Preamble

In this section, we consider the approximation of systems with time-delay by systems of ordinary differential equations. This approach follows the foundational work of Krasovskii \cite{1, 2, 3} regarding the stability and control of systems with hereditary effects.

Section 1. Problem Formulation

Consider the system of differential equations for $t_0 < t \leq T$:
$$\frac{dx(t)}{dt} = \sum_{i=1}^{k} A_i(t)x(t - h_i(t)) \tag{1.1}$$
where $x(t)$ is an $n$-dimensional vector, $h_i(t)$ are time-varying delays, and $A_i(t)$ are $n \times n$ matrices. The initial condition is given by $x(t) = \phi(t)$ for $t_0 - h \leq t \leq t_0$, where $h = \max_i \sup_t h_i(t)$.

To approximate the delay system (1.1), we introduce a system of ordinary differential equations of dimension $n(m+1)$:
$$\begin{aligned}
\frac{dy^{(0)}(t)}{dt} &= \sum_{i=1}^{k} A_i(t) \sum_{j=0}^{m} c_{ij}(t) y^{(j)}(t) \
\frac{dy^{(1)}(t)}{dt} &= \frac{m}{h}(y^{(0)}(t) - y^{(1)}(t)) \
&\vdots \
\frac{dy^{(m)}(t)}{dt} &= \frac{m}{h}(y^{(m-1)}(t) - y^{(m)}(t))
\end{aligned} \tag{1.2}$$
where the coefficients $c_{ij}(t)$ are defined based on the position of the delay $h_i(t)$ within the partitioned interval $[0, h]$. Specifically, $c_{ij}(t) = 1$ if $jh/m \leq h_i(t) < (j+1)h/m$, and $c_{ij}(t) = 0$ otherwise. The initial conditions for (1.2) are set as:
$$y^{(j)}(t_0) = \phi(t_0 - jh/m), \quad j = 0, \dots, m \tag{1.2'}$$

We assume that the delays $h_i(t)$ are continuously differentiable and satisfy the condition:
$$\frac{dh_i(t)}{dt} \leq 1 - \rho, \quad \rho > 0 \tag{1.3}$$
Under these conditions, we aim to show that the solution $y^{(0)}(t)$ of the approximating system (1.2) converges to the solution $x(t)$ of the original system (1.1) as $m \to \infty$.

Section 2. Convergence Analysis

To prove the convergence, we first analyze the case of a single delay:
$$\frac{dx(t)}{dt} = A(t)x(t) + B(t)x(t - h(t)) \tag{2.1}$$
The solution to (2.1) can be represented in integral form:
$$x(t) = x(t_0) + \int_{t_0}^t A(\tau)x(\tau)d\tau + \int_{t_0}^t B(\tau)x(\tau - h(\tau))d\tau \tag{2.2}$$
By performing a change of variables $\tau^ = \tau - h(\tau)$, and utilizing the condition (1.3), we can rewrite the delayed integral term. Let $\tau = f(\tau^)$ be the inverse function. Then:
$$x(t) = x(t_0) + \int_{t_0}^t A(\tau)x(\tau)d\tau + \int_{f(t_0)}^{f(t)} B(f(\tau^))x(\tau^) \frac{df}{d\tau^} d\tau^ \tag{2.3}$$

The approximating system corresponding to (2.1) is:
$$\begin{aligned}
\frac{dy^{(0)}(t)}{dt} &= A(t)y^{(0)}(t) + B(t) \sum_{j=0}^{m} c_j(t) y^{(j)}(t) \
\frac{dy^{(j)}(t)}{dt} &= \frac{m}{h}(y^{(j-1)}(t) - y^{(j)}(t)), \quad j = 1, \dots, m
\end{aligned} \tag{2.4}$$
The solution for the intermediate variables $y^{(j)}(t)$ can be expressed using the kernel of the $m$-th order delay operator:
$$y^{(j)}(t) = \int_{t_0}^t \frac{m^j (t-\tau)^{j-1}}{(j-1)! h^j} \exp\left(-\frac{m}{h}(t-\tau)\right) y^{(0)}(\tau) d\tau + \dots \tag{2.5}$$

As $m \to \infty$, the kernel $\frac{m^j (t-\tau)^{j-1}}{(j-1)! h^j} \exp(-\frac{m}{h}(t-\tau))$ behaves like a Dirac delta function centered at $\tau = t - jh/m$. Specifically, using Stirling's approximation and properties of the Gamma distribution, we observe that for large $m$:
$$\frac{m^m (t-\tau)^{m-1}}{(m-1)! h^m} \exp\left(-\frac{m}{h}(t-\tau)\right) \to \delta\left(\tau - \left(t - \frac{mh}{m}\right)\right) \tag{2.7}$$
This allows us to approximate the sum $\sum c_j(t) y^{(j)}(t)$ by the delayed state $x(t - h(t))$.

Section 3. Main Results

Theorem 1.1. Let the coefficients $A_i(t)$ be continuous and the delays $h_i(t)$ satisfy condition (1.3). Then for any $\epsilon > 0$, there exists an $M(\epsilon)$ such that for all $m > M$, the solution of the approximating system (1.2) satisfies:
$$|x(t) - y^{(0)}(t)| < \epsilon, \quad t \in [t_0, T]$$

Theorem 1.2. If the initial function $\phi(t)$ is Lipschitz continuous, the convergence is uniform on the interval $[t_0, T]$. Furthermore, the intermediate variables $y^{(j)}(t)$ converge to the values of the solution at the corresponding delayed time points:
$$|y^{(j)}(t) - x(t - jh/m)| \to 0 \text{ as } m \to \infty \tag{3.1}$$
uniformly for $j = 1, \dots, m$.

The proof relies on the integral representation (2.3) and (2.5). By estimating the difference between the exact delay kernel and the approximating polynomial-exponential kernel, we show that the error term vanishes as $m$ increases. The condition (1.3) ensures that the transformation of the time scale is well-behaved, preventing the "accumulation" of delay effects that could lead to divergence.

References

1. Krasovskii, N. N. Stability of Motion. Moscow, 1959.
2. Repin, Yu. M. "On the approximation of systems with delay by ordinary differential equations." PMM, Vol. 29, No. 2, 1965.
3. Solodovnikov, V. V. Statistical Dynamics of Linear Automatic Control Systems. Moscow, 1960.
4. Neimark, Yu. I. Stability of Linearized Systems. Leningrad, 1949.