In this paper, we propose a new version of a projection algorithm for the generalized variational inequalities problem (GVIP) involving a multi-valued mapping. In this work, we introduce a new combination of the main algorithmic steps proposed in projection methods that have been developed previously for classical or generalized variational inequalities problem with the aim of significantly reducing the computational cost. For the convergence, the assumptions required for the underlying mapping are continuity and a condition that is weaker than the pseudo-monotonicity. We show that this method is well-defined and its corresponding algorithm is globally convergent. Preliminary comparative experiments on several test problems to validate the effectiveness of the proposed method.

In this paper we develop a novel approach to the analysis and study of one class optimization problems with non-coercive objective functionals. With this in mind we introduce a special class fo anisotropic functional spaces. We give a precise definition of such spaces and show that they can be considered as a natural generalization of the standard Sobolev spaces. Bases on this concept, we relax of a special class of non-coercive minimization problems in Sobolev spaces W1,2(Ω), provide a rigorous mathematical analysis of the proposed relaxed version, establish sufficient conditions of its solvability, show that the objective functional is coercive, and derive the corresponding optimality conditions. To demonstrate the validity of the obtained results, we apply the proposed approach to the relaxation of the well-know variational model for removing multiplicative noise in image processing.

A nonlinear boundary value problem is considered for a system of parabolic equations in an inhomogeneous region, that requires conjugation conditions. The boundary value problem serves as a mathematical model of the nonlinear non-isothermal filtrationconsolidation process in a heterogeneous soil mass. A distinctive feature at the physical level is the application of a nonlinear Darcy’s law incorporating the threshold gradient, where this gradient depends on the thermal state of the porous medium. These features are also reflected in the conjugation conditions. The finite element method is applied to obtain approximate discontinuous solutions of the corresponding system of quasilinear parabolic equations. The existence and uniqueness of an approximate generalized solution are proven. Estimates for the accuracy of the finite element solutions are also established in terms of total approximation. A model test example is used to analyze the differences in pressure head and temperature distributions between the case studied in this article and the classical formulation.

We provide a mathematical and numerical analysis for solving an eigenvalue boundary value problems (Dirichlet and Neumann) for Helmholtz operator, which describes a regular shielded double-sided microstrip transmission line problem. The paper presents a new adapted method for numerical calculation of dispersion characteristics of this problem, and, in particular, balanced microstrip line problem. The boundary value problem for a regular planar transmission line is solved in a rigorous formulation considering the singularity of the behavior of the current density at the edges of the microstrips. The method of discretization by Galerkin approximations of the boundary value problem using an additional basis from Chebyshev polynomials of the 1st and 2nd kind was adapted for the case of a doubly connected domain (two electrodynamically coupled microstrip lines). In an explicit form, the matrix elements for the system of linear algebraic equations were obtained, the solution algorithm of which was implemented in a high-performance computing environment. The dispersion characteristics of a double-sided microstrip and balanced transmission line made on the RO3010 substrate material were calculated using the developed algorithm. As an example of using the obtained dispersion characteristics, a circuit was designed and the scattering characteristics of tapered transition with an exponential profile from a double-sided microstrip with a characteristic impedance Z0 = 50 Ohm to a balanced transmission line with a impedance of Z0 = 100 Ohm were obtained. According to the results of the analysis of exponential transitions with exponents n = 0.707 and n = 1.0 in the function of the strip width versus the longitudinal coordinate, the reflection coefficient is below |Γ| < 0.1.

The method of reflected waves was extended to the finite vibrating string, subject to concentrated (at both its ends) and distributed (along its length) forces. To this end, the periodic odd continuation along the infinite space-time strip was implemented to all functions, influencing the above vibrations: 1) the initial, 2) the boundary (converted into the double layers) and 3) the right-hand side of the 1D non-homogeneous wave equation, resulting to a weak formulation of the IBVP, instead of the original strong one. The weak solution to the IBVP as the sum of regular and singular distributions was obtained by convolving the fundamental solution of the 1D wave operator and the above continued functions, similarly to obtaining the weak solution to the Cauchy problem for the 1D non-homogeneous wave equation. The weak solution to the IBVP was shown can be transformed (through some intermediate representations) into the well known strong one, obtained by the method of separation of variables. The opposite is also true, that is the strong solution to the IBVP can be transformed into a representation of the weak one by two successive integrations by parts, provided that the convergence of some Fourier series is interpreted in the weak sense (as distributions).

A combined approach to identifying the coordinates of point impulse sources, using observation data at certain time intervals at characteristic points, has been transferred to the case of anisotropy. The corresponding mathematical model is constructed based on the assumption that the movement of particles in the domain is quasiideal and satisfies the generalization of Darcy’s law; at the same time, the diffusion component of the movement of pollutants is neglected. The initial problem is divided into two smaller ones: it is assumed to construct a hydrodynamic mesh and, based on this, to identify the coordinates of pollution sources. In the first case, numerical methods of quasiconformal mappings are applied. In the second case, based on the use of the method of characteristics with respect to the convection equation, an integral equation is constructed. Numerical experiments were conducted, confirming the effectiveness of the previously developed approach in the case of anisotropy. The most significant residuals between the a priori known and calculated coordinates of pollution sources occur on those streamlines that pass near stagnant zones and intersect areas in the vicinity of so-called “key” points (critical points of quasiconformal mapping violation) at the boundary of the domain. In certain cases of filtration tensor distribution, this can have a significant negative impact on the accuracy of the solution.

This note presents an analytical framework for decision-making in drone swarm systems operating under uncertainty, based on the integration of Partially Observable Markov Decision Processes (POMDP) with Deep Deterministic Policy Gradient (DDPG) reinforcement learning. The proposed approach enables adaptive control and cooperative behavior of unmanned aerial vehicles (UAVs) within a cognitive AI platform, where each agent learns optimal energy management and navigation policies from dynamic environmental states. We extend the standard DDPG architecture with a belief-state representation derived from Bayesian filtering, allowing for robust decision-making in partially observable environments. In this paper, for the Gaussian case, we numerically compare the performance of policies derived from DDPG to optimal policies for discretized versions of the original continuous problem. Simulation results demonstrate that the POMDP-DDPG-based swarm control model significantly improves mission success rates and energy efficiency compared to baseline methods. The developed framework supports distributed learning and decision coordination across multiple agents, providing a foundation for scalable cognitive swarm autonomy. The outcomes of this research contribute to the advancement of energy-aware control algorithms for intelligent multi-agent systems and can be applied in security, environmental monitoring, and infrastructure inspection scenarios. The results were partially supported by the National Research Foundation of Ukraine, grant No. 2025.06/0022 “AI platform with cognitive services for coordinated autonomous navigation of distributed systems consisting of a large number of objects.

This paper proposes an extension of Zoutendijk’s Method of Feasible Directions (MFD) for solving linearly constrained multi-objective optimization problems. The approach integrates a Hit-and-Run sampling technique to generate a diverse set of feasible points and uses an iterative two-step process. This process involves determining an improving feasible direction by solving a quadratic subproblem and adjusting the step size using the multi-objective Armijo rule. As a result, the method generates a sequence of feasible points that converge to a critical Pareto point, and under convexity assumptions, to a Pareto-optimal point. The process is applied to all points in the generated population, enabling an adequate approximation of the Pareto front. The convergence of the method is proven. A comparative study, conducted in MATLAB on 25 linearly constrained multi-objective optimization benchmark problems, evaluates our MFD against the NSGA-II algorithm and another variant of MFD that uses a different direction search subproblem and sampling procedure. The quality of the approximated Pareto front is assessed using four metrics: purity, spacing, hypervolume, and generalized spread. The results show that our MFD outperforms the other methods in terms of both convergence speed and the quality of the Pareto front for the tested problems

A new type of chaotic attractors, whose basin of attraction is the entire phase space, is considered. The main difference between these attractors and the known ones is that any trajectory starting from the basin of attraction first enters a unique transport channel (which is a straight line), and then the trajectory reaches the attractor itself along this channel. For any quadratic dynamic system generating the mentioned attractor, a new concept of a uniquely defined degenerate autonomous quadratic dynamic system with exactly one real double equilibrium point is introduced. It is shown that if the degenerate system exhibits chaotic behavior, then the original (non-degenerate) system also exhibits similar chaotic behavior.

In this paper, we introduce and study a modified inertial subgradient extragradient iterative algorithm for solving bilevel split quasimonotone variational inequality problems with a fixed point constraint of demimetric mappings in the framework of real Hilbert spaces. The method involves a strongly monotone operator at the upper-level problem and quasimonotone mapping at the lower level. We obtain a strong convergence result of the proposed method under some mild conditions on the algorithm parameters without the prior knowledge of the operator norm or the coefficient of the underlying operator in the scope of infinite dimensional real Hilbert spaces. Finally, some numerical demonstrations are given to illustrate the gains of this method. Our results generalize and improve some well-known results in the literature.