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

and Applied Analysis 3 Rn×m : {M Mij ∈ Rn×m : Mij > 0; ∀ i, j ∈ n×m} is the set of all n×m real matrices of positive entries. If M ∈ Rn×m then M 0 is used as a simpler notation for M ∈ Rn×m . The superscript T denotes the transpose, M i 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: R0 : {v v1, v2, . . . , vn T ∈ R : vi ≥ 0; ∀i ∈ n} is the set of all n real vectors of nonnegative components. If v ∈ R then v ≥ 0 is used as a simpler notation for v ∈ Rn0 . R : {0/ v v1, v2, . . . , vn T ∈ R : vi ≥ 0; ∀i ∈ n} is the set of all n real nonzero vectors of nonnegative components i.e., at least one component is positive . If v ∈ R then v > 0 is used as a simpler notation for v ∈ R . R : {v v1, v2, . . . , vn T ∈ R : vi > 0; ∀i ∈ n} is the set of all n real vectors of positive components. If v ∈ R then v 0 is used as a simpler notation for v ∈ R . M Mij ∈ Rn×n is a Metzler matrix if Mij ≥ 0; for all i, j / i ∈ 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 → C for some real interval a, b ⊂ R satisfiesf ∈ Ck−2 a, b ,R and, furthermore, dk−1f τ /dτk−1 exists everywhere in a, b for k Reα 1 for some α ∈ C0 . Then, the Riemann-Liouville left-sided fractional derivative Dα a f of order α ∈ C0 of the vector function f in a, b is pointwise defined in terms of the Riemann-Liouville integral as ( Dα a f ) t : 1 Γ k − α ( d dt ∫ t a f τ t − τ α 1−k dτ ) , t ∈ a, b , 2.1 where the integer k is given by k Reα 1 and Γ : C\Z0− → C, whereZ0− : {n ∈ Z : n ≤ 0}, is the Γ-function defined by Γ z : ∫∞ 0 τ z−1e−τdτ ; z ∈ C \ Z0−. If f ∈ Ck−1 a, b ,R and, furthermore, f k τ ≡ df τ /dτ exists everywhere in a, b , then the Caputo left-sided fractional derivative Dα a f of order α ∈ C0 of the vector function f in a, b is pointwise defined in terms of the Riemann-Liouville integral as ( Dα a f ) t : 1 Γ k − α ∫ t a f k τ t − τ α 1−k dτ, t ∈ a, b , 2.2 where k Reα 1 if α / ∈ Z0 and k α if α ∈ Z0 . The following relationship between both fractional derivatives holds provided that they exist i.e., if f : a, b → C possesses Caputo left-sided fractional derivative in a, b , 1 ( Dα a f ) t Dα a ⎡ ⎣f τ − k−1 ∑ j 0 f j a τ − a j j! ⎤ ⎦ t , t ∈ a, b . 2.3 4 Abstract and Applied Analysis Since Reα ≤ k, 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: ( Dα 0 x ) t 1 Γ k − α ∫ t a f k τ t − τ α 1−k dτ p ∑ i 0 Aix t − hi Bu t 3.1 with k − 1 < α ∈ R ≤ k; k − 1, k ∈ Z0 , 0 h0 < h1 < h2 < · · · < hp h < ∞ being distinct constant delays, A0, Ai ∈ Rn×n i ∈ p : {1, 2, . . . , p} , are the matrices of dynamics for each delay hi, i ∈ p ∪ {0}, B ∈ Rn×m is the control matrix. The initial condition is given by k n-real vector functions φj : −h, 0 → R, with j ∈ k − 1 ∪ {0}, which are absolutely continuous except eventually in a set of zero measure of −h, 0 ⊂ R of bounded discontinuities with φj 0 xj 0 x j 0 xj0. The function vector u : R0 → R 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 1 Aix t − hi Bu t . Theorem 3.1. The linear and time-invariant differential functional fractional differential system 3.1 of any order α ∈ C0 has a unique solution onR0 for each given set of initial functions φj : −h, 0 → R, j ∈ k − 1∪{0} being absolutely continuous except eventually in a set of zero measures of −h, 0 ⊂ R of bounded discontinuities with φj 0 xj 0 x j 0 xj0; j ∈ k − 1 ∪ {0} and each given control u : R0 → R being a bounded piecewise continuous control function. Such a solution is given by xα t k−1 ∑ j 0 ( Φαj0 t xj0 p ∑ i 1 ∫hi 0 Φα t − τ Aiφj τ − hi dτ )