Preamble

This work, published in 1967 (Vol. III, No. 12), builds upon the foundational research of S. T. Zavalishchin \cite{6} and other recent developments in the field \cite{7, 8}. Specifically, we address the operator-based approaches discussed in \cite{9} and the methodologies established in \cite{1}. The study also incorporates insights from E. A. Barbashin regarding the stability of motion. Our primary objective is to refine the concept of "generalized solutions" in the context of differential equations with distributions, particularly focusing on the results presented by S. T. Zavalishchin in \cite{4}. We aim to clarify the relationship between the spaces of solutions and the spaces of input perturbations.

§ 1. General Definitions and Stability

Consider the linear differential equation:
$$\dot{x} = A(t)x, \tag{1.1}$$
where $x$ is an $n$-dimensional vector and $A(t)$ is an $n \times n$ matrix. Let $U(t)$ be the fundamental matrix of the homogeneous system (1.1). We extend this to the non-homogeneous case involving a distribution $\eta$:
$$d\mu = A(t)\mu \, dt + d\eta, \tag{1.2}$$
The solution to (1.2) can be represented using the Cauchy formula:
$$\mu(t) = W(t, 0)\mu_0 + \int_0^t W(t, s) \, d\eta(s), \tag{1.3}$$
where $W(t, s) = U(t)U^{-1}(s)$ is the transition matrix. For an initial condition $\mu_0 = 0$, the behavior of the solution on the interval $0 \le t < \infty$ is determined by the properties of the integral term. Following the conventions in \cite{1, 3}, we analyze the mapping from the space of perturbations $\eta$ to the space of solutions $\mu$.

Let $V(\infty)$ denote the space of functions of bounded variation on $[0, \infty)$. We define the operator $Q$ that maps a perturbation $\eta \in V(\infty)$ to a solution $\mu$ via the relation:
$$\mu = Q\eta = \int_0^t W(t, s) \, d\eta(s). \tag{1.4}$$
The stability of this mapping is characterized by the norm:
$$|\eta|V = \sup$$} |\eta(t)| \tag{1.5
and the corresponding norm for the solution space. We say the system is $(D_0, D)$-stable if the operator $Q$ maps the space $D_0$ into $D$ continuously. As shown in \cite{1}, a necessary condition for the boundedness of the operator $Q$ is:
$$\sup_{t \ge t_0} \int_{t_0}^t |W(t, s)| \, ds < \infty. \tag{1.7}$$
Under the assumption that the matrix $A(t)$ is bounded, i.e.,
$$\sup_{t \ge 0} |A(t)| < \infty, \tag{1.8}$$
the stability condition (1.7) is equivalent to the existence of constants $M, \alpha > 0$ such that:
$$|W(t, s)| \le M e^{-\alpha(t-s)} \quad (0 \le s \le t < \infty). \tag{1.9}$$

§ 2. Generalized Solutions and Distributions

In \cite{4}, Zavalishchin introduced the space of distributions $K_+$ and defined generalized solutions for systems where the input $\eta$ belongs to this class. Let $V K_+$ be the space of distributions whose primitives belong to $V(\infty)$. We consider the mapping $Q$ in the context of these generalized functions:
$$\mu = Q\eta, \quad \eta \in V K_+. \tag{2.1}$$
The relationship between the classical solution (1.4) and the generalized solution (2.2) is central to our analysis. If the system (1.1) is exponentially stable, then the operator $Q$ defined by (2.3) provides a continuous mapping between the respective distribution spaces.

Specifically, for any $\eta \in V(\infty)$, the solution $\mu$ defined by (1.4) satisfies:
$$|\mu(t)| \le W_0 \cdot \text{Var}(\eta), \tag{1.10}$$
where $W_0$ is a constant depending on the stability of $A(t)$. This confirms that the integral operator $Q$ is well-behaved under the conditions of exponential stability (1.11).

Conclusion

The results presented here demonstrate that the framework of $(D_0, D)$-stability provides a robust mechanism for analyzing differential equations driven by distributions. By utilizing the transition matrix $W(t, s)$ and the operator $Q$, we can ensure that generalized solutions remain within predictable bounds, provided the underlying homogeneous system (1.1) satisfies standard stability criteria \cite{1, 4, 5}.

References

1. Barbashin, E. A. Introduction to the Theory of Stability. Nauka, 1967.
2. Gel'fand, I. M., and Shilov, G. E. Generalized Functions. Fizmatgiz, 1964.
3. Krasovskii, N. N. Stability of Motion. Stanford University Press, 1963.
4. Zavalishchin, S. T. Siberian Mathematical Journal, Vol. 2, No. 7, pp. 872–881, 1966.
5. Zavalishchin, S. T. Differential Equations, Vol. 3, No. 2, pp. 171–179, 1967.
6. Roitenberg, Ya. N. Automatic Control. Nauka, 1966.
7. Filippov, A. F. Differential Equations with Discontinuous Right-Hand Sides. Mat. Sbornik, 1960.
8. Wexler, D. Rev. Roum. Math. Pures et Appl., Vol. X, No. 8, pp. 1163–1199, 1965.

Submitted October 25, 1966.