Extended Mixed Delays in Dynamical Systems

• Extended Mixed Delays are defined as a generalization of delay constructs integrating point, distributed, and noisy-memory delays in dynamic systems.
• They influence system stability, positivity, and optimal control by altering state arguments and leveraging the spectral properties of characteristic matrices.
• Applications span both deterministic and stochastic fields, including nonzero-sum games and linear-quadratic control, demonstrating delay-independent stability.

Extended mixed delays represent a generalization of delay constructs in dynamic systems, encompassing point delays, distributed delays, and stochastic memory effects in both deterministic and stochastic frameworks. In positive linear coupled systems and stochastic differential equations, extended mixed delays may be unbounded, time-varying, or induced by noise, and can influence all system coefficients, including drift, diffusion, and cost terms. Recent research establishes rigorous conditions for existence, uniqueness, positivity, stability, and optimal control in systems with extended mixed delays, demonstrating that spectral properties of the associated characteristic matrices govern qualitative behavior independently of delay magnitudes or functional forms (Tuan et al., 2022, Li et al., 16 Jan 2026).

1. Mathematical Formulation of Extended Mixed Delays

A broad class of systems with extended mixed delays is described as follows. Consider a deterministic mixed-order positive linear coupled system: $$\begin{cases} {}^C D^{\hat\alpha}x(t) =\,A\,x(t) + B\,x(t-\tau_1(t)) + E\,y(t-\tau_2(t)) + f(t), \quad t>0,\ y(t) = C\,x(t) + D\,y(t-\tau_3(t)) + g(t), \quad t\geq 0, \end{cases}$$ with Caputo derivatives ${}^C D^{\hat\alpha}$, time-varying delays $\tau_k(t)$, and history functions on $[-r,0]$. Extended mixed delays appear by shifting arguments of the state and auxiliary variables in each equation, where delays may be state-dependent, unbounded, or irregular.

In the stochastic context, the controlled state equation with extended mixed delays is: $$dx(t) = b(t, x(t), y(t), z(t), \kappa(t), u(t), \mu(t), \nu(t), \lambda(t))\,dt + \sigma(\cdots)\,dW(t),$$ where delay arguments include:

• Point delay: $y(t) = x(t-\delta)$
• Distributed delay: $z(t) = \int_{t_0}^t \phi_1(t,s)x(s)ds$
• Noisy memory: $\kappa(t) = \int_{t_0}^t \psi_1(t,s)x(s)dW(s)$ (and similarly for control variables). Coefficients $\phi_i, \psi_i$ encode weighting kernels (Li et al., 16 Jan 2026).

2. Existence, Uniqueness, and Positivity Conditions

Existence and uniqueness of solutions to mixed-order systems with extended mixed delays is guaranteed under: (i) $\|C\|_\infty < 1$, $\|D\|_\infty < 1$ (invertibility and nonnegativity of $I_n-D$), (ii) continuity of delays on finite intervals, (iii) continuity of external inputs. The solution is found in $C([-r,T];\mathbb R^{d+n})$ via weighted-norm integral equations and Banach fixed-point arguments.

Positivity—the property that nonnegative history and inputs guarantee nonnegative solutions—is characterized by the following conditions:

• System matrix $A$ Metzler,
• Matrices $B, E, C, D$ entry-wise nonnegative,
• $\|C\|_\infty < 1$, $\|D\|_\infty < 1$. These are necessary and sufficient for positivity, and remain valid independent of the scale or growth of delays (Tuan et al., 2022).

3. Spectral and Stability Properties

Stability and attractivity in systems with extended mixed delays hinge primarily on spectral properties of the characteristic matrix

$\Lambda := A + B + E(I_n - D)^{-1}C$

rather than the explicit form or magnitude of delay functions. Specifically:

• Theorem (Global Attractivity): The system with decaying inputs ($\lim_{t\to \infty} f(t) = 0$, $\lim_{t\to \infty} g(t) = 0$) is globally attractive if $\Lambda$ is Hurwitz (all eigenvalues have negative real part).
• Theorem (Delay-independent Stability): In the homogeneous case, attractivity and stability are governed exclusively by the Hurwitz condition on $\Lambda$, with no constraints on delays (bounded, unbounded, time-varying are all permissible) (Tuan et al., 2022).

For systems with persistent bounded disturbances, the smallest asymptotic bound is computed as: $$x^* = -\Lambda^{-1}(F + E(I_n-D)^{-1}G), \quad y^* = (I_n-D)^{-1}(Cx^*+G)$$ where $F, G$ are suprema of input bounds, and $(x^*, y^*)$ is minimal over all admissible inputs.

4. Comparison Principle and Delay Independence

A fractional-order comparison principle allows ordering of solutions by comparing corresponding right-hand sides of differential inequalities. In particular, if initial histories and inputs to one system are dominated by those of another, the solutions retain the order for all future times. This principle formalizes the intuition that delay effects in positive systems act only via shifting state arguments, and do not inherently destabilize the system provided spectral conditions are met.

A plausible implication is that the explicit size or variability of delays (extended, mixed, noisy, or distributed) is quantitatively negligible in governing ultimate system stability, so long as matrices satisfy the positivity/Metzler-Hurwitz conditions (Tuan et al., 2022).

5. Stochastic Control with Extended Mixed Delays

In controlled stochastic systems with extended mixed delays, the functional coefficients (drift, diffusion, cost) depend on point delay, distributed delay, and noisy-memory terms of both states and controls. The optimal control problem is formulated over adapted processes with prescribed histories, and cost functionals depend on both delayed and delayed-memory arguments.

Key advances include the transformation of delay variational equations into forward stochastic Volterra integral equations (SVIE) via the stochastic Fubini theorem. Adjoint equations involve both traditional backward stochastic differential equations (BSDE) and backward stochastic Volterra integral equations (BSVIE) with Malliavin derivatives, necessitating new duality and coefficient-decomposition methods.

The stochastic maximum principle is derived, establishing necessary and sufficient conditions for optimality of controls under concavity. The adjoint system can be further reduced, via the Clark-Ocone formula, to anticipated BSDEs—showing that only adapted future expectations and deterministic memory kernels need be considered (Li et al., 16 Jan 2026).

6. Applications: Games, Linear-Quadratic Control, and Robustness

Extended mixed delays are crucial for modeling feedback, memory, and distributed stochastic effects in various applications:

• Nonzero-sum stochastic games: Each player’s control and cost involve all forms of extended mixed delays; Nash equilibrium strategies satisfy adapted maximum principles parameterized by individual delayed-memory terms.
• Linear-quadratic (LQ) control: Explicit solutions for optimal control with extended mixed delays yield time-varying policies that combine point, distributed, and noisy-memory feedback (Li et al., 16 Jan 2026).
• Numerical studies: Mixed-order deterministic systems with unbounded, state-dependent, or distributed delays show that spectral conditions alone guarantee convergence, attractivity, and bounding, confirming theoretical results (Tuan et al., 2022).

7. Summary of Analytical Features

Feature	Deterministic Component	Stochastic Component
Delay Types	Point, time-varying, unbounded	Point, distributed, noisy memory
Governing Condition	$\Lambda$ Hurwitz	Adjoint anticipated BSDE, spectral kernels
Solution Existence	Small-gain, nonnegativity	Adaptedness, kernel regularity
Delay Dependency in Stability	None (delay-independent)	None (memory kernels deterministic)

This synthesis highlights that extended mixed delays, encompassing a wide class of delay, memory, and distributed effects, interact with system dynamics purely through the argument structure of states and controls, with qualitative behaviors—positivity, stability, optimality—governed entirely by matrix conditions and appropriate kernel regularity, independent of delay magnitudes or structures (Tuan et al., 2022, Li et al., 16 Jan 2026).