Representation of solutions of linear differential systems with pure delay and multiple delays with linear parts given by non-permutable matrices

Representation of solutions for linear fractional systems with pure delay and multiple delays

“…Although we considered the exact solutions of (1.1) and (1.2) with order 𝛼 ∈ (1, 2), our results can be applied in the case of 𝛼 = 1 and 𝛼 = 2. In other words, if 𝛼 = 1 or 2, then our results coincide with the corresponding results in previous works 8,19,20,27,28 . Moreover, according to these papers, we can conjecture that Theorem 2 holds true even if the function f is not exponentially bounded.…”

Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay

“…( 25) is finite time stable. In this work, using new conformable delayed matrix functions, we derived explicit solutions of linear conformable fractional delay systems of order α ∈ (1, 2], which extend and improve the corresponding and existing ones in [12,13] in the case of α = 2 without any restrictions on the matrix coefficient of the linear part, by removing the condition that B is a nonsingular matrix and replacing the matrix coefficient of the linear part B 2 in [12] by an arbitrary, not necessarily squared, matrix. In addition, using the formula of general solutions and a norm estimation of the conformable delayed matrix functions, we established some sufficient conditions for the finite time stability results, which extend and improve the existing ones in [27] in the case of α = 2.…”

“…In 2008, Khusainov et al [12] adopted this approach to represent the solutions of an oscillating system with pure delay by establishing a delayed matrix sine and a delayed matrix cosine. This pioneering research yielded plenty of novel results on the representation of solutions, which are applied in the stability analysis and control problems of time-delay systems; see for example [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] and the references therein. Thereafter, in 2021, Xiao et al [29] obtained the exact solutions of linear conformable fractional delay differential equations of order α ∈ (0, 1] by constructing a new conformable delayed exponential matrix function.…”

Analytical solutions of systems of linear retarded and neutral delay differential equations by the Laplace transform: featuring limit cycles

“…Later on, the authors improved their approach to account for delay systems with non-commuting matrices [33]. Recently there have been improvements in regard to methods to solve linear RDDEs [37][38][39][40]. For instance in [37] the authors deal with linear differential systems with a single delay and multiple delays with linear parts given by non-permutable matrices.…”