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

On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials

2026-07-08T16:35:23+00:00

A real scalar polynomial whose roots lie in the left half--plane of the complex plane is called a Hurwitz polynomial. This notion goes back to the works of J. C. Maxwell, E. J. Routh, and A. Hurwitz, and was later studied using techniques such as Sturm sequences, Markov parameters, and continued fractions; see, for instance, \emph{The Theory of Matrices}, Vol. II, Chapter XV, AMS Chelsea (2000), by F. Gantmacher. A $q\times q$ matrix polynomial $P(z)$ is called Hurwitz if $\det P(z)$ is a Hurwitz polynomial. Every matrix polynomial $f_n$ can be written in the form $f_n(z)=h_n(z^2)+z\,g_n(z^2)$. The matrix polynomial $f_{2m}$ is said to be of Hurwitz type if the expression $g_{2m}(z)h_{2m}^{-1}(z)$ admits a representation as a finite continued fraction with positive definite matrix coefficients. Similarly, the odd-degree matrix polynomial $f_{2m+1}$ is of Hurwitz type if $\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$ has the same property. The concept of Hurwitz-type matrix polynomials was introduced in \emph{On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials}, Linear Algebra Appl. 476 (2015), by A. E. Choque Rivero.

In the present work, we derive an explicit form of the Bezoutian associated with Hurwitz-type matrix polynomials. The fact that Hurwitz-type matrix polynomials are Hurwitz matrix polynomials was suggested and partially proved using Bezoutians in \emph{On generalization of classical Hurwitz stability criteria for matrix polynomials}, J. Comput. Appl. Math. 383 (2021), by X. Zhan and A. Dyachenko.

In contrast to that work, we employ the decomposition of the Bezoutian form introduced in \emph{Some Questions in the Theory of Moments}, Translations of Mathematical Monographs~2, AMS, 1962, by N. I. Akhiezer and M. G. Krein, for scalar polynomials.

Additionally, we propose a method to enlarge the class of Hurwitz-type matrix polynomials by adding to a given polynomial a matrix polynomial that is not of Hurwitz type, so that the resulting polynomial becomes of Hurwitz type.

On stabilization of canonical nonlinear systems in a special case

2026-07-08T16:22:10+00:00

The paper addresses the stabilization problem for canonical nonlinear systems, which in the triangular case correspond to systems in p--normal form. These systems are inherently nonlinear because they are not feedback linearizable, and their linear approximation is degenerate and does not characterize the stability of the equilibrium at the origin. In the triangular case, backstepping is a standard feedback design tool due to its recursive nature. Backstepping provides a powerful framework for stabilizing a wide class of nonlinear systems in strict-feedback form. However, this recursive approach typically leads to feedback controllers of increasing complexity as the system dimension grows. This motivates the investigation of simpler control design methods. Several notable results on non-recursive stabilization have been obtained in recent years. The classes of polynomial control laws have been constructed for the case of strictly decreasing exponents of power terms in the right-hand side of the system. In particular, it has been shown that the system can be asymptotically stabilized by a linear feedback control law in the strictly decreasing case. Moreover, control coefficients can be chosen as arbitrary positive numbers. It is also established that it is possible to achieve stabilization in a certain special case of non-strictly decreasing exponents under additional conditions on the control coefficients. In this work, we solve the linear stabilization problem in a previously unexplored case of non-strictly decreasing exponents for a three-dimensional system. We propose constructive method of finding conditions for the control coefficients to guarantee asymptotic stabilization. Our approach is based on the Lyapunov function method. We also use stabilization through nonlinear approximation to generalize our result. The effectiveness of the proposed approach is demonstrated and confirmed by numerical examples.

Correct boundary value problems with integral condition in half-space for partial differential equations

2026-07-07T20:52:40+00:00

It is well known that the Cauchy problem is ill posed for partial differential equations with constant coefficients if they do not satisfy the Petrovsky well-posedness condition.

In this work, an integral condition is proposed under which the resulting boundary value problem becomes well-posed in the Schwartz space, as well as in spaces of functions with polynomial growth in spatial variables. The presented integral condition is an integral over a semi-axis with an exponential weight, which ensures the convergence of this integral.

By considering this integral as the Laplace transform of the function $\exp(-t^2/2)$, one can obtain its asymptotics, which allow to estimate the resolving function. This estimate makes it possible to prove a theorem on the well-posedness of the resulting boundary value problem in the Schwartz space, as well as in a scale of functions of finite smoothness with polynomial growth.

Furthermore, the problem is considered for an inhomogeneous differential equation where the right-hand side belongs to the Schwartz space in the spatial variables and is compactly supported in the time variable. For the resulting problem, a Green's function is found and estimated from above. Using this estimate, a theorem on the well-posedness of this problem is proved in the Schwartz space and its dual space.

Examples of equations that are ill posed in the sense of Petrovsky are provided, and specific integral conditions are indicated under which the resulting problems will be well-posed in the Schwartz space. \\

 Examples of Petrovsky-incorrect equations are given and specific integral conditions are indicated under which the resulting problems will be correct in the L. Schwartz space.

 Consider the equation

 $$\displaystyle\frac{\partial u(x_1,x_2,t)}{\partial t}=\displaystyle\frac{\partial^2 u(x_1,x_2,t)}{\partial x_1^2}-\displaystyle\frac{\partial^2 u(x_1,x_2,t)}{\partial x_2^2}. $$

 For this equation, the Petrovsky conditions are not satisfied, since the polynomial $P(s_1,s_2)=-s_1^2+s_2^2$ is unbounded. But if we add the condition

 $$\int\limits_0^{\infty} \exp\left(-\displaystyle\frac{t^2}{2}\right) u(x_1,x_2,t)dt=\varphi(x_1,x_2),$$

 then which the boundary value problem becomes well-posed in the L.\,Schwartz space, as well as in the scale of Banach spaces.

Formal power series solutions of the differential equation ay'=by^m over Dedekind domains of characteristic zero

2026-07-07T20:52:43+00:00

Let D be a Dedekind domain of characteristic zero, let K be the fraction field of D, and consider the Cauchy problem ay′ = byᵐ, y(0) = c₀, in the ring D[[x]], where a, b, c₀ ∈ D, a, b ≠ 0, and m ∈ ℕ. The paper studies when the unique formal solution y ∈ K[[x]] with initial value c₀ actually has all coefficients in D, and therefore belongs to D[[x]]. The argument starts from the coefficient recursion in K[[x]] and derives an explicit formula for the coefficients of the unique solution. This reduces the existence problem in D[[x]] to an arithmetic integrality question governed by the valuations attached to nonzero prime ideals of D. The main point is that over a Dedekind domain the relevant obstruction is not ordinary divisibility by prime elements, but comparison of valuations of fractional ideals after localization at prime ideals.

For the linear case m = 1, the zero initial value always gives the zero solution. For c₀ ≠ 0, the coefficients contain factorial denominators, and this produces a global obstruction unless only finitely many rational primes remain nonunits in D. In that finite-prime situation the exact criterion for a nonzero initial value is the ideal containment (b) ⊆ (a)r₁, where r₁ is an explicitly described correction ideal built from the prime ideals lying over the rational primes that are nonunits in D. For the nonlinear case m ≥ 2, writing d = m − 1, the coefficients are expressed through the product Cₖ(d) = ∏(di + 1), where i runs from 0 to k − 1. The paper shows that prime ideals over rational primes not dividing d create no additional obstruction beyond the basic condition (bc₀ᵈ) ⊆ (a), while the prime ideals over rational primes dividing d contribute a correction ideal r(d). As a result, the exact existence criterion becomes (bc₀ᵈ) ⊆ (a)r(d). In particular, for m = 2 one has r(1) = D, so the condition reduces to (bc₀) ⊆ (a). Several examples are included to illustrate the theorem over ℤ, localizations of ℤ, Gaussian integers, and the Dedekind domain ℤ[√−5], which is not a unique factorization domain.

First-order implicit linear difference equation over finite commutative rings with identity

2026-07-07T20:52:46+00:00

The paper studies an implicit first-order linear difference equation $BX_{n+1}=AX_n+F_n,\quad n=0,1,2,\ldots$ over a finite commutative ring $R$ with identity, that is, an equation with a noninvertible element $B$ of the ring $R$. In contrast to the classical (explicit) linear difference equation, an implicit linear difference equation over the finite ring $R$ may have no solutions, and may also have infinitely many solutions. Since any finite commutative ring with identity is isomorphic to a finite direct sum of local commutative rings with identity, the equation decomposes into a system of equations over local finite commutative rings with identity. It is shown that the condition that the ideal $(A,B)$ generated by the elements $A,B\in R$ coincides with $R$ is necessary and sufficient for the existence of a finite number of solutions of this equation; the number of solutions in the case of their existence is counted and a formula for the general solution is provided. The condition $(A,B)\ne R$ is a necessary and sufficient condition for the existence of an infinite number of solutions of the corresponding homogeneous equation $BX_{n+1}=AX_n,\quad n=0,1,2,\ldots$. It is also established that in the case $(A,B)\ne R$ the condition $F_n\in (A,B),\quad n=0,1,2,\ldots$ is necessary for the solvability of the nonhomogeneous implicit linear difference equation, but it is not sufficient, as examples show. Under the additional restriction that $(A,B)$ is a proper principal ideal of the ring $R$, this condition is also sufficient for the existence of a solution of the considered nonhomogeneous equation; in this situation the equation has infinitely many solutions. Under the assumption that $(A,B)$ is a principal ideal, first a criterion for the existence of a solution is proved in the case of a local finite commutative ring with identity, and then in the case of an arbitrary finite commutative ring with identity. As in Fredholm theory, it is shown that if the corresponding homogeneous equation has only the trivial solution, then the studied nonhomogeneous equation has a unique solution. The work of the proved theorems is demonstrated by concrete examples.

Naum Il’ich Akhiezer on the 125th Anniversary of His Birth

2026-07-07T20:52:49+00:00

The article is dedicated to the 125th anniversary of the birth of Naum Illich Akhiezer, an outstanding mathematician, professor of Kharkiv University, and leader of the Kharkiv school of mathematicians of the mid-twentieth century. It briefly outlines his life, scientific and pedagogical work, his role in the development of mathematics and mathematical education in Kharkiv, and the preservation of his memory.