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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 867932, 19 pages doi:10.1155/2011/867932 Research Article About Robust Stability of Caputo Linear Fractional Dynamic Systems with Time Delays through Fixed Point Theory M. De la Sen Faculty of Science and Technology, University of the Basque Country, 644 de Bilbao, Leioa, 48080 Bilbao, Spain Correspondence should be addressed to M. De la Sen, manuel.delasen@ehu.es Received 9 November 2010; Accepted 31 January 2011 Academic Editor: Marl` ne Frigon e Copyright q 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 investigates the global stability and the global asymptotic stability independent of the sizes of the delays of linear time-varying Caputo fractional dynamic systems of real fractional order possessing internal point delays. The investigation is performed via ﬁxed point theory in a complete metric space by deﬁning appropriate nonexpansive or contractive self-mappings from initial conditions to points of the state-trajectory solution. The existence of a unique ﬁxed point leading to a globally asymptotically stable equilibrium point is investigated, in particular, under easily testable suﬃciency-type stability conditions. The study is performed for both the uncontrolled case and the controlled case under a wide class of state feedback laws. 1. Introduction Fractional calculus is 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. There is a good compendium of related results with examples and case studies in 1 . Also, there is an existing collection of results in the background literature 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 for instance 1, 10 and so forth. On the other hand, there is also an increasing interest in the recent mathematical related to dynamic fractional diﬀerential systems oriented towards several ﬁelds of science like physics, chemistry or control theory. Perhaps the reason of interest in fractional calculus is that the numerical value of the fraction parameter allows
2 Fixed Point Theory and Applications a closer characterization of eventual uncertainties present in the dynamic model. We can also ﬁnd, in particular, abundant literature concerned with the development of Lagrangian and Hamiltonian formulations where the motion integrals are calculated though fractional calculus and also in related investigations concerned 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, for instance, 16–18 , 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, for instance 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 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, 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 delays and also being independent of eventual coincidence of some values of delays if those ones are, in particular, multiple related to the associate matrices of dynamics. Most of the results are centred in characterizations via Caputo fractional diﬀerentiation although some extensions presented are concerned with the classical Riemann-Liouville diﬀer-integration. It is proved that the existence 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. On the other hand, ﬁxed point theory is a very powerful mathematical tool to be used in many applications where stability knowledge is needed. For instance, the concepts of contractive, weak contractive, asymptotic contractive and nonexpansive mappings have been investigated in detail in many papers from several decades ago see, for instance, 32–34 and references therein . It has been found, for instance, that contractivity, weak contractivity and asymptotic contractivity ensure the existence of a unique ﬁxed pointing complete metric or Banach spaces. Some theory and applications of some types of functional equations in the context of ﬁxed point theory have been investigated in 35, 36 . Fixed point theory has also been employed successfully in stability problems of dynamic systems such as time-delay and continuous-time/digital hybrid systems and in those involving switches among diﬀerent parameterizations. This paper is concerned with the investigation of ﬁxed points in Caputo linear fractional dynamic systems of real order α which involved delayed dynamics subject to a ﬁnite set of bounded point delays which can be of arbitrary sizes. The self-mapping deﬁned in the state space from initial conditions to points of the state—trajectory solution are characterized either as nonexpansive or as contractive. The ﬁrst case allows to establish global stability results while the second one characterizes global asymptotic stability. 1.1. Notation C , R, and Z are the sets of complex, real, and integer numbers, respectively.
Fixed Point Theory and Applications 3 R and Z are the sets of positive real and integer numbers, respectively, C is the set of complex numbers with positive real part. C0 : C ∪ {iω : ω ∈ R}, where i is the complex unity, R0 : R ∪ {0} and Z 0 : Z ∪ {0}. R− and Z− are the sets of negative real and integer numbers, respectively; and C− is the set of complex numbers with negative real part. C0− : C− ∪{iω : ω ∈ R}, where i is the complex unity, R0 : R− ∪{0} and Z0− : Z− ∪{0}. N : {1, 2, . . . , N } ⊂ Z0 , “∨” is the logic disjunction, and “∧” is the logic conjunction. t/h is the integer part of the rational quotient t/h. σ M denotes the spectrum of the real or complex square matrix M i.e., its set of distinct eigenvalues . denotes any vector or induced matrix norm. Also, m p and M p are the p -norms of the vector m or induced real or complex matrix M, and μp M denote the p measure of the square matrix M, 20 . The matrix measure μp M is deﬁned In ε X p − ε /ε which has the property as the existing limit μp M : limε → 0 max − M p , maxi∈n reλi M ≤ μp M ≤ M p for any square n-matrix M of spectrum {λ i M ∈ C : 1 ≤ i ≤ n}. An important property for the investigation of this σM paper is that μ2 M < 0 if M is a stability matrix: that is, if re λ i M < 0; 1 ≤ i ≤ n. ∞ denotes the supremum norm on R0 , or its induced supremum metric, for functions or vector and matrix functions without speciﬁcation of any pointwise particular vector or matrix norm for each t ∈ R0 . If pointwise vector or matrix norms are speciﬁed, the corresponding particular supremum norms are deﬁned by using an extra subscript. Thus, m p∞ : supt∈R0 m t p and M p∞ : supt∈R0 M t p are, respectively, the supremum norms on R0 for vector and matrix functions of domains in R0 ×Rn , respectively, in R0 ×Rn×m deﬁned from their p pointwise respective norms for each t ∈ R0 . In is the nth identity matrix. Kp M is the condition number of the matrix M with respect to the p -norm; k : {1, 2, . . . , k}. 1.1 The sets BPC i dom, codom and PC i dom, codom are the sets of functions of a certain domain and codomain which are of class C i−1 dom, codom and with the ith derivative is bounded piecewise continuous, respectively, piecewise continuous in the deﬁnition domain. 2. Caputo Fractional Linear Dynamic Systems with Point Constant Delays and the Contraction Mapping Theorem Consider the linear functional Caputo fractional dynamic system of order α with r delays: t xk τ 1 α D0 x t : dτ Γ k−α t − τ α 1−k 0 r Ai t x t − ri Btut 2.1 i0 r r Ai x t − ri Ai t x t − ri Btut, i0 i0
4 Fixed Point Theory and Applications with k − 1 < α ∈ R ≤ k; k ∈ Z , 0 r0 < r1 < r2 < · · · < rr h < ∞ being distinct constant delays, where ri i ∈ r are the r in general incommensurate delays 0 r0 < ri i ∈ r subject to the system piecewise continuous bounded matrix functions of delayed dynamics Ai : R0 → Rn×n i ∈ r ∪ {0} which are decomposable as a nonunique sum of a constant Ai Ai t , for all t ∈ R0 , and matrix plus a bounded matrix function of time, that is, Ai t B : R0 → Rn×m is the piecewise continuous bounded control matrix. The initial condition is given by k n-real vector functions ϕj : −h, 0 → Rn , with j ∈ k − 1 ∪ {0}, which are absolutely continuous except eventually in a set of zero measure of −h, 0 ⊂ R of bounded xj 0 , j ∈ k − 1 ∪ {0}. The function vector xj 0 discontinuities with ϕj 0 xj 0 u : R0 → R is any given bounded piecewise continuous control function. The following m result is concerned with the unique solution on R0 of the above diﬀerential fractional system 3.1 . The proof, which is based on Picard-Lindelof theorem, follows directly from a parallel ¨ existing result from the background literature on fractional diﬀerential systems by grouping all the additive forcing terms of 2.1 in a unique one see for instance 1, 1.8.17 , 3.1.34 – 3.1.49 , with f t ≡ r 1 Ai x t − hi Bu t . For the sake of simplicity, the domains of initial i conditions and controls are all extended to −h, 0 ∪ R0 by zeroing them on the irrelevant intervals of −h, 0 so that any solution for t ∈ R0 of 2.1 is identical to the corresponding one under the above given deﬁnition domains of vector functions of initial conditions and controls. Theorem 2.1. The linear and time- invariant diﬀerential functional fractional dynamic system 2.1 of any order α ∈ C0 has a unique continuous solution on −h, 0 ∪ R0 satisfying a x ≡ ϕ ≡ k−0 ϕj on R0 with ϕj 0 1 xj 0 ; j ∈ k − 1 ∪ {0}; for all t ∈ xj 0 xj 0 j −h, 0 for each given set of initial functions and ϕj : −h, 0 → Rn , j ∈ k − 1 ∪ {0} being bounded piecewise continuous with eventual discontinuities in a set of zero measure of −h, 0 ⊂ R of bounded discontinuities, that is, ϕj ∈ BPC 0 −h, 0 , Rn ; j ∈ k − 1 ∪ {0} and each given bounded piecewise continuous control u : R0 → Rm , with u t 0 for t ∈ −h, 0 , being a bounded piecewise continuous control function, and b k −1 ri r Φαj t xj 0 Φα t − τ Ai ϕj τ − ri dτ xα t 0 j0 i1 ri r Φα t − τ Ai τ ϕj τ − ri dτ 2.2 0 i1 t t r r Φα t − τ Ai τ xα τ − ri dτ Φα t − τ Ai τ xα τ − ri dτ ri ri i0 i1 t Φα t − τ B τ u τ dτ, t ∈ R0 , 0 which is time-diﬀerentiable satisfying 2.1 in R with k 1 if α ∈ Z and k α if α ∈ Z , Re α / and Φα t : tα− 1 Eαα A0 tα , Φαj t : tj Eα,j A0 t α , 1 2.3 ∞ A0 t α j ∈ k − 1 ∪ {0, α}, Eαj A0 tα : , Γα j 0 for t ∈ R and Φα0 t Φα t 0 for t < 0, where Eα,j A0 tα are the Mittag-Leﬄer functions. 0
Fixed Point Theory and Applications 5 A technical result about norm upper-bounding functions of the matrix functions 2.3 - 2.4 follows Lemma 2.2. The following properties hold. i There exist ﬁnite real constants KEαj ≥ 1, KΦαj ≥ 1; j ∈ k − 1 ∪ {0} and KΦα ≥ 1 such that for any α ∈ R < 1 Eαj A0 tα ≤ KEαj eA0 t , Φαj t ≤ KΦαj tj eA0 t , j ∈ k − 1 ∪ {0, α}, 2.4 1−α Φα t ≤ KΦα for t ∈ R ≥ 1. e A0 t , 1/t ii If α ∈ R ≥ 1 then ∞ A 0 tα ! ! tα ≤ sup Eαj A0 tα e A0 Γ Γα ! 1 j ∈Z0 0 tα j ∈ k − 1 ∪ {0}, t ∈ R0 , e A0 , ! tα tα Φαj t ≤ sup ≤ t j e A0 j ∈ k − 1 ∪ {0}, t ∈ R0 , t j e A0 , Γα j 1 ∈Z0 ! tα− 1 e A0 ≤ tα−1 eA0 tα tα Φα t ≤ sup t ∈ R0 . , Γ 1α ∈Z0 2.5 If, in addition, A0 is a stability matrix then eA0 t ≤ Ke− λt and eA0 ≤ Ke− λt ≤ Ke−λt ; t ∈ R0 tα α for some real constants K ≥ 1, λ ∈ R . Then, one gets from 2.5 ≤ Ke−λt , ≤ tj e−λ t , Φαj t j ∈ k − 1 ∪ {0}, Eαj A0 tα 2.6 ≤ tα−1 e−λt Φα t for t ∈ R0 , and the fractional dynamic system in the absence of delayed dynamics is exponentially stable if the standard fractional system for α 1 is exponentially stable. iii The following inequalities hold. ≤ tk−α Φα t Φα, k−1 t for α ∈ k − 1, k ∩ R for k ∈ Z , t ∈ R0 , 1−k Φα t ≤ tα Φα,k−2 t for α ∈ k − 1, k ∩ R , t ∈ R0 , 2.7 Φk t ≡ Φk,k−1 t k ∈ Z , t ∈ R0 . for α
6 Fixed Point Theory and Applications Proof. Note from 2.3 - 2.4 for 0 < α ∈ R
Fixed Point Theory and Applications 7 The inequalities 2.4 hold since the above matrix norms are bounded on the real interval 1, ∞ and their limit superior is upper-bounded by the given formulas and Property i is proved. On the other hand, if R α ≥ 1 then ∞ A 0 tα ! Eαj A0 tα Γα ! j 0 ! tα tα ≤ sup j ∈ k − 1 ∪ {0}, t ∈ R0 e A0 e A0 , Γ 1 ∈Z0 ! tα tα Φαj t ≤ sup ≤ t j e A0 j ∈ k − 1 ∪ {0}, t ∈ R0 , t j e A0 , Γα j 1 ∈Z0 ! tα−1 eA0 ≤ tα−1 eA0 tα tα Φα t ≤ sup t ∈ R0 . , Γ 1α ∈Z0 2.10 α If, in addition, A0 is a stability matrix then eA0 t ≤ Ke−λt and eA0 t ≤ Ke−λt ≤ Ke−λt , t ∈ α α R0 for some real constants K ≥ 1 and λ ∈ R since tα ≥ t, for all t ∈ R0 . Properties i - ii have been proved. iii It is proved as follows. Note from 2.3 - 2.4 that 1 α−1 ∞ ∞ 1 α t1 j − α A0 t α j Γ A0 t Φαj t ≤ sup t ∈ R0 , , Γ Γα Γα 1α j 1 j1 ∈Z0 0 0 2.11 j ∈ k − 1 ∪ {0}, so that if k − 1 < α ∈ R ≤ k, then 1 α−1 ∞ j −α Γ 1 α t1 A0 t ≤ tk−α Φα t , Φα,k−1 t ≤ sup t ∈ R0 2.12 Γα Γ k 1α ∈Z0 0 Also, 1 α−1 ∞ ∞ Γα A0 t α j A0 t j1 ≤ tα−j −1 sup Φα t Γ Γ Γα 1α 1α j 1 ∈Z0 2.13 0 0 α−j −1 ≤t Φαj t t ∈ R0 , , Γα 1 /Γ < ∞. This implies that if sup j 1α ∈Z0 1−k Φα t ≤ tα Φα,k−2 t for α ∈ k − 1, k ∩ R , 2.14 Φk t ≡ Φk,k−1 t for α ∈ k − 1, k ∩ R .
8 Fixed Point Theory and Applications 3. Fixed Point Results A technical deﬁnition is now given to facilitate the subsequent result about ﬁxed point. Property ii has been proven. Deﬁnition 3.1. S ϕ, u is the set of all the piecewise continuous n-vector function from −h, 0 ∪ R0 to Rn being time- diﬀerentiable in R which are solutions of 2.1 for all admissible k-tuples of initial conditions ϕ : ϕ0 , ϕ1 , . . . , ϕk−1 with ϕj ∈ BPC 0 −h, 0 , Rn and controls u ∈ BPC 0 R0 , Rn with ϕj 0 xj 0 ; for all j ∈ k − 1 ∪ {0}. xj 0 xj 0 A ﬁxed point theorem is now given for the Caputo fractional system 2.1 . Theorem 3.2. Assume any set of r given ﬁnite delays 0 r0 < r1 ≤ . . . ≤ rr h < ∞. The following properties hold. δ i Assume that Φαj ∈ L∞ R0 , Rn×n and 0 Φα δ − τ dτ A0 ∞ < 1; let gh : R → R0 be deﬁned by −1 δ 1− Φα δ − τ dτ gh δ : A0 ∞ 0 ⎛ ⎞ 3.1 k −1 δ r ×⎝ ⎠ ≤ 1, Φαj δ Φα δ − τ dτ δ∈R . Ai ∞ 0 j0 i1 Then, the mapping fh : −h, 0 × Rn → R × Rn deﬁned by the state trajectory solution 2.2 of the uncontrolled system from any initial conditions in the admissible set is nonexpansive, and the solution is bounded fulﬁlling supt∈R0 xα t ∞ ≤ supt∈ −h,0 k−0 ϕj t ∞ . If gh δ ≤ Kc δ < 1 j 1; for allδ ∈ R then fh : −h, 0 × R → R × R is contractive and possesses a unique ﬁxed point, n n irrespective of the delays, in some bounded subset of Rn . Such a ﬁxed point is 0 ∈ Rn which is also a globally asymptotically stable equilibrium point. δ δ ii Assume that Φαj ∈ L∞ R0 , Rn×n , Φα ∈ L2 R0 , Rn×n and 0 Φα δ −τ dτ 0 A0 t τ 2 dτ 1/2 < 1; for all t ∈ R0 and deﬁne pointwise gh : R0 × R → R0 as follows δ δ −1 1− Φα δ − τ dτ 2 1/2 g h t, δ : A t τ dτ 0 0 0 ⎛ k −1 δ ×⎝ Φαj δ Φα δ − τ 2 1/2 dτ 3.2 ∞ 0 j0 δ r 1/2 × τ − ri δ∈R . 2 Ai t dτ , 0 i1 Then, Property (i) still holds by replacing their corresponding constraints on gh by corresponding ones on gh . r i 0 Ki xt , t x t − ri is injected to 2.1 where Ki : iii Assume that a control u t Rn × R0 → Rm is in BPC R0 , Rm , xit : max 0, t − ri , t → Rn , for all i ∈ r − 1 ∪ {0}, for all
Fixed Point Theory and Applications 9 t ∈ R0 is a strip of the state-trajectory solution of 2.1 . Assume also that ∀i ∈ r − 1 ∪ {0}, ∀t ∈ R0 , Φαj ∈ L∞ R0 , Rn×n , ≤ Ki0 < ∞, Ki xit , t ∞ 3.3 R0 , Rn×n , Φα ∈ L1 and deﬁne gf : R → R0 as −1 δ 1− Φα δ − τ dτ 0 gh δ : A0 B ∞ K0 ∞ 0 ⎛ ⎞ k −1 3.4 δ r ×⎝ ⎠ ≤ 1, Φαj δ Φα δ − τ dτ 0 Ai B ∞ Ki ∞ 0 j0 i1 δ∈R , δ provided that 0 Φα δ − τ dτ A0 ∞ 0 B ∞ K0 < 1. Then, for any given set of ﬁnite delays, the mapping ff : −h, 0 × Rn × Rm × R0 → R × Rn deﬁned by the state trajectory solution 2.2 of the controlled system from any initial conditions in the admissible set and any given admissible control is a nonexpansive mapping if gh δ ≤ 1; for all δ ∈ R and contractive and the zero equilibrium is the unique ﬁxed point, irrespective of the delays and control, if gh δ ≤ Kc δ < 1; for all δ ∈ R which is also a globally asymptotically stable equilibrium point. iv Assume that ∃ ε < 1 ∈ R that gh δ < 1 − ε; for all t ∈ R0 . Then, state trajectory solution 2.2 of the forced system from any initial conditions in the admissible set is deﬁned by a contractive self-mapping with a unique ﬁxed point in some bounded subset of Rn for all controls of the δ r i 0 Ki xit , t x t − ri fulﬁlling Ki xit , t ∞ ≤ ε/ r 1 Φα δ − τ dτ B ∞ , for form u t 0 all i ∈ r − 1 ∪ {0}. v Assume that Φαj ∈ L∞ R0 , Rn×n ; for all j ∈ k − 1 ∪ {0}, Φα ∈ L2 R0 , Rn×n and BKi ∈ L R0 , Rn×n ; for all i ∈ r − 1 ∪{0}, instead of the hypotheses 3.3 , and deﬁne gf : R0 × R → R0 2 as: ⎛ δ gf t , δ : ⎝ 1 − Φα δ − τ dτ 0 ⎛ ⎞⎞−1 1/2 1/2 δ δ ×⎝ ⎠⎠ 2 2 A0 t τ dτ Bt τ K0 xt τ , t τ dτ 0 0 ⎛ 1/2 k −1 δ ×⎝ Φαj δ Φα δ − τ 2 dτ ∞ 0 j0 ⎛ ⎞⎞ 1/2 1/2 δ δ r ×⎝ ⎠⎠, τ − ri 2 2 Ai t dτ Bt τ Ki xt τ , t τ dτ 0 0 i1 ∀t, δ ∈ R , 3.5
10 Fixed Point Theory and Applications provided that the inverse exists on R0 . Then, Property (iii) still holds by replacing their corresponding constraints on gf by corresponding ones on gf . If, in addition, ∃ε < 1 ∈ R , δ δ ε ∈ R such that gh δ < 1 − ε; for all t ∈ R0 then the mapping ff : −h, 0 × Rn × Rm × R0 → R × Rn deﬁning the state-trajectory solution from any set of admissible initial conditions and all controls r i 0 Ki xit , t x t − ri being subject to ut δ r ε ≤ ∀t ∈ R 2 1/2 Bt τ Ki xt τ , t τ dτ , 3.6 1/2 δ Φα δ − τ 0 2 dτ i1 B ∞ 0 is contractive with a unique ﬁxed point, irrespective of the delays, which is 0 ∈ Rn being a globally asymptotically stable equilibrium point. Proof. The pointwise diﬀerence between two solutions x t and z t of 2.1 subject to respective piecewise continuous initial conditions ϕx : −h, 0 → Rn and ϕz : −h, 0 → Rn and respective controls ux , uy ∈ BPC 0 R0 , Rn is according to 2.2 k −1 ri r xα t − zα t Φαj t xj 0 − zj 0 Φα t − τ Ai ϕxj τ − ri − ϕzj τ − ri dτ 0 j0 i1 k −1 r ri Φα t − τ Ai τ ϕxj τ − ri − ϕzj τ − ri dτ 0 j 0i 1 t r Φα t − τ Ai xα τ − ri − zα τ − ri dτ ri i1 ri r Φα t − τ Ai τ xα τ − ri − zα τ − ri dτ 0 i0 t Φα t − τ B τ ux τ − uz τ dτ, t ∈ R0 . 0 3.7 Note from 2.3 that Φαj 0 In /j !; for all j ∈ k − 1 ∪ {0} what is used in the deﬁnition of the metric space M, · ∞ with the supremum metric · ∞ ⎧ ⎛ ⎞ ⎨ k −1 −h, 0 ∪ R0 , Rn : φ ∈ S φ, u , φ ≡ ⎝ φj ⎠ ∈ BPC 0 φ ∈ PBC 0 −h, 0 , Rn , M: ⎩ j0 ⎫ ⎬ ∀j ∈ k − 1 ∪ {0}, φu ∈ Mu , ⎭ 3.8
Fixed Point Theory and Applications 11 where φ ∈ PBC 0 R0 , Rn : φ ∈ S 0, u , u ∈ BPC 0 R0 , Rn Mu : , 3.9 where BPC 0 R, Rn is the set of bounded continuous n-vector functions on R. Now, deﬁne P : M → M as the subsequent piecewise bounded continuous function on −h, 0 ∪ R0 , which is bounded continuous on R , that is, φ ∈ PBC 0 −h, 0 , Rn , φ ∈ BC 0 R , R n and satisﬁes 2.1 on R . One gets for any bounded piecewise continuous solution of 2.1 k −1 t Φαj t φj 0 Φα t − τ A0 τ φ τ dτ P φ, u t: 0 j0 k −1 r ri t Φα t − τ Ai τ φj τ − ri dτ Φα t − τ Ai τ φj τ − ri dτ 0 ri j0i1 t Φα t − τ B τ uφ τ dτ. 0 3.10 Note that the supremum metric on −h, 0 ∪ R0 is induced by the supremum norm on −h, 0 ∪ R0 so that it is then coincident with the supremum norm. Deﬁne the truncated φa,b ∈ M as φ τ , τ ∈ a, b and φa,b τ 0, τ ∈ 0, a ∪ b, ∞ ⊂ R0 ; for all a, b > a ∈ a, b ⊆ φa,b τ R0 , for all t ∈ R0 , for all φ ∈ M and note that φ a,b : supτ ∈ a,b φ τ φa,b ∞ ≤ φ ∞ and φt ∈ M is a simpliﬁed notation for the truncated φ ∈ M on 0, ∞ . Norms without subscripts mean, depending on context, vector or correspondingly induced matrix norms as, for instance, the 2 -vector or induced matrix norms or pointwise values of such norms for vector or matrix functions in the subsequent developments. Let Mt be the space of truncated functions φt ∈ M. Note that any truncated solution of 2.1 on any ﬁnite interval is always in M so that one gets for any δ ∈ R from 3.10 in the most general controlled case with control r i 0 Ki xt , t x t − ri ut δ − P η, uη P φ, uφ t t δ k −1 δ ≤ Φαj δ φj − ηj Φα δ − τ B t u φ − uη τ dτ t tδ 0 j0 δ δ r Φα δ − τ Ai t Φα δ − τ A0 t φ−η τ dτ τ dτ tδ 0 0 t δ−ri i1 3.11
12 Fixed Point Theory and Applications ⎛ k −1 δ r ≤⎝ Φαj δ Φα δ − τ B t φ−η τ Ki xi,t τ , t τ dτ t δ−ri 0 j0 i1 δ r Φα δ − τ Ai t φ−η τ dτ 3.12 t δ−ri 0 i1 ⎞ δ ⎠ Φα δ − τ dτ φ − η A0 t τ Bt τ K0 t τ tδ 0 k −1 δ r ≤ Φαj δ φ−η Φα δ − τ dτ tδ 0 j0 i1 r × φ−η φ−η 0 0 Ai B ∞ Ki A0 B ∞ K0 ∞ t δ−ri tδ ∞ i1 ⎛ ⎞ k −1 δ r r ≤⎝ ⎠ Φαj δ Φα δ − τ dτ 0 0 Ai B ∞ Ki A0 B ∞ K0 ∞ ∞ 0 j0 i1 i1 × φ−η t δ, 3.13 where the property that A0 is constant has been used to rewrite the limits of the involved integral is the most convenient fashion to simplify the related expressions. Equation 3.13 leads to φ−η tδ −1 δ ≤ 1− Φα δ − τ dτ 0 A0 B ∞ K0 ∞ ∞ 0 ⎛ ⎞ k −1 δ r r ×⎝ ⎠ Φαj δ Φα δ − τ dτ φ−η Ki0 Ai B ∞ ∞ t δ−ri ∞ 0 j0 i1 i1 −1 δ ≤ 1− Φα δ − τ dτ 0 A0 B K0 ∞ ∞ ∞ 0 ⎛ ⎞ k −1 δ r r ×⎝ ⎠, Φαj δ Φα δ − τ dτ φ−η Ki0 Ai B ∞ ∞ t δ−r1 ∞ 0 j0 i1 i1 ∀δ ∈ R , ∀t ∈ R0 , 3.14
Fixed Point Theory and Applications 13 δ Φα δ − τ dτ < 1, since r1 ≤ ri i ∈ r , so that 0 provided that A0 B ∞ K0 ∞ 0 φ−η t −1 δ ≤ 1− Φα δ − τ dτ 0 A0 B ∞ K0 ∞ 0 ⎛ ⎞ k −1 δ r r ×⎝ ⎠ Φαj δ Φα δ − τ dτ φ−η Ki0 Ai B ∞ ∞ t−r1 ∞ 0 j0 i1 i1 φ−η ∀δ ∈ R , ∀t ∈ R0 . gh δ t−r1 ; 3.15 Then, the mapping fh : −h, 0 × Rn → R × Rn deﬁning the state trajectory solution from admissible initial conditions is nonexpansive if gh δ ≤ 1. Furthermore, the state trajectory solution is globally Lyapunov stable since by taking the trivial solution η ≡ 0 in 3.15 , it follows that any solution φ of 2.1 generated from any set of admissible initial conditions is uniformly bounded on R0 . If, in addition, gh δ ≤ Kc δ < 1 then it follows also from 3.15 as t → ∞ that any real sequence of the form {v kr1 τ }k∈Z0 ,τ ∈ 0,r1 ∩R0 is a convergent Cauchy sequence to zero in the metric space M, · ∞ of the solutions of 2.1 under the · ∞ is complete. class of given initial conditions and controls with the supremum metric Therefore, a unique ﬁxed point exists on some bounded set of Rn from Banach contraction principle. Since φ−η ≤ φ−η k lim lim Kc δ 0, 3.16 k r1 τ ,τ ∈ 0,r1 ∩R0 k 1 r1 τ Z0 k → ∞,τ ∈ 0,r1 ∩R0 Z0 k→∞ it follows by taking one of the solutions to be the trivial solution that the only ﬁxed point is the equilibrium point zero which is a globally asymptotically stable attractor. Property i has been proven. By zeroing the control and considering the uncontrolled system, one proves Property i as a particular case of Property iii . Property ii and its particular case Property iv for the case of controller gains satisfying Ki xit , t ∞ ≤ Ki0 < ∞ and Property v are proved by using similar technical tools to those involved in the above proofs by replacing the basic inequality 3.13 by t − P η, uη P φ, uφ t ⎛ 1/2 k −1 t r ≤⎝ Φαj t Φα t − τ 2 dτ t−ri j0 i1 t−r1 ,t ⎛ ⎞ 1/2 3.17 t t 1/2 2 ×⎝ ⎠ 2 Ai τ dτ B τ Ki xτ , τ dτ t−ri t−ri ⎞ 1/2 t ⎠ φ−η ∀δ ∈ R , ∀t ∈ R0 . 2 A0 τ dτ , t−r1 t−r1
14 Fixed Point Theory and Applications If all the delays are zero, it is more convenient to discuss the adhoc solution version of 2.2 : k −1 t t r Φαj t xj 0 Φα t − τ Ai τ xα τ dτ Φα t − τ B τ u τ dτ, t ∈ R0 , xα t 0 0 j0 i0 3.18 r where Φαj t and Φα t are similar to Φαj t from 2.3 - 2.4 by replacing A0 → i 0 Ai . The following result is a counterpart to Theorem 3.2 for the case of absence of delays. Theorem 3.3. Assume that Φαj t ≤ K0j t ; t ∈ R0 , for all j ∈ k − 1 ∪ {0} with max0≤j ≤k−1 supt∈R0 K0j t ≤ 1 K 0 < ∞. 2 Φα ∈ L1 R0 , Rn×n with supt∈R0 Φα t ≤ K 1 < ∞. Then, the Caputo delay-free fractional dynamic system 2.1 of real order α has the following properties. r i 0 Ki xit , t x t subject to Ki xit , t ∞ ≤ i It is globally stable under a control u t r Ki < ∞, for all i ∈ r − 1{0} if K 1 < 1/ i 0 Ai ∞ B ∞ Ki0 , for all i ∈ r − 1{0}. 0 If, in addition, K0j t → 0 as t → ∞; for all j ∈ k − 1 ∪ {0} then the system is globally asymptotically stable to the zero equilibrium point. K0 is constant if K 1 < 1/ r 0 Ai ∞ r 0 0 ii Property (i) holds if K0 t B∞ i 1 Ki i where Φαj t and Φα t are similar to Φαj t from 2.3 -(2.4) by replacing A0 → r 0 BK0 . i 0 Ai Proof. i One gets, after taking norms in 3.18 , that k −1 t r ≤ Φαj t xj 0 Φα t − τ dτ B Ki0 xα t Ai sup xα τ ∞ τ ∈ 0,t 0 j0 i0 k −1 t r ≤ Φα t − τ dτ B Ki0 3.19 K0j t xj 0 Ai sup xα τ ∞ τ ∈ 0,t 0 j0 i0 k −1 r ≤ B Ki0 K0j t xj 0 K1 Ai sup xα τ ∞ τ ∈ 0,t j0 i0 ⎛ ⎞ k −1 r ≤ K0 ⎝ xj 0 ⎠ t ∈ R0 , B Ki0 K1 Ai sup xα τ , 3.20 ∞ τ ∈ 0,t j0 i0
Fixed Point Theory and Applications 15 ≤ xα with supτ ∈ 0,t xα τ xα . Thus, one gets from 3.20 ∞ t ≤ sup xα τ xα t τ ∈ 0,t ⎛ ⎞ −1 k −1 r K0 ⎝ xj 0 ⎠ ≤ K 2 < ∞, ≤ 1 − K1 t ∈ R0 . 0 Ai B ∞ Ki ∞ i0 j0 3.21 Since K 1 r 0 Ai ∞ B ∞ Ki0 < 1, where K 2 : supt∈R0 xα t < ∞. As a result, the i Caputo fractional system of real order α is globally stable under zero delays since any state trajectory solution generated from any admissible initial conditions is bounded for all time. The proof of Property ii is similar to that of i under the modiﬁed constraints. Now, assume that if, in addition, K0j t → 0 as t → ∞; for all j ∈ k − 1 ∪ {0}, then ⎛⎡ ⎤⎞ k −1 r ≤ min⎝1, ⎣ K 1 ⎦⎠K 2 , 0 xα t K0j t Ai B ∞ Ki 3.22 ∞ j0 i0 so that ⎛ ⎡ ⎤⎞ k −1 r ≤ min⎝1, lim sup⎣ K 1 ⎦⎠K 2 0 lim sup xα t K0j t Ai B ∞ Ki ∞ t→∞ t→∞ j0 i0 3.23 r 0 Ai B ∞ Ki K1 K2 < K2 . ∞ i0 0; for all j ∈ k − 1 ∪ {0} and r 0 Ai ∞ B Ki0 K 1 < 1. Equation Since limt → ∞ K0j t i 3.23 implies that the supremum xα t on R0 is reached by the ﬁrst time at some ﬁnite time t0 ∈ R0 . Thus, one gets from 3.19 that ≤ lim lim xα t sup xα τ t→∞ t→∞ τ ∈ t0 ,t ⎛ ⎞ −1 k −1 r ⎝ ⎠ ≤ 1 − K1 lim K0j t − t0 0 Ai B ∞ Ki xαjt0 ∞ t→∞ i0 j0 0 3.24 provided that K0j t → 0 as t → ∞; for all j ∈ k − 1 ∪ {0} which proves the global asymptotic stability. Property i has been proven. Property ii follows in a similar way under the K0 K 1 < 1/ r 0 Ai ∞ B ∞ r 1 Ki0 , Φαj t , Φα t being 0 modiﬁed constraints K0 t i i r similar to Φαj t from 2.3 - 2.4 by replacing A0 → 0 Ai BK0 . i0
16 Fixed Point Theory and Applications The subsequent stability result is based on a transformation of the matrix A0 to its diagonal Jordan form which allows an easy computation of the 2 -matrix measure of its diagonal part. Theorem 3.4. Assume that JA0 JA0d JA0 is the Jordan form of A0 with JA0d being diagonal and JA0 being oﬀ-diagonal such that the above decomposition is unique with A0 T −1 JA0 T where T is a unique nonsingular transformation matrix. The following properties hold. i The Caputo fractional diﬀerential system 2.1 is globally Lyapunov stable independent of the delays if the 2 -matrix measure of JA0d is negative, that is, 1 ∗ μ2 JA/α : 1 λmax JA/α 1 JA1/α max Re λ1/α < 0, ∀λk ∈ σ JA/α , 1 3.25 k 2 k∈n 0d 0d 0d 0d where σ JA/α is the spectrum of JA/α and, furthermore, 1 1 0d 0d 1 −1 1 1 1 T JA0 T, T −1 A1 T, T −1 A2 T, . . . , T −1 Ar T 1/α ≤ μ 2 J A0 d 3.26 β0 β1 β2 βr 2 for some set of numbers βi ∈ R i ∈ p ∪ {0} satisfying r 0 βi2 1. The fractional system is globally i asymptotically Lyapunov stable for one such set of real numbers if μ2 JA/α < 0, and 1 0d 1 −1 1 1 1 T JA0 T, T −1 A1 T, T −1 A2 T, . . . , T −1 Ar T 1/α < μ 2 J A0 d . 3.27 β0 β1 β2 βr 2 ii A necessary condition for μ2 JA/α < 0 is that A0 should be a stability matrix with 1 0d | arg λ | < απ/2 ; for all λ ∈ σ A0 . Such a condition holds directly if α > 2ϕ/π where −ϕ, ϕ ⊆ −π/2, π/2 is the symmetric maximum real interval containing the arguments of all λ ∈ σ A0 . It also holds, in particular, if A0 is a stability matrix and α ∈ R ≥ 1. Proof. It follows by using the matrix similarity transformation A0 T −1 JA0 T T −1 JA0d JA0 T and using the homogeneous transformed Caputo fractional diﬀerential system from 2.1 r C C Ai T x t − hi ⇐⇒ α α D0 z t D0 T x t i0 p r T −1 Ai T x t − hi T − 1 A0 T x t T −1 Ai T x t − hi C α D0 x t 3.28 i0 i1 r T − 1 J A0 d T x t T −1 Ai T x t − hi , i0
Fixed Point Theory and Applications 17 T x t ; for all t ∈ R0 , h0 where z t 0 plays the role of an additional delay. A0 J A0 ∗ and Ai Ai i ∈ r by noting also that since JA0d JA0d is diagonal with real eigenvalues by construction, one has 1 ∗ μ2 JA/α 1 λmax JA/α 1 J 1/α JA/α 1 λ max A0 d 2 0d 0d 0d 3.29 Re λmax JA/α 1 Re λ1/α JA0d Re λ1/α A0d max max 0d 1/α μ 2 J A0 d . Then, the remaining part of the proof of Property i is similar to that quoted as a suﬃcient condition for stability independent of the delays in 27 . Property i has been proven. To prove property ii , note that 1 1/α ∗ ∗ λmax JA/α 1 JA1/α ≥ if μ2 JA/α < 0 so that 1 λmax JA0d J A0 d 2 0d 0d 0d 1 1 1/α ∗ ∗ 0 > μ2 JA/α 1 λmax JA/α 1 JA1/α ≥ λmax JA0d J A0 d 2 2 0d 0d 0d 3.30 JA/α 1 max Re λ : λ ∈ σ 0d 1 1/α ∗ ≥ max Re λ1/α : λ ∈ σ A0 ≡ σ JA0d λmax JA0d J A0 d . 2 Thus, A0 is a stability matrix if and only if arg λ ∈ −θ1 , θ2 ⊆ −π/2, π/2 ; for all λ ∈ σ A0 . If also JA/α is a stability matrix with μ2 JA/α < 0, then 1/α arg λ ∈ −θ1 /α, θ2 /α ⊆ 1 1 0d 0d −π/2, π/2 so that | arg λ | < απ/2 ; for all λ ∈ σ A0 , which is also a necessary condition for the fulﬁllment of the suﬃciency-type condition 3.27 for global asymptotic stability of 2.1 , which implies the stability of the matrix JA/α with the further constraint that μ2 JA/α < 1 1 0d 0d 0. It follows after inspecting the solution 2.2 , subject to 2.3 - 2.4 , and Lemma 2.2 that the stability properties for arbitrary admissible initial conditions or admissible bounded controls are lost in general if α ≥ 2. However, it turns out that the boundedness of the solutions can be obtained by zeroing some of the functions of initial conditions. Note, in particular that ϕj is required to be identically zero on its deﬁnition domain for k − 1 ∪ {0} j < α − 1 α ≥ 2 in order that the Γ- functions will be positive note that Γ x is discontinuous at zero with an asymptote to −∞ as x → 0− . This observation combined with Theorem 3.4 leads to the following direct result which is not a global stability result. Theorem 3.5. Assume that α ≥ 2 and the constraint 3.27 holds with negative matrix measure μ2 JA/α . Assume also that ϕj : −h, 0 → Rn are any admissible functions of initial conditions for 1 0d k − 1 ∪ {0} j ≥ α − 1 while they are identically zero if k − 1 ∪ {0} j < α − 1. Then, the unforced solutions are uniformly bounded for all time independent of the delays. Also, the total solutions for admissible bounded controls are also bounded for all time independent of the delays.
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