceed by contradiction by assuming thereexist𝑗𝑘1 such that 𝐴0𝑀𝐑𝑛×𝑛Φ𝛼𝑗0(𝑡)<0, some 𝑡𝐑0+. Note that 𝐴0𝑀𝐑𝑛×𝑛𝑒𝑇𝑖𝑒𝐴0𝑡(𝑡)𝑒𝑚=𝑒𝑇𝑖Φ𝛼00(𝑡)𝑒𝑚<0, some 𝑡𝐑0+, some (𝑖,𝑚𝑖)𝑛. Then, one gets Φ𝛼𝑗0𝑖𝑚(𝑡)=𝑒𝑇𝑖Φ𝛼𝑗0(𝑡)𝑒𝑚𝑒𝑇𝑖𝑒𝐴0𝑡𝑒𝑚max0𝑡(𝛼1)!Γ(𝛼+𝑘)<0(4.8) which contradicts (Φ𝛼𝑗0(𝑡))>0; for all𝑡𝐑0+𝐴0𝑀𝐑𝑛×𝑛; for all 𝑗𝑘1, some 𝑡𝐑0+. Thus, the proof of Properties (i)-(ii) becomes complete since the above proven property (a) extends to any 𝑗𝑘1{0} as follows.
(c) 𝐴0𝑀𝐑𝑛×𝑛Φ𝛼𝑗0(𝑡)>0; for all 𝑗𝑘1{0}, 𝑘=[𝛼]+1 if 𝛼𝐙+ and 𝑘=𝛼𝐙+, forall𝑡𝐑0+; for all 𝛼𝐑+so that the unforced solution for any set of nonnegative point initial conditions is nonnegative for all time and, furthermore, 𝜑𝑖(𝑡)0(𝑖𝑘1{0}); for all 𝑡[,0],𝑢(𝑡)𝐑n0+; for all 𝑡𝐑0+,𝐴𝑖0(𝑖𝑝)and𝐵0; for all 𝑡𝐑0+ implies that (3.2) is everywhere nonnegative within its definition domain. The converse is also true as it follows by contradiction arguments. If there is one entry of B or 𝐴𝑖(𝑠ome𝑖𝑝) which is negative, or if 𝐴0𝑀𝐑𝑛×𝑛, it can always be found a control 𝑢(𝑡)𝐑𝑛0+ of sufficiently large norm along a given time interval such that some component of the solution is negative for some time. It can be also found that some nonnegative initial condition of sufficiently large norm at 𝑡=0 such that some component of the solution is negative at 𝑡=0+. Thus, Property (iii) is proven.

The following result is obvious from the proof of Theorem 4.1.

Corollary 4.2. Theorem 4.1(iii) is satisfied also independent of the delays for any given set of delays satisfying the constraint 0=012𝑝=<.

Proof. It follows directly since Theorem 4.1 is an independent of the delay size type result and, under the delay constraint 0=012𝑝=<, it has also to be fulfilled for any combination of delays satisfying the stronger constraint 0=0<1<2<<𝑝=<.

Corollary 4.3. Any solution (3.8), subject to (3.9), to the Caputo fractional differential system (3.1) under the delay constraint 0=012𝑝=< is nonnegatively independent of the delays within a first interval, that is, it satisfies 𝑥𝛼(𝑡)𝐑𝑛0+; for all 𝑡[,𝑡)𝐑0+ for some sufficiently small 𝑡𝐑0+ for any given absolutely continuous functions of initial conditions 𝜑𝑗[,0]𝐑𝑛0+, 𝑗𝑘1{0} and any given piecewise continuous vector function 𝑢𝐑0+𝐑𝑛0+ with 𝑘=[𝛼]+1 if 𝛼𝐙+ and 𝑘=𝛼𝐙+, for all 𝑡𝐑0+; for all 𝛼𝐑+ if and only if 𝐴0𝑀𝐑𝑛×𝑛, 𝐴𝑖𝐑𝑛×𝑛0+(𝑖𝑝), and 𝐵𝐑𝑛×𝑚0+. In addition, 𝑥𝛼[,0)𝐑0+𝐑𝑛0+ if, in addition, either 𝐴00 or if 𝐴0 is nilpotent or if 0<𝛼𝑘=1. Furthermore, Ψ𝛼𝑗0(𝑡)>0 (with at least n entries being positive), detΨ𝛼𝑗0(𝑡)>0(𝑗𝑘1{0}) and Ψ𝛼(𝑡)0; for all𝑡𝐑0+(if𝐵𝐑+𝑛×𝑚thenΨ𝛼(𝑡)>0;𝑡𝐑0+).

Proof. The solution (3.8) is identical to the unique solution (3.2) for (3.1) thus it is everywhere nonnegative under the same conditions that those of Theorem 4.1 which have been extended in Corollary 4.2.

Note that the conditions of nonnegativity of the solution of the above theorem also imply the excitability of all the components of the state-trajectory solution; that is its strict positivity for some 𝑡𝐑+ provided that 𝐵0 and the control 𝑢𝐑0+𝐑𝑛0+ is admissible (i.e., piecewise continuous) and nonidentically zero since Ψ𝛼(𝑡)>0 and nonsingular for all 𝑡𝐑+. It is now seen that the positivity conditions for the Riemann-Liouville fractional differential system (3.5) are not guaranteed in general by the above results for any given absolutely continuous functions of initial conditions 𝜑𝑗[,0]𝐑𝑛0+, 𝑗𝑘1{0} and any given piecewise continuous vector function 𝑢𝐑0+𝐑𝑛0+ with 𝑘=[𝛼]+1 if 𝛼𝐙+ and 𝑘=𝛼𝐙+, for all 𝑡𝐑0+; for all 𝛼𝐑+. The following two results hold by using Corollary 3.2 and Corollary 3.4.

Theorem 4.4. Any solution (3.7), subject to (3.3)-(3.4), to the Riemann-Liouville fractional differential system (3.5) under the delay constraint 0=012𝑝=< is everywhere nonnegative independent of the delays, that is, it satisfies 𝑥𝛼[,0)𝐑0+𝐑𝑛0+, for any given absolutely continuous functions of initial conditions 𝜑𝑗[,0]𝐑𝑛0+, 𝑗𝑘1{0} and any given piecewise continuous vector function 𝑢𝐑0+𝐑𝑛0+ with 𝑘=[𝛼]+1 if 𝛼𝐙+ and 𝑘=𝛼𝐙+, forall𝑡𝐑0+; for all 𝛼𝐑+ if 𝐴0𝑀𝐑𝑛×𝑛, 𝐴𝑖𝐑𝑛×𝑛0+(𝑖𝑝), (𝐸𝛼,𝑗+1(𝑡)(1/𝑗!)𝐼𝑛)0; forall𝑗𝑘1{0}, for all 𝑡𝐑0+ and 𝐵𝐑𝑛×𝑚0+. The conditions 𝐴0𝑀𝐑𝑛×𝑛, 𝐴𝑖𝐑𝑛×𝑛0+(𝑖𝑝) and 𝐵𝐑𝑛×𝑚0+ are also necessary for 𝑥𝛼[,0)𝐑0+𝐑𝑛0+ for any nonnegative function of initial conditions and nonnegative controls. The condition (𝐸𝛼,𝑗+1(𝑡)(1/𝑗!)𝐼𝑛)0; for all 𝑗𝑘1{0}, for all 𝑡𝐑0+ is removed for initial conditions 𝜑𝑗[,0]𝐑𝑛0+ subject to 𝜑𝑗(0)=𝑥𝑗0=0.

Proof. The proof follows in a similar way as the sufficiency part of the proof of Theorem 4.1(iii) by inspecting the nonnegative of the solution Corollary 3.2, (3.7) for a nonnegative function of initial conditions and any nonnegative control.

Theorem 4.5. Any solution (3.10), subject to (3.3)-(3.4), to the Riemann-Liouville fractional differential system (3.5) under the delay constraint 0=012𝑝=< is everywhere nonnegatively independent of the delays, that is, it satisfies 𝑥𝛼[,0)𝐑0+𝐑𝑛0+, for any given absolutely continuous functions of initial conditions 𝜑𝑗[,0]𝐑𝑛0+, 𝑗𝑘1{0} and any given piecewise continuous vector function 𝑢𝐑0+𝐑𝑛0+ with 𝑘=[𝛼]+1 if 𝛼𝐙+ and 𝑘=𝛼𝐙+, for all 𝑡𝐑0+; forall𝛼𝐑+ if and only if 𝐴0𝑀𝐑𝑛×𝑛, 𝐴𝑖𝐑𝑛×𝑛0+(𝑖𝑝), (Ψ𝛼𝑗0(𝑡)(𝑡𝑗/𝑗!)𝐼𝑛)0; for all 𝑗𝑘1{0}, for all 𝑡𝐑0+ and 𝐵𝐑𝑛×𝑚0+. The condition (Ψ𝛼𝑗0(𝑡)(𝑡𝑗/𝑗!)𝐼𝑛)0; for all 𝑗𝑘1{0}, for all 𝑡𝐑0+ is removed for initial conditions 𝜑𝑗[,0]𝐑𝑛0+ subject to 𝜑𝑗(0)=𝑥𝑗0=0.

Proof. The proof of sufficiency follows in a similar way as the sufficiency part of the proof of Theorem 4.1(iii) (see also the proof of Theorem 4.5) by inspecting the nonnegativity of the solution Corollary 3.2, (3.7) for a nonnegative function of initial conditions and any nonnegative control. The proof necessity follows by contradiction by inspecting the solution (3.10) as follows.
(a) Assume that 𝐴0𝑀𝐑𝑛×𝑛 and the solution is nonnegative for all time for any nonnegative function of initial conditions and controls. Take initial conditions 𝜑𝑗(𝑡)=0; for all 𝑡[,0], for all 𝑗𝑘1; 𝜑0(𝑡)=0; for all 𝑡[,0), 𝜑0(0)=𝑥00(0)0and 𝑢0 on 𝐑0+. Then (3.10) becomes 𝑥𝛼Ψ(𝑡)=𝛼00(𝑡)𝐼𝑛𝑥00=𝐸𝛼00(𝑡)𝐼𝑛𝑥00for𝑡0,1(4.9) since Ψ𝛼00(𝑡)=Φ𝛼00(𝑡) for 𝑡[0,1]. Since 𝐴0𝑀𝐑𝑛×𝑛, there exist 𝑡[0,1] and (𝑖=𝑖(𝑡),𝑚(𝑡)𝑖)𝑛 such that (Φ𝛼00(𝑡))𝑖𝑚<0. Otherwise, if 𝐴0𝑀𝐑𝑛×𝑛 and Φ𝛼00(𝑡)>0; for all 𝑡[0,1], it would follow from (4.3) that Φ𝛼00(𝑡)>0; for all 𝑡𝐑0+ since 𝑒𝐴0𝑡=𝑒𝜒𝐴01+𝐴0𝛿=𝑒𝜒𝐴01𝑒𝐴0𝛿=𝑒𝐴01𝜒𝑒𝐴0𝛿>0(4.10) from the semigroup property of (𝑒𝐴0𝑡,𝑡𝐑0+) with 𝜒=𝜒(𝑡,1)=[𝑡/1] and (0,1)𝛿=𝛿(𝑡,1)=𝑡𝜒1 what implies Φ𝛼00(𝑡)>0; for all 𝑡𝐑0+ from (4.3). Thus, 𝐴0𝑀𝐑𝑛×𝑛 which contradicts 𝐴0𝑀𝐑𝑛×𝑛. It has been proven that 𝐴0𝑀𝐑𝑛×𝑛𝑒𝑇𝑖Φ𝛼00(𝑡)𝑒𝑚<0; forall𝑡(0,1] for some (𝑖=𝑖(𝑡),𝑚(𝑡)𝑖)𝑛. Now, take 𝑥00𝑗=𝛿𝑗𝑚(𝑗𝑛) where 𝛿𝑗𝑚 denotes the Kronecker delta. Then, 𝑥𝛼𝑖(𝑡)=𝑒𝑇𝑖Ψ𝛼00𝑡(𝑡)𝑗𝐼𝑗!𝑛𝑥00=𝑒𝑇𝑖Φ𝛼00𝑡(𝑡)𝑗𝐼𝑗!𝑛𝑒𝑚=𝑒𝑇𝑖Φ𝛼00(𝑡)𝑒𝑚<0.(4.11) As a result, 𝐴0𝑀𝐑𝑛×𝑛 is a necessary condition for the solution to be nonnegative for all time irrespective of the delay sizes.
(b) Assume that the solution is nonnegative for all time for any nonnegative function of initial conditions and controls. Assume that 𝑒𝑇𝑖𝐴𝑒𝑗<0 and 𝑖; for all 𝑖()𝑝 for some 𝑖,𝑗𝑛, 𝑝. Take initial conditions 𝑥𝑗0=𝜑𝑗(0)=0; forall𝑡[,0]; forall𝑗𝑘1{0}, 𝜑𝑗0; forall𝑗𝑘1 and 𝑢0. One gets from (3.2) 𝑥𝛼𝑖(𝑡)=0𝑒𝑇𝑖Φ𝛼(𝑡𝜏)𝑗()𝑝𝐴𝑗𝜑0𝜏+𝑑𝜏0𝑒𝑇𝑖Φ𝛼(𝑡𝜏)𝐴𝜑0𝜏𝑑𝜏;𝑡0,𝑖(4.12) for the case =𝑖; forall𝑖(𝑝). Now, if =>, take a further specification of initial conditions as follows: 𝜑0(𝑡)=0; forall𝑡[0,], and 𝜑0(𝜏)=(𝑘1,,𝑘𝑛)𝑇0; forall𝑡(,] then 𝑥𝛼𝑖(𝑡)=𝑒𝑇𝑖Φ𝛼(𝑡𝜏)𝐴𝜑0𝜏=𝑑𝜏𝑛𝑛𝑟=1𝑚=1Φ𝛼𝑖𝑟(𝑡𝜏)𝐴𝑟𝑚𝑘𝑑𝜏𝑚=𝑛𝑗=1𝑘𝑗Φ𝑇𝛼𝑖𝐴(𝑡𝜏)𝑑𝜏𝑗=(𝑚𝑗)𝑛𝑘𝑚𝑚𝑟=1Φ𝛼𝑖𝑟𝐴(𝑡𝜏)𝑑𝜏𝑟𝑚+𝑘𝑗𝑚𝑟=1Φ𝛼𝑖𝑟𝐴(𝑡𝜏)𝑑𝜏𝑟𝑗,𝑡0,𝑖.(4.13) As a result, 𝐴𝑖0(𝑖𝑝) is a necessary condition for the solution to be nonnegative for all time irrespective of the delay sizes.
(c) Assume that the solution is nonnegative for all time for any nonnegative function of initial conditions and controls, and 𝐵0 is not fulfilled so that it exists at least an entry 𝐵𝑗<0 of 𝐵. Then, one has under identically zero initial conditions the following unique solution: 𝑥𝛼𝑖(𝑡)=𝑡0𝑒𝑇𝑖Ψ𝛼(=𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏𝑚𝑖=1𝑡0Ψ𝑇𝛼𝑖(𝑡𝜏)𝐵𝑖𝑢𝑖=(𝜏)𝑑𝜏𝑚𝑛𝑖=1=1𝑡0Ψ𝛼𝑖(𝑡𝜏)𝐵𝑖𝑢𝑖=(𝜏)𝑑𝜏(𝑖𝑗)𝑚𝑛=1𝑡0Ψ𝛼𝑖(𝑡𝜏)𝐵𝑖𝑢𝑖(𝜏)𝑑𝜏𝑛=1𝑡0Ψ𝛼𝑗(||𝐵𝑡𝜏)𝑗||𝑘𝑑𝜏𝑢𝑗<0,𝑡𝐑0+(4.14) provided that 𝑘𝑢𝑗>(𝑖𝑗)𝑚𝑛=1𝑡0Ψ𝛼𝑖(𝑡𝜏)𝐵𝑖𝑢𝑖(𝜏)𝑑𝜏/(𝑛=1𝑡0Ψ𝛼𝑗(𝑡𝜏)|𝐵𝑗|𝑑𝜏) by assuming that 𝐵0 fails because 𝐵𝑗<0 for some (,𝑗)𝑛×𝑚 and a constant control component 𝑢𝑗𝑘𝑢𝑗>0 is injected on the time interval [0,𝑡] for some arbitrary 𝑡𝐑+ for the remaining control components being chosen 𝑡  be nonnegative for all time. This contradicts that the solution is nonnegative for all time if the condition 𝐵0 fails.

Remark 4.6. Note that Theorem 4.1 can be extended as a necessary condition for 𝑡[0,1] since Ψ𝛼𝑗0(𝑡)=Φ𝛼𝑗0(𝑡) for 𝑡[0,1]; forall𝑗𝑘1{0}, forall𝑡𝐑0+.

Remark 4.7. Note by simple calculation that (𝐴0𝑀𝐑𝑛×𝑛𝐴𝑖0(𝑖𝑝))(𝑝𝑖=0𝐴𝑖)𝑀𝐑𝑛×𝑛. This is a necessary and sufficient condition for the nonnegativity of the solutions of the Caputo fractional differential system (3.1) of arbitrary order 𝛼𝐑+ under arbitrary nonnegative controls and initial conditions in the absence of delays; that is, for 𝑖=0; forall𝑖𝜔{0} and any 𝜔𝐙+.

Remark 4.8. The given conditions to guarantee that the solution is everywhere nonnegative under any given arbitrary nonnegative initial conditions and nonnegative controls are independent of the sizes of the delays type; that is, for any given set of 𝑝 delays. However, the conditions are weakened for particular situations involving repeated delays as follows. Note from Theorem 4.5 that the various given conditions 𝐴𝑖0 of necessary type to guarantee the nonnegativity of the solution under any admissible nonnegative controls and nonnegative initial conditions are weakened to (𝜈𝑖=1𝐴(𝑖1𝑗=0𝜈𝑗+)) if there is some repeated delay 𝑖 of multiplicity 𝜈𝑖2 (i.e., the number of distinct delays is 0𝑞<𝑝). Also, if 0=0 is repeated with multiplicity 𝜈02 then the condition 𝐴0𝑀𝐑𝑛×𝑛 for 𝜈0=1 is replaced by (𝜈01=0𝐴)𝑀𝐑𝑛×𝑛.

Remark 4.9. Note that there is a duality of all the given results of sufficiency type or necessary and sufficiency type in the sense that the solutions are guaranteed to be nonpositive for all time under similar conditions for the cases when all components of the controls and initial conditions are nonpositive for all time.

5. Asymptotic Behavior of Unforced Solutions for 𝛼𝐑+

The asymptotic behaviour and the stability properties of the Caputo fractional differential system (3.1) can be investigated via the extension of the subsequent formulas for 𝛼𝐑+, (see (1.8.27)–(1.8.29), [1]).(1)If 0<𝛼<2 then for |𝑧| and some 𝜇𝐑 satisfying 𝜇<𝜋min(1,𝛼):𝐸𝛼𝛽1(𝑧)=𝛼𝑧(1𝛽)/𝛼𝑒(𝑧1/𝛼)𝑁𝑗=111Γ(𝛽𝛼𝑗)𝑧𝑗1+𝑂𝑧𝑁+1(5.1) with |arg𝑧|𝜇<𝜋min(1,𝛼), any 𝑁𝐙+, and 𝐸𝛼𝛽(𝑧)=𝑁𝑗=111Γ(𝛽𝛼𝑗)𝑧𝑗1+𝑂𝑧𝑁+1(5.2) with 𝜋|arg𝑧|𝜇<𝜋=𝜋min(1,𝛼), any 𝑁𝐙+.(2)If 𝛼2 then for |𝑧|𝐸𝛼𝛽1(𝑧)=𝛼𝑗𝑄𝑧1/𝛼𝑒2𝑗𝜋𝐢/𝛼1𝛽𝑒(𝑒2𝑗𝜋𝐢/𝛼𝑧1/𝛼)𝑁𝑗=111Γ(𝛽𝛼𝑗)𝑧𝑗1+𝑂𝑧𝑁+1(5.3) for any 𝑁𝐙+ with 𝛽𝑘1{0}{𝛼}, |arg𝑧|𝛼𝜋/2, 𝑄={𝑛𝐙|arg𝑧+2𝜋𝑛|𝛼𝜋/2} and 𝐢=1 being the complex imaginary unit. The above formulas are extendable to the Mittag-Leffler matrix functions 𝐸𝛼,𝑗+1(𝐴0𝑡𝛼)==0(𝐴0𝑡𝛼)/Γ(𝛼+𝑗+1); forall𝑗𝑘1{0}, respectively, 𝐸𝛼𝛼(𝐴0𝑡𝛼) by identifying 𝑧𝐴0𝑡𝛼, 𝑧1/𝛼(𝐴0)1/𝛼𝑡 (if (𝐴0)1/𝛼exists) and 𝑧1𝐴01𝑡𝛼 (if 𝐴0 is non-singular), 𝛽𝑗+1, respectively, 𝛽𝛼. Irrespective of the existence of (𝐴0)1/𝛼and of 𝐴00 being singular or nonsingular, it is possible to identify 𝑧1𝐴01𝑡𝛼 and 𝑧𝐴0𝑡𝛼 and to use𝐸𝛼,𝑗+1𝐴0𝑡𝛼𝐸𝛼,𝑗+1𝐴0𝑡𝛼==0𝐴0𝑡𝛼Γ(𝛼+𝑗+1),𝑗𝐸𝑘1{0},𝛼𝛼𝐴0𝑡𝛼𝐸𝛼𝛼𝐴0𝑡𝛼==0𝐴0𝑡𝛼.Γ(𝛼+𝛼)(5.4)

The method may be used to calculate an asymptotic estimate of the solution (3.2) if 𝐴0 is non-singular (or an upperbounding function for any nonzero 𝐴0) of the Caputo fractional differential system (3.1), via (3.3)-(3.4), or, equivalently (3.8), via (3.9) and (3.3)-(3.4). The estimations may be extended with minor modification to the Riemann-Liouville fractional differential system (3.5). Note that if all the complex eigenvalues of 𝐴0 appear by conjugate pairs 𝐴0 then 𝐴0=𝑇1𝐽0𝐴0 where 𝐽0 is its real canonical form. First, consider two separate cases as follows.

(A) Assume that 𝛼𝐑+, 𝐴0 is real non-singular and (𝐴0)1/𝛼 exists; that is, there exist 𝑀 such that 𝑀𝛼=𝐴0 and 𝐴𝑖(𝑖𝑝) is real. Then, one gets from (5.1)–(5.3):𝐸𝛼,𝑗+1𝐴0𝑡𝛼=1𝛼𝐴0𝑗1/𝛼𝑡𝑗𝑒(𝐴01/𝛼)𝑡𝑁=11𝐴Γ((1𝛼)+1)0𝑡𝛼𝐴+𝑂0(𝑁+1)𝑡(𝑁+1)𝛼,𝑗𝐸𝑘1{0},𝛼𝛼𝐴0𝑡𝛼=1𝛼𝐴0(1𝛼)/𝛼𝑡1𝛼𝑒(𝐴01/𝛼)𝑡𝑁=11𝐴Γ((1)𝛼)0𝑡𝛼𝐴+𝑂0(𝑁+1)𝑡(𝑁+1)𝛼(5.5) as 𝑡 if 0<𝛼<2, for any 𝑁𝐙+, 𝐸𝛼,𝑗+1𝐴0𝑡𝛼=1𝛼𝑄𝐴01/𝛼𝑡𝑒2𝜋𝐢/𝛼𝑗𝑒(𝑒2𝜋𝐢/𝛼(𝐴01/𝛼)𝑡)𝑁=11𝐴Γ((1𝛼)+1)0𝑡𝛼𝐴+𝑂0(𝑁+1)𝑡(𝑁+1)𝛼,𝑗𝐸𝑘1{0},𝛼𝛼𝐴0𝑡𝛼=1𝛼𝑄𝐴01/𝛼𝑡𝑒2𝜋𝐢/𝛼1𝛼𝑒(𝑒2𝜋𝐢/𝛼(𝐴0