Short abstract: Necessary and sufficient conditions for complete controllability of systems of linear partial differential equations with constant coefficients in L. Schwartz space are obtained. A number of sufficient conditions for complete controllability have also been found for special cases.Examples are given for all of these special cases.
Extended abstract: In a number of papers, the controllability theory was recently studied. But quite a few of them were devoted to control systems described by ordinary differential equations. In the case of systems described by partial differential equations, they were studied mostly for classical equations of mathematical physics. For example, in papers by G. Sklyar and L. Fardigola, controllability problems were studied for the wave equation on a half-axis. In the present paper, the complete controllability problem is studied for systems of linear partial differential equations with constant coefficients in the Schwartz space of rapidly decreasing functions. Necessary and sufficient conditions for complete controllability are obtained for these systems with distributed control of the special form:\linebreak $\textbf{u}(x,t)= e^{-\alpha t}u(x)$. To prove these conditions, other necessary and sufficient conditions obtained earlier by the author are applied (see ``Controllability of evolution partial differential equation''. Visnyk of V. N. Karasin Kharkiv National University. Ser. ``Mathematics, Applied Mathematics and Mechanics''. 2016. Vol. 83, p. 47--56). Thus, the system \begin{equation*} \frac{\partial w(x,t)}{\partial t} = P\left(\frac\partial{i\partial x} \right) w(x,t)+ e^{-\alpha t}u(x),\quad t\in[0,T], \ x\in\mathbb R^n, \eqno{(1)} \end{equation*} is completely controllable in the Schwartz space if there exists $\alpha>0$ such that $$ \det\left( \int_0^T \exp\big(-t(P(s)+\alpha E)\big)\, dt\right)\neq 0,\quad s\in\mathbb R^N. $$ This condition is equivalent to the following one: there exists $\alpha>0$ such that $$ \exp\big(-T(\lambda_j(s)+\alpha)\big)\neq 1 \quad \text{if}\ (\lambda_j(s)+\alpha)\neq0,\qquad s\in\mathbb R^n,\ j=\overline{1,m}, $$ where $\lambda_j(s)$, $j=\overline{1,m}$, are eigenvalues of the matrix $P(s)$, $s\in\mathbb R^n$. The particular case of system (1) where $\operatorname{Re} \lambda_j(s)$, $s\in\mathbb R$, $j=\overline{1,m}$, are bounded above or below is studied. These systems are completely controllable. For instance, if the Petrovsky well-posedness condition holds for system (1), then it is completely controllable. Conditions for the existence of a system of the form (1) which is not completely controllable are also obtained. An example of a such kind system is given. However, if a control of the considered form does not exists, then a control of other form solving complete controllability problem may exist. An example illustrating this effect is also given in the paper.
Keywords: complete controllability; the Cauchy problem; Fourier transform.
2010 Mathematics Subject Classification: 35M10. [ Full-text available (PDF) ] Top of the page.
Short abstract: Let $RG$ be a group algebra of a finite group $G$ over the field $R$ of real numbers. A probability $P(g)$ on the group $G$ corresponds to an element \linebreak $p=\sum\limits_{g}^{} P(g)g \in RG$; we call it probability on the algebra $RG$. For a natural number $n$, $n$-fold convolution of probability $P$ on $G$ corresponds to $p^n \in RG$. Let $e \in RG$ be the probability that corresponds to the uniform probability $E(g)=|G|^{-1} (g \in G)$. Two probabilities $p, p_1\in RG$ are called complementary, if their convex linear combination equals to $e$, i.e. $ \alpha p +(1-\alpha)p_1 = e$ for some $\alpha$, $0<\alpha <1$. We find condition for existence of such $\alpha$ and compare $\parallel p^n-e \parallel$ and $\parallel {p_1}^n-e \parallel$ for arbitrary norm $\|.\|$.
Extended abstract: Let function $P$ be a probability on a finite group $G$, i.e. $P(g)\geq0\ $ $(g\in G),\ \sum\limits_{g}P(g)=1$ (we write $\sum\limits_{g}$ instead of $\sum\limits_{g\in G})$. Convolution of two functions $P, \; Q$ on group $G$ is \linebreak $ (P*Q)(h)=\sum\limits_{g}P(g)Q(g^{-1}h)\ \ (h\in G)$. Let $E(g)=\frac{1}{|G|}\sum\limits_{g}g$ be the uniform (trivial) probability on the group $G$, $P^{(n)}=P*...*P$ ($n$ times) an $n$-fold convolution of $P$. Under well known mild condition probability $P^{(n)}$ converges to $E(g)$ at $n\rightarrow\infty$. A lot of papers are devoted to estimation the rate of this convergence for different norms. Any probability (and, in general, any function with values in the field $R$ of real numbers) on a group can be associated with an element of the group algebra of this group over the field $R$. It can be done as follows. Let $RG$ be a group algebra of a finite group $G$ over the field $R$. A probability $P(g)$ on the group $G$ corresponds to the element $ p = \sum\limits_{g} P(g)g $ of the algebra RG. We denote a function on the group $G$ with a capital letter and the corresponding element of $RG$ with the same (but small) letter, and call the latter a probability on $RG$. For instance, the uniform probability $E(g)$ corresponds to the element $e=\frac{1}{|G|}\sum\limits_{g}g\in RG. $ The convolution of two functions $P, Q$ on $G$ corresponds to product $pq$ of corresponding elements $p,q$ in the group algebra $RG$. For a natural number $n$, the $n$-fold convolution of the probability $P$ on $G$ corresponds to the element $p^n \in RG$. In the article we study the case when a linear combination of two probabilities in algebra $RG$ equals to the probability $e\in RG$. Such a linear combination must be convex. More exactly, we correspond to a probability $p \in RG$ another probability $p_1 \in RG$ in the following way. Two probabilities $p, p_1 \in RG$ are called complementary if their convex linear combination is $e$, i.e. $ \alpha p + (1- \alpha) p_1 = e$ for some number $\alpha$, $0 <\alpha <1$. We find conditions for existence of such $\alpha$ and compare $\parallel p ^ n-e \parallel$ and $\parallel {p_1} ^ n-e \parallel$ for an arbitrary norm $ \|.\| $.
Keywords: probability; finite group; convergence; convolution; group algebra.
2010 Mathematics Subject Classification: 60B15; 60B10; 20D99. [ Full-text available (PDF) ] Top of the page.
Short abstract: The present article gives sufficient conditions for the existence and uniqueness of the solution of an implicit linear difference equation of an arbitrary order over a certain class of non-Archimedean rings, in particular a ring of formal power series. It is shown that this solution can beA.B. Goncharuk found using the Cramer rule. Some results on such equations over a ring of polynomials are also given.
Extended abstract: Over any field an implicit linear difference equation one can reduce to the usual explicit one, which has infinitely many solutions -- one for each initial value. It is interesting to consider an implicit difference equation over any ring, because the case of implicit equation over a ring is a significantly different from the case of explicit one. The previous results on the difference equations over rings mostly concern to the ring of integers and to the low order equations. In the present article the high order implicit difference equations over some other classes of rings, particularly, ring of polynomials, are studied. To study the difference equation over the ring of integer the idea of considering $p$-adic integers -- the completion of the ring of integers with respect to the non-Archimedean $p$-adic valuation was useful. To find a solution of such an equation over the ring of polynomials it is naturally to consider the same construction for this ring: the ring of formal power series is a completion of the ring of polynomials with respect to a non-Archimedean valuation. The ring of formal power series and the ring of $p$-adic integers both are the particular cases of the valuation rings with respect to the non-Archimedean valuations of some fields: field of Laurent series and field of $p$-adic rational numbers respectively. In this article the implicit linear difference equation over a valuation ring of an arbitrary field with the characteristic zero and non-Archimedean valuation are studied. The sufficient conditions for the uniqueness and existence of a solution are formulated. The explicit formula for the unique solution is given, it has a form of sum of the series, converging with respect to the non-Archimedean valuation. Difference equation corresponds to an infinite system of linear equations. It is proved that in a case the implicit difference equation has a unique solution, it can be found using Cramer rules. Also in the article some results facilitating the finding the polynomial solution of the equation are given.
Keywords: difference equations; non-Archimedean valuation; ring of polynomials.
2010 Mathematics Subject Classification: 12J25; 39A06. [ Full-text available (PDF) ] Top of the page.
Short abstract: May 24, 2021 turned 75 years since the birth of the famous mathematician, academician of the National Academy of Sciences of Ukraine Oleksandr Andriyovych Borisenko. The editorial board welcomes. According to the interviews of O.A. Borisenko: ''Half a century in geometry. To the 75th anniversary of corresponding member of NAS of Ukraine O.A. Borisenko. (2021). Bulletin of the National Academy of Sciences of Ukraine, (5), 95–102.''
2010 Mathematics Subject Classification: 01A70. [ Full-text available (PDF) ] Top of the page.
Short abstract: On December 3, 2020, a senior researcher at the Department of Applied Mathematics of V.N. Karazin Kharkiv National University Ivan Dmitrovich Borisov passed away. The bright memory of Ivan Dmitrievich Borisov, a real scientist and a wonderful man, forever remains in the hearts of his colleagues, students and friends.
2010 Mathematics Subject Classification: 01A70. [ Full-text available (PDF) ] Top of the page.