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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 161246, 25 pages
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.
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.
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 . There are a lot of results concerning the exact and approximate solutions of fractional differential equations of Riemann-Liouville and Caputo types, [1–4], fractional derivatives involving products of polynomials, [5, 6], fractional derivatives and fractional powers of operators, [7–9], 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 [11–17] 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., [16–18]) and Finance . 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., [20–22]). 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, . 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, [25–31]. 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, [31–33]. 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, . 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.
, , and are the sets of integer, real, and complex numbers, and are the positive integer and real numbers, and
The following notation is used to characterize different levels of positivity of matrices: is the set of all real matrices of nonnegative entries. If then is used as a simpler notation for .
is the set of all nonzero real matrices of nonnegative entries (i.e., at least one of their entries is positive). If then is used as a simpler notation for .
is the set of all real matrices of positive entries. If then is used as a simpler notation for . The superscript denotes the transpose, and are, respectively, the th row and the th column of the matrix .
A close notation to characterize the positivity of vectors is the following: is the set of all real vectors of nonnegative components. If then is used as a simpler notation for . is the set of all real nonzero vectors of nonnegative components (i.e., at least one component is positive). If then is used as a simpler notation for .
is the set of all real vectors of positive components. If then is used as a simpler notation for .
is a Metzler matrix if ; for all . is the set of Metzler matrices of order .
The maximum real eigenvalue, if any, of a real matrix , is denoted by . 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 for some real interval satisfies and, furthermore, exists everywhere in for for some . Then, the Riemann-Liouville left-sided fractional derivative of order of the vector function in is pointwise defined in terms of the Riemann-Liouville integral as where the integer is given by and , where , is the -function defined by ; . If and, furthermore, exists everywhere in , then the Caputo left-sided fractional derivative of order of the vector function in is pointwise defined in terms of the Riemann-Liouville integral as where if and if . The following relationship between both fractional derivatives holds provided that they exist (i.e., if possesses Caputo left-sided fractional derivative in ),  Since , the above formula relating both fractional derivatives proves the existence of the Caputo left-sided fractional derivative in if the Riemann-Liouville one exists in .
3. Solution of a Fractional Differential Dynamic System of Any Order with Internal Point Delays
Consider the linear and time-invariant differential functional Caputo fractional differential system of order : with ; , , being distinct constant delays, , are the matrices of dynamics for each delay , is the control matrix. The initial condition is given by -real vector functions , with , which are absolutely continuous except eventually in a set of zero measure of of bounded discontinuities with . The function vector is any given bounded piecewise continuous control function. The following result is concerned with the unique solution on 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 ).
Theorem 3.1. The linear and time-invariant differential functional fractional differential system (3.1) of any order has a unique solution on for each given set of initial functions , being absolutely continuous except eventually in a set of zero measures of of bounded discontinuities with ; and each given control being a bounded piecewise continuous control function. Such a solution is given by with if and if , for and for , where 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 as follows: 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 , one gets Since the Caputo left-sided fractional derivative and the Riemann-Liouville fractional integral of order are inverse operators (what is not the case if ), (see [1, Lemma 2.21(a)]), one gets from (3.6), (2.3), and (3.2) if the subsequent result for the fractional differential system (3.5) on .
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 and the evolution operator defined in [2, 3] for the standard (nonfractional differential system), that is, in (3.1). Thus, another equivalent expression for the unique solution of the Caputo fractional differential system of order is given in the subsequent result.
Corollary 3.4. If (3.5) being of order is replaced with (3.1) under the same initial conditions then its unique solution on is given by with if and if 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 which yields the solution:(2)a further particular case which yields the solution: since , which is the unique solution of under any almost everywhere absolutely continuous function (except eventually in some subset of zero measure of of bounded discontinuities) of initial conditions . Use for this case, the less involved notations for the smooth evolution operator from to , and , for the exponential matrix function from to , which defines a -semigroup of infinitesimal generator from to . Then, the unique solution for the given function of initial conditions is and for , where satisfies , and with (the -identity matrix) and , which has a unique solution for , [2, 3, 25, 26].
A problem of interest when considering a set of delays in is the case of potentially repeated delays, then subject to , with of them being distinct, each being repeated times so that
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 with (potentially repeated) delays and distinct delays: on for the given set of initial conditions on are given by for any with if and if , and, respectively by for any with if and if , where and , for .
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 , .
Theorem 4.1. The Caputo fractional differential system (3.1) under the delay constraint for any given absolutely continuous functions of initial conditions , and any piecewise continuous vector function if if and ; for all has following properties:
(i) is nonsingular; and ; for all (if then ; for all ),(ii) (1); for all ,(2); for all for some sufficiently small with , for all . This property holds for all (i.e., ; for all ) if, in addition, either or if is nilpotent or if . Furthermore, there are at least n entries (one per row) of being positive; ; (iii)Any solution (3.2) to any Caputo fractional differential system (3.1) is nonnegative independent of the delays; that is, ; for all for some , for any set of delays satisfying and any absolutely continuous functions of initial conditions , for all and any piecewise continuous control , if and only if for being sufficiently small. Furthermore, ; for all if, in addition, either or if is nilpotent or if and .
Proof. It is now proven that ; for all ; for all for any . First, note the following. If then if from the above part of the proof and also ; for all . This follows by contradiction. Assume that for some . Consider the positive differential system , , so that which contradicts the system being positive. Thus, ; for all . Furthermore, since 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:
is non-singular; for all . Since is nonsingular; for all at least of its entries (one per-row) is positive. Property (i) has been proven. Now, one gets from (3.3)-(3.4): Let the th unit Euclidean vector of whose th component is 1. Then, one obtains for all , irrespective of the value of and being if and , provided that : for all , for all since ; for all , for some and is finite if and only if is nilpotent (of degree ). Equation (4.3) implies that and then , for all in the following cases:
(a) , , since and some sufficiently small , since ; for all and for some sufficiently small for any .
(b) and since ; , for all for any . It follows from inspection of (4.2) since for all , since . This implies ; for all .
(c) ; for all for any so that , for all , since , irrespectively of or not, what follows from (4.3). This implies ; for all .
(d) , . Then, so that since implies As a result, from (4.2); for all . Also, direct calculations with (3.3)-(3.4) lead to and similar developments to the above ones yield ; for all , for all under the same conditions as above in the cases (a) to (d) for . On the other hand, one gets from (3.2)–(3.4) for the unforced system with point initial conditions at : which leads to by taking point initial conditions , so that is nonsingular for all 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 , , 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 , is nonsingular for all , all and, in addition, ; , for all if either or if is nilpotent or if or without these restricting condition within some first interval . The following properties have been proven:
(a) ; for all ,
(b) ; for all , if and , for all ).
It remains to prove ; for all ; for all , some . This is equivalent to its contrapositive logic proposition. Proceed by contradiction by assuming such that , some . Note that , some , some . Then, one gets which contradicts ; for all; for all , some . Thus, the proof of Properties (i)-(ii) becomes complete since the above proven property (a) extends to any as follows.
(c) ; for all , if and , ; for all so that the unforced solution for any set of nonnegative point initial conditions is nonnegative for all time and, furthermore, ; for all ; for all ; for all 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 which is negative, or if , it can always be found a control 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 such that some component of the solution is negative at . 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 .
Proof. It follows directly since Theorem 4.1 is an independent of the delay size type result and, under the delay constraint , it has also to be fulfilled for any combination of delays satisfying the stronger constraint .
Corollary 4.3. Any solution (3.8), subject to (3.9), to the Caputo fractional differential system (3.1) under the delay constraint is nonnegatively independent of the delays within a first interval, that is, it satisfies ; for all for some sufficiently small for any given absolutely continuous functions of initial conditions , and any given piecewise continuous vector function with if and , for all ; for all if and only if , , and . In addition, if, in addition, either or if is nilpotent or if . Furthermore, (with at least n entries being positive), and ; for all.
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 provided that and the control is admissible (i.e., piecewise continuous) and nonidentically zero since and nonsingular for all . 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 , and any given piecewise continuous vector function with if and , for all ; for all . 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 is everywhere nonnegative independent of the delays, that is, it satisfies , for any given absolutely continuous functions of initial conditions , and any given piecewise continuous vector function with if and , ; for all if , , ; , for all and . The conditions , and are also necessary for for any nonnegative function of initial conditions and nonnegative controls. The condition ; for all , for all is removed for initial conditions subject to .
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 is everywhere nonnegatively independent of the delays, that is, it satisfies , for any given absolutely continuous functions of initial conditions , and any given piecewise continuous vector function with if and , for all ; if and only if , , ; for all , for all and . The condition ; for all , for all is removed for initial conditions subject to .
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 and the solution is nonnegative for all time for any nonnegative function of initial conditions and controls. Take initial conditions ; for all , for all ; ; for all , and on . Then (3.10) becomes since for . Since , there exist and such that . Otherwise, if and ; for all , it would follow from (4.3) that ; for all since from the semigroup property of with and what implies ; for all from (4.3). Thus, which contradicts . It has been proven that ; for some . Now, take where denotes the Kronecker delta. Then, As a result, 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 and ; for all for some , . Take initial conditions ; ; , ; and . One gets from (3.2) for the case ; . Now, if , take a further specification of initial conditions as follows: ; , and ; then As a result, 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 is not fulfilled so that it exists at least an entry of . Then, one has under identically zero initial conditions the following unique solution: provided that by assuming that fails because for some and a constant control component is injected on the time interval for some arbitrary for the remaining control components being chosen be nonnegative for all time. This contradicts that the solution is nonnegative for all time if the condition fails.
Remark 4.6. Note that Theorem 4.1 can be extended as a necessary condition for since for ; , .
Remark 4.7. Note by simple calculation that . This is a necessary and sufficient condition for the nonnegativity of the solutions of the Caputo fractional differential system (3.1) of arbitrary order under arbitrary nonnegative controls and initial conditions in the absence of delays; that is, for ; and any .
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 delays. However, the conditions are weakened for particular situations involving repeated delays as follows. Note from Theorem 4.5 that the various given conditions of necessary type to guarantee the nonnegativity of the solution under any admissible nonnegative controls and nonnegative initial conditions are weakened to if there is some repeated delay of multiplicity (i.e., the number of distinct delays is ). Also, if is repeated with multiplicity then the condition for is replaced by .
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
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 , (see (1.8.27)–(1.8.29), ).(1)If then for and some satisfying : with , any , and with , any .(2)If then for for any with , , and being the complex imaginary unit. The above formulas are extendable to the Mittag-Leffler matrix functions ; , respectively, by identifying , (if exists) and (if is non-singular), , respectively, . Irrespective of the existence of and of being singular or nonsingular, it is possible to identify and and to use
The method may be used to calculate an asymptotic estimate of the solution (3.2) if is non-singular (or an upperbounding function for any nonzero ) 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 appear by conjugate pairs then where is its real canonical form. First, consider two separate cases as follows.