In the article, we investigate the inverse problem for the integro-differential time-fractional diffusion equation with initial-boundary and integral overdetermination conditions on a rectangular domain. Initially, we give a definition to the classical solution for the original problem. Subsequently, the direct problem is transformed into an equivalent integral equation using the Fourier method. We prove the existence and uniqueness of the solution to this equivalent problem by applying estimates of the Mittag-Leffler function and the successive approximation method. In the second part, we address the inverse problem, which is reduced to an equivalent auxiliary problem. We utilize the contraction mapping principle for proving local existence and uniqueness of the inverse problem solution. Additionally, we provide a practical overview of numerical solutions to the original problem using the finite difference method. The results of a numerical experiment on finer grids and at smaller time steps are demonstrated. The developed algorithm allows one to simultaneously determine the time-dependent kernel in the integral term and the solution to the problem. In conclusion, a test example is given to illustrate the effectiveness of the proposed numerical algorithm.
We consider a linear hybrid discrete-continuous system $$\left\{\begin{aligned}&\dot x(t)=A_{11}(t)x(t)+A_{12}(k)y(k),\\ &y(k+1)=A_{21}(k)x(k)+A_{22}(k)y(k).\end{aligned}\right.\ \qquad (1)$$ The concepts of the Lyapunov spectrum of system (1) and its stability with respect to small perturbations of the system coefficients are introduced. An example of a system of type (1) with an unstable Lyapunov spectrum is constructed.
In a finite-dimensional Euclidean space, we consider the problem of pursuit by a group of pursuers of a single evader, described by a system of the form $$\dot z_i = f(t)z_i + a_i(t) u_i - v, \quad u_i\in U_i(t), \quad v\in V(t),$$ where the functions $a_i(t)$ are equal to 1 for all $t$, except for a certain segment of a given length, on which they are equal to zero (a separate segment for each pursuer). This fact can be interpreted as meaning that each pursuer may experience a control device failure at any previously unknown point in time, and the length of the time interval required to fix the failure is given. During the process of fixing the failure, the pursuers are unable to capture the evader. The target sets are convex compact sets. Sufficient conditions for the solvability of the pursuit problem are obtained.
The present paper deals with the problem of simple pursuit with equal opportunities $$\begin{array}{rlllllcccc}P_i\colon& \dot x_i = u_i,& u_i(t) \in U(t), & x_i(t_0) = X_i^0, & i = 1,2, \dots, n, \\ E\colon & \dot y = v, & v(t) \in U(t) , & y(t_0) = Y^0, & t \in [t_0, \infty).\\ \end{array}$$ We say that a multiple capture in the problem of pursuit holds if the specified number of pursuers catch evader, possibly at different times: $$x_\alpha (\tau_\alpha) = y(\tau_\alpha), \quad \alpha \in \Lambda, \quad \Lambda \subset \{1,2, \dots, n\}, \quad |\Lambda| = b\quad (n \geqslant b \geqslant 1).$$ The problem of nonstrict simultaneous multiple capture requires that the capture moments coincide: $$x_\alpha (\tau) = y(\tau), \quad \alpha \in \Lambda.$$ The problem of a simultaneous multiple capture requires that the lowest capture moments coincide: $$x_\alpha (\tau) = y(\tau),\quad x_\alpha(s) \ne y(s),\quad s \in [t_0, \tau), \quad\alpha \in \Lambda.$$ We obtain necessary and sufficient conditions for simultaneous multiple capture of the evader in terms of initial positions of the participants and other parameters.
Lattice models with a small number of sites are analytically studied. They represent sections of the infinite Bogolyubov–de Gennes model. For the zero eigenvalue (zero energy), conditions for the existence and analytical expressions for the eigenfunctions are found, including those describing Majorana bound states.
We study a linear differential game involving a single evader and $m$ pursuers, $m\geq2$, in $\mathbb R^n$. The pursuers' control sets are unit balls, while the evader's control set is a ball of radius $\sigma$, where $\sigma>1$. We say that evasion is possible if the state of the evader does not coincide with the state of any pursuer for all $t\geq0$. To solve the evasion problem, a strategy is proposed for the evader, and it is shown that evasion is possible from any given initial positions of players. Using this strategy, we show that the maximum number of approach times of the pursuers to the evader is bounded above by $m(m+1)/2$.
The paper continues the exposition for systems with the delays of the completed, that is, completely analogous to the finite-dimensional systems of ordinary differential equations, theory of positional differential games by N.N. Krasovskii and A.I. Subbotin. The theory is based on the approach developed by the author that allows one to constructively transfer all the results of the theory of ordinary differential equations to systems with delays.
We consider representations of a commutative strongly Rickart semiring and its total semiring of fractions by sections of their Pierce sheaves. It is established that the basis spaces of these sheaves coincide. The main theorem provides a characterization of the strogly Rickart semiring and its total semiring of fractions in terms of Pierce stalks. This generalizes G.M. Bergman's result on the characterization of a commutative Rickart ring.
For linear third-order difference equations with delay $$x(n+3) + ax(n+2) + bx(n+1) + dx(n-1) = 0, \quad n\in\mathbb{Z}, \quad x\in\mathbb{R},$$ $$x(n+3) + bx(n+1) + c x(n) + dx(n-1) = 0, \quad n\in\mathbb{Z}, \quad x\in\mathbb{R},$$ stability regions are studied and constructed in three-dimensional spaces $\{(a,b,d)\}\subset \mathbb{R}^3$ and $\{(b,c,d)\} \subset \mathbb{R}^3$, respectively.
This paper studies the controllability problem for systems of functional inclusions with causal operators and impulse characteristics in Banach spaces. The main result of the paper is a global existence theorem for trajectories for systems described by functional inclusions with impulse characteristics. The proof is based on topological degree theory for condensing multivalued mappings. As applications of the main result, generalized existence theorems are obtained for systems of two important classes: first-order semilinear differential inclusions of fractional order 0 < q < 1.
We introduce and investigate a new class of associative rings, called n-del U rings, characterized by the condition that for every unit u in the ring, the element u(n)-1 belongs to a distinguished subset del(R) of del-nilpotent elements. This subset consists of elements x is an element of R such that 1 - ux is invertible for all units u commuting with x. We explore the structural properties of n-del U and pi-del U rings, provide illustrative examples, and examine their behavior under various ring-theoretic constructions including direct products, quotient rings, and trivial extensions. Our results establish connections between the n-del U condition and classical notions such as regularity, cleanness, and Dedekind-finiteness, offering new insights into the interplay between unit powers and nilpotency in ring theory.
The differential properties of the minimax solution are investigated in a class of plane Dirichlet problems for the Bellman equation. The class of problems is defined by closed non-convex solid boundary sets whose boundaries contain pseudovertices, which are singular points associated with the singularity of the minimax solution. The differential properties of the solution depend on the order of smoothness of the boundary of the boundary set at the pseudovertices and on the cardinality of the values of the metric projection operator onto this set. The paper distinguishes between situations where the operator has single-point values and when the number of projections is greater than one. Using tools from the theory of alpha sets and Efimov-Stechkin support balls, the features of the characteristic function of a non-convex set are investigated. Formulas for its limit values are found, which in a fairly general case facilitate the construction of a Chebyshev layer of the boundary set, which is a region adjacent to the boundary set in which the minimax solution is differentiable. An example and its meaningful interpretation from the point of view of optimal control are given.
For a Cauchy problem associated with a nonlinear ordinary differential equation in a Hilbert space X, we obtain sufficient conditions for exact controllability to a given final state (as well as to given intermediate states at intermediate times) over arbitrarily fixed (without additional conditions) time interval under a constraint on the control norm value. This is a generalization of a similar result previously obtained by the author for the case of an operator differential equation with a stationary linear operator and linearly incoming control without a constraint on the norm. As before, the Minty-Browder theorem is used, as well as the chain technology for sequentially continuing the solution of the control system to intermediate states. As an example (of independent interest), a strongly nonlinear pseudoparabolic partial differential equation describing the evolution of an electric field in a semiconductor is considered.
This article presents an approach to deductive program synthesis using Gentzen's sequent calculus within the framework of logic programming. By leveraging sequent calculus as a formal system for structured logical inference, our method automates the derivation of provably correct programs from specifications expressed in negation-free first-order predicate logic. We formalize the syntax and semantics of sequent calculus, implementing its core inference rules (introduction and elimination rules) as predicates in logic programming to enable scalable synthesis. Practical examples demonstrate the transformation of logical specifications into executable programs. The approach ensures formal correctness through a constructive semantics inspired by Kleene's realizability, with synthesized programs operating in a subrecursive language to guarantee termination. We evaluate the method's strengths, including its reliability for safety-critical systems, and its limitations, such as computational complexity for unbounded constructions. Compared to AI-driven synthesis, our approach prioritizes formal guarantees, complementing modern trends like relational programming. Future research directions include optimizing computational efficiency and extending applicability to complex real-world problems.
This paper is devoted to the study of one well-known problem of B. N. Pshenichnyi, namely the problem of simple group pursuit, when players make step-by-step movements. The paper considers two separate cases. In the first case, a discrete pursuit game is solved, when only one pursuer and one evader participate in the game. To solve this problem, an algorithm for applying the fl-strategy is given. According to the proposed method, the players first approach each other and eventually coincide exactly. In the second case, the proposed solution method is extended to the game of group pursuit. The obtained results are verified using animation models created in the Visual C# programming language using ScottPlot.WinForms technology.
This paper addresses the problem of simple pursuit of one evader by a group of pursuers in a differential game described by equations with Caputo fractional derivatives from the interval (0, 1). Integral constraints are imposed on the players' controls, and the pursuers use quasi-strategies. The goal of the group of pursuers is to bring at least one of them within any predetermined distance of the evader. It is proven that if the total energy of the pursuers is greater than the energy of the evader, then the capture occurs in the game.
A nonlinear controlled system in a finite-dimensional Euclidean space is considered. Some control tasks are formulated for it. An approach to solving problems based on the use of reachability sets and integral funnels of controlled systems and corresponding differential inclusions is discussed. Due to the complexity of the control tasks under consideration, an analytical representation of solutions for non-trivial controlled systems is impossible, and therefore this paper focuses on the issues of approximate design of solutions to problems. These issues are primarily related to the approximate construction of reachability sets and integral funnels of controlled systems. The paper also examines the problems of optimal performance of some nonlinear controlled systems, in particular, problems with phase constraints. The paper provides examples.
In regard to our recent studies of rings with (strongly, weakly) nil-clean-like properties, we explore indepth both the structural and characterization properties of those rings whose elements that are not units are weakly nil-clean. Group rings of this sort are considered and described as well. This somewhat supplies our recent results of this branch when the units are weakly nil-clean published in Punjab University Journal of Mathematics (2024).
A two-player differential game with an unfixed endpoint is considered. A special feature of the game is the presence of not only a target set but also a lifeline. If the second player steers the lifeline, then the payoff equals infinity. The payoff functional depends on the trajectory of the players and their controls. Special cases of the differential game under consideration are the pursuit-evasion game and time-optimal game. Universal positional strategies are constructed for the game under consideration under the assumption that the Dirichlet problem for the Hamilton-Jacobi equation, related to the differential game, admits a viscosity proximal solution. The construction of universal strategies is based on the concept of a proximal gradient and utilizes the Krasovsky-Subbotin approach. The universality of positional strategies means that for any initial point from a compact set, the feedback strategy is equally effective. In addition, theorems on the evaluation of the guaranteed result of the players are proved.
This paper reports on the simulation of the motion of an aquatic robot with an internal spinning rotor. We develop two mathematical models of robot motion in a fluid: the model of motion based on the Kirchhoff equations for the motion of a rigid body in a fluid and a model based on the Navier-Stokes equations. In addition to the simulation, we develop a prototype of the aquatic robot with a spinning rotor, with which we conduct real experiments. In this paper, we present the results of real experiments and simulations and draw conclusions based on them.
