Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics

Return condition for oscillating systems

2025-10-03T17:05:06+00:00

This paper is devoted to the problem of null-controllability for the oscillating linear system $\dot{x}_{2i-1} = x_{2i}, \dot{x}_{2i} = - x_{2i-1} + u$, $i = \overline{1,n}$ with constraints on the control $u \in [c, 1]$ and $u \in \{c, 1\}$ in the case when the origin is not an equilibrium point. Null-controllability means that there exists a moment of time $T_0$ such that for any time $T > T_0$ we are able to reach the origin in precisely this time. The criterion of controllability into a non-equilibrium point was obtained by V. I. Korobov and a new condition called the return condition on the interval was introduced, which must be satisfied, together with the classical conditions for controllability into an equilibrium point. This condition means that there exists a time interval $I = [T, T + \alpha]$, $\alpha > 0$ so that a trajectory starting from the origin may return there at any time $T\in I$.
The aim of this paper is to show that the return conditions are satisfied for the considered oscillatory system, and to obtain the analytical solution for the control that solves the return condition problem. The considered approach involves constructing a piecewise-constant control using values $u = c$ and $u = 1$ and transforming the problem into a trigonometric moment problem on an interval. This problem has a non-unique solution, and in our paper we present one involving $2n$ switching points and another with only $2$ in the case when $c \le \frac{1}{2}$. The solution with $2$ switching moments is especially interesting since it does not depend on the dimensionality of the system. We also generalize the problem to the case where the eigenvalues are of the form $\lambda_{2j}, \lambda_{2j-1} = \pm \nu_k i$, where $\nu_k$ are rational numbers. Additionally, we discuss some partial cases where $c > \frac{1}{2}$ and where the eigenvalues are irrational.

Implicit linear difference equations over finite commutative rings of order p^2 with identity

2025-09-18T12:16:57+00:00

It is known that, up to isomorphism, there are exactly four finite commutative rings with identity, whose order is equal to $p^2$, where p is a prime number. Namely, these rings are the residue class ring modulo $p^2$, the direct sum of two residue class rings $\mathbb{Z}_p$ modulo $p$, the field of order $p^2$ and the ring $\mathcal{S}_p = \mathbb{Z}_p[t]/(t^2)$. Recently, a solvability criterion was established for the first-order linear difference equation over the residue class ring modulo $m \ge 2$. Considering this, it appears necessary to solve the solvability problem for the linear difference equation over the ring $\mathcal{S}_p$ of order $p^2$.

This paper investigates first-order implicit linear difference equations over the ring $\mathcal{S}_p$. The paper presents the solvability criterion for the mentioned equation over this ring. In addition, the obtained results describe both the number of solutions and the form of the general solution of this equation. Analogous results were obtained for the initial problem over the ring $\mathcal{S}_p$. In particular, it was established that, unlike in the case of an integral domain, the initial problem over the ring $\mathcal{S}_p$ may have infinitely many solutions. Moreover, if it has a finite number of solutions, then the solution of this initial problem is unique. We obtain several corollaries of the solvability criterion for the implicit linear difference equation over the ring $\mathcal{S}_p$. In particular, as in Fredholm theory, we show that if a homogeneous equation, which corresponds to the non-homogeneous equation, has only the trivial solution, then the non-homogeneous equation, which is being investigated, has a unique solution.

The article includes an example demonstrating the application of the obtained theoretical results to solving a certain equation over the ring $\mathcal{S}_p$ and the corresponding initial problem.

The results may be applied to further studies of linear difference equations over finite rings, and also to the general theory of discrete dynamical systems.

Controllability conditions for evolutionary systems of linear partial differential equations

2025-09-10T13:04:22+00:00

Many works are devoted to control theory but most of them are related to ordinary differential equations. Of the partial differential equations, mathematical physics equations are most often considered, for example, wave equations.

In the article by Makarov O.A. "Controllability of an evolutionary system of partial differential equations. Visnyk of V. N. Karazin Kharkiv National University, Series "Mathematics, Applied Mathematics and Mechanics", 2016, Vol. 83. pp. 47-56" a system of partial differential equations has previously been considered. The complete controllability of such systems in the L. Schwartz space under certain control conditions has been investigated in this work. In particular, it was proved that if the eigenvalues of the system matrix are real, then there is a time-independent control. In addition, the case when the eigenvalues are imaginary was investigated.

The purpose of this article is to study the controllability of a system of linear partial differential equations for an arbitrary matrix of the system under the constraints on the search for control in the form $u(x,t)=u(x)\cdot\exp(-\alpha t)$, where the vector of the function $u(x)$ belongs to the L. Schwartz space.

Necessary and sufficient conditions for the complete controllability of this system are obtained and examples of both controllable and uncontrollable systems are given. As a consequence of this theorem, sufficient conditions for the complete controllability of the system are obtained, the real parts of the eigenvalues of the matrix $P(s)$ are bounded from above or below. In addition, it is proved that if the spatial variable belongs to the space $\mathbb{R}$, then such a system is completely controllable. Examples are given for each case.

A partial differential equation of the second order in time is also considered. It has been proven that if the roots of the characteristic equation satisfy the conditions of the criterion, then this equation is completely controllable. Thus, the Helmholtz equation are completely controllable.

Convolution of probabilities constant in a subgroup and outside a subgroup

2025-09-09T16:18:21+00:00

Let be $P$ the probability on a finite group $G$, i.e. the function $P\left(g\right)$ takes non-negative values and $\sum _{g}P\left(g\right) =1$ $\left(g\in G\right)$. For any two functions $F_{1} \left(g\right)$ and $F_{2} \left(g\right)$ their $G$ convolution
\[\left(F_{1} *F\right)_{2} \left(t\right)=\sum _{h\in G}F_{1} \left(h\right)F_{2} \left(h^{-1} t\right), t\in G\]
is also a function on $G$.

In recent years, the topic of studying random walks (and not only on groups) has become very popular. From an analytical point of view, the study of random walks is only the study of their transition function, that is, the n-fold convolution of probability measures. It is well known that under simple conditions, imposed on the probability support $P$ on the group $G$, n-fold convolution $P^{(n)} =P*...*P$ ($n$ times) with $n\to \infty $ converges to uniform probability $U\left(g\right)=\frac{1}{\left|G\right|} $( \textbf{$g\in G$) }, which is obviously a constant on $G$.

We study convolution of functions that may be different but are constant in or outside some subgroup. In short, we study the cases where the convolution of such functions has the same properties of constancy with respect to some subgroup.

The article considers the convolution of probabilities (and, in general, real functions) constant outside (or inside) a subgroup of $H$ the finite group $G$. Let $D=G\backslash H$. For a given function $F$ on $G$ the subgroup $H$, for which $F$ is constant on $D$, is unique in the following sense: there is at most one such number $c$ and one smallest subgroup $H$ of the group $G$ such that $F\left(g\right)=c$ for all elements $g\in D$.

It is proved that if the functions $F_{1} ,\ldots $, $F_{n} $ are constants on $D$, $F_{i} (x)=c_{i} $ for arbitrary $x\in D$ $\left(i=1,\ldots ,n\right)$, then their convolution $F_{1} *\ldots *F_{n} $ is also a constant on $D$. An expression for this constant is found in terms of the numbers $c_{i} ,\; \; i=1,\ldots ,n$. Functions on the group $G$ that are constant on $D$ form a semigroup with respect to the convolution.

In previous statements we studied the case, when all functions $F_{1},\ldots, F_{n}$, except one, are constant on the subgroup $H$, and this one is constant on $D=G\backslash H$. Under these conditions it is proved that the convolution $F_{1} *\ldots *F_{n} $ is constant on $H$.

In the above statements about a convolution, at least one of its factors is constant outside the subgroup. But if all factors are constant on some subgroup, then their convolution does not necessarily have the same property. An example is given of two functions that are constant on subgroup $H$, but their convolution is not constant on $H$.

This example is not presented in the language of probabilities (or functions) on a group (as in all other parts of the article), but in the language of group algebras. The group algebra $KG$ of the group $G$ over the field $K$ appears in questions related to convolutions of functions on the group $G$ in the following way: each function $F\left(g\right)$ on $G$ with values in an arbitrary field $K$ determines an element $\sum _{g}F\left(g\right)g$ $\left(g\in G\right)$ of the algebra $KG$; the convolution of probabilities corresponds to the product of elements of the algebra $KG$. If the function $F\left(g\right)$ is a probability, then $K$ is the field of real numbers.