FLMEs with Pure Delay

• FLMEs with Pure Delay are systems where the matrix state evolution is governed by fixed delay segments and noncommutative interactions, defining their unique analytical framework.
• The explicit solution representations use recursively constructed auxiliary matrices, handling both continuous and discrete cases with convolution-type formulations.
• These equations are pivotal in control theory and signal processing, enabling advanced analysis of delayed feedback and complex iterative learning algorithms.

First-Order Linear Matrix Equations (FLMEs) with pure delay are systems in which the evolution of a matrix-valued state depends exclusively on its history, typically at a fixed delay, and potentially with noncommutative coefficients. The canonical forms appear in both continuous and discrete time and serve as fundamental models in control theory, signal processing, iterative learning, and applied mathematics where delay effects and matrix-valued interactions are central. The analysis and explicit solution of such systems require specialized techniques to accommodate structural properties, especially when underlying matrices do not commute.

1. Mathematical Formulation of FLMEs with Pure Delay

In continuous time, a general FLME with pure delay and noncommutative coefficients is given by

$$\dot{X}(\vartheta) = A_0 X(\vartheta-\sigma) + X(\vartheta-\sigma) A_1 + G(\vartheta), \quad \vartheta \geq 0$$

with initial data

$$X(\vartheta) = \Psi(\vartheta), \quad \vartheta \in [-\sigma, 0]$$

where $A_0, A_1 \in \mathbb{R}^{d \times d}$ satisfy $A_0A_1 \neq A_1A_0$, $\sigma > 0$ is the delay, $G(\vartheta)$ is a matrix-forcing term, and $\Psi$ is a prescribed history.

The discrete-time analogue is formulated as

$$\Delta X(u) = A_0 X(u-m) + X(u-m)A_1 + G(u), \quad X(u) = \Psi(u), \; u \in \mathbb{Z}_{-m}^0$$

where $\Delta X(u) = X(u+1) - X(u)$, $m$ is the discrete delay.

Noncommutativity ($A_0A_1 \neq A_1A_0$) structurally distinguishes these FLMEs from classical, commutative-delay systems, affecting the algebraic and analytical behavior of fundamental solutions and convolution operators.

2. Explicit Solution Representation and the Role of Auxiliary Matrices

A central result is the derivation of explicit solution formulae using recursively constructed auxiliary matrices. For the continuous case, the fundamental solution $Z(\vartheta)$ is defined recursively to encode the delay propagation; for the discrete case, a sequence of matrices $Q_u$ and a function $Z(u)$ play analogous roles.

For continuous time, the explicit representation (assuming sufficient regularity for $\Psi$) is

$$X(\vartheta) = Z(\vartheta) \Psi(-\sigma) + \int_{-\sigma}^0 Z(\vartheta-\sigma-s) \frac{d\Psi(s)}{ds} ds$$

and in presence of nonhomogeneity $G$,

$$X(\vartheta) = Z(\vartheta) \Psi(-\sigma) + \int_{-\sigma}^0 Z(\vartheta-\sigma-s) \frac{d\Psi(s)}{ds} ds + \int_0^\vartheta Z(\vartheta-\sigma-s) G(s) ds$$

for $\vartheta \geq 0$.

Discrete-time solution for homogeneous equations: $$X(u) = Z(u) \Psi(-m) + \sum_{r=-m+1}^0 Z(u - m - r) \Delta\Psi(r-1)$$ with the nonhomogeneous part appended analogously for general $G(u)$.

The matrices $Q_{u+1}(k)$ are built iteratively: $$Q_{u+1}(l\sigma) = A_0 Q_u((l-1)\sigma) + Q_u((l-1)\sigma)A_1$$ reflecting the noncommutative ordering essential in the foundation of $Z(\vartheta)$ or $Z(u)$. When $A_0$ and $A_1$ commute, these reduce to binomial structures; with noncommutativity preserved, the expansion becomes hierarchical, encoding matrix order deeply in the solution layers.

3. Structural Impact of Noncommutativity

In the commutative regime ($A_0A_1 = A_1A_0$), solutions admit simplifications: binomial forms, exponential mappings, or direct closed-form sums. The noncommutative case, as emphasized in (Asadzade et al., 21 Sep 2025) and complemented by (Mahmudov, 2018), requires distinct tracking of multiplication order in each summand. Auxiliary tables provided in the literature enumerate the entries of $Q_{u+1}(k)$ explicitly, detailing how action by $A_0$ and $A_1$ is layered.

This structuring yields solutions as sums of terms embodying specific sequence orderings of coefficient matrices. Analytically, the solution Z-function encapsulates these structural dependencies, leading to representations that are sharply differentiated from commutative cases and are necessary for correct stability or control analysis.

4. Fundamental Solution Construction and Convolution-Type Formulae

The construction of the fundamental solution $Z(\vartheta)$ (continuous) or $Z(u)$ (discrete) is based on recursive relations governed by delay segments and matrix application order. For continuous problems, $Z(\vartheta)$ is piecewise:

• For $\vartheta \in [-\sigma, 0)$: $Z(\vartheta) = 0$
• For $\vartheta \in [0, \sigma)$: $Z(\vartheta) = I$
• For $\vartheta \in [k\sigma, (k+1)\sigma)$,

$Z(\vartheta) = \sum_{j=1}^{k} Q_{k+1}(j\sigma)$

Discrete analogues use binomial combinatorics weighted by the $Q$ matrices, resulting in

$$Z(u) = \sum_{r} \binom{u - (r - 1)m}{r} Q_{r+1}(rm)$$

For both cases, noncommutative recursion is indispensable.

Including general inhomogeneity requires convolution-type integrals (continuous) or sums (discrete) against the appropriate Z-function, allowing the inclusion of arbitrary forcing terms in the solution structure.

5. Applications and Computational Implications

The representation formulas for FLMEs with pure delay presented in (Asadzade et al., 21 Sep 2025) directly impact control theory (delayed feedback control, iterative learning), signal processing (memory systems, prediction filters), and adaptive algorithms subject to delayed updating. In control applications, explicit solution forms facilitate analysis of stability under delay, even when system matrices are nonpermutable, a feature crucial for robust controller synthesis.

In computational settings, hierarchical and recursive solution representations permit efficient simulation and numerical analysis, while recursion in $Q_u$ and $Z(\cdot)$ forms the basis for schemes in time-marching algorithms, especially for high-dimensional matrix systems where direct exponentiation is prohibitively costly or ill-defined due to noncommutativity.

Analytical insights from this framework reconcile prior results for commutative systems with a broader, structurally accurate methodology for noncommutative cases.

6. Comparative Analysis with Classical Delay System Representations

Classical solution representations for delay-differential or delay-difference equations generally rely heavily on the ability to combine exponentials neatly (e.g., $e^{A}e^{B} = e^{A+B}$ if $AB = BA$) or binomial expansions. In general FLMEs with nonpermutable matrices, as addressed in (Asadzade et al., 21 Sep 2025) and previously (Mahmudov, 2018), one must forgo such simplifications; instead, solution formulae must systematically account for orderings imposed by delay structure.

The recursive framework developed enables extension of classical results, including the construction and application of “delayed exponential” or fundamental matrix functions, to general matrix algebraic settings. When matrices do commute, these approaches smoothly collapse to traditional forms, highlighting their generality.

7. Research and Application Outlook

The explicit, unified treatment offered for both continuous and discrete FLMEs with pure delay and arbitrary coefficient matrices reveals a foundational analytic and computational toolkit, opening the way for further theoretical investigations, enhanced control strategies, and broader application in complex systems with inherent delays and nonpermutability.

Relevant research domains include robust controller synthesis for uncertain or variable delays, iterative learning control, and matrix-valued signal processing systems. The analytic distinctions between commutative and noncommutative cases—manifest in the solution structure—prompt further study into spectral properties, stability criteria, and approximation algorithms tailored to general matrix delayed systems.