This paper investigates the effect of green infrastructures in mitigating air pollution in urban environments. This complex issue needs a thorough examination of traffic-related emissions and the atmospheric dispersion of chemical species. A macroscopic second-order model is employed to simulate traffic conditions, followed by a microscopic approach to estimate emission levels. Finally, partial differential equations-based atmospheric dispersion, driven by chemistry ordinary differential equations for particulate/nitrogen oxides/ozone interactions, handles vertical transport under varying conditions. Numerical results show that different configurations of green barriers yield varying pollutant concentrations in metropolitan regions.

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.

Semi-open queueing networks are suitable for modeling complex manufacturing, health care, and logistics systems. Such networks are different from more well-known open queueing networks because the number of users, that can be serviced in the network simultaneously is restricted by a finite constant. The network loses customers who arrive when its capacity reaches its limit. This paper examined an analytical model characterized by features like the possibility to capture potential correlations in the arrival process by assuming the marked Markov arrival process and modify service rates in the network's nodes depending on the number of users currently processed in the network. A hysteresis strategy for dynamic service rate selection was assumed. Fixing the thresholds of this strategy, the behavior of the network was determined by a continuous-time multidimensional Markov chain with a finite state that is a quasi-birth-and-death process. An explicit formula for the generator of this process was obtained. Expressions for the computation of network performance measures were derived. Numerical results highlight the dependence of some measures on thresholds defining the control policy, and their use to optimize the system is illustrated.

Mostly motivated by the crop field classification problem and the automated computational methodology for extracting agricultural crop fields from satellite data, we proposed in a bounded variation (BV) space a new approach to the piecewise smooth approximation of the slope-based vegetation indices and the closely related crop field segmentation problem of multi-band satellite images.

This paper focuses on differential inclusions for measures, that are differential relations whose solutions are time-evolving measures. The definition of evolution equations for measures attracted a lot of attention recently. We start by recalling the main concepts developed in the latest literature and comparing them. In particular, we show how the definition of Measure Differential Inclusion is the most general allowing to model phenomena as diffusion from a Dirac delta. Then we pass to Lyapunov-type stability proposing two concepts of stability, based on the measure support and first moment, and show relationships between such definitions depending on the assumptions on the evolution equation used.

In this paper, we analyze the numerical aspects of the practical implementation of the generalized active contour model, that has been recently proposed in the literature, for extracting agricultural crop fields with a high level of inhomogeneity from satellite data. We also derive the corresponding Euler–Lagrange equation and discuss its relaxation method.

We study a thermal model associated with a heat-conducting material based on a three-phase-lag constitutive equation for the heat flux, a model that leads to a Moore–Gibson–Thompson type equation for the thermal displacement. We are researching the compatibility of the three-phase-lag constitutive equation in concern with the second law of thermodynamics, thus discovering restrictions to be imposed on the involved thermal coefficients. On this basis, we manage to obtain the well-posedness problem of the model as the uniqueness of the solutions and their continuous dependence on the given data. Finally, we show that such a model not only allows the propagation of damped in time waves but also exponentially decaying in time thermal standing mode waves. We also show that if the thermodynamic restrictions are not fulfilled, then we can be led to instability. Through the present treatment of the thermal model in question, we obtain important information on the associated Moore–Gibson–Thompson type equation for the thermal displacement.

Recently, the concept of measure differential equation was introduced in [17]. Such a concept allows for deterministic modeling of uncertainty, finite -speed diffusion, concentration, and other phenomena. Moreover, it represents a natural generalization of ordinary differential equations to measures. In this paper, we deal with the stability of fixed points for measure differential equations. In particular, we discuss two concepts related to classical Lyapunov stability in terms of measure support and first moment. The two concepts are not comparable, but the latter implies the former if the measure differential equation is defined by an ordinary one. Finally, we provide results concerning Lyapunov functions.

In this paper, we introduce a new idea to generalize the concept of relaxed control to the framework of measure differential equations, recently introduced in [15]. A relaxed control is defined as a probability measure on the space of controls, and, similarly, a measure control is a feedback relaxed control which depends on the measure distribution on the state space representing the state of the system. Relaxed controls are useful to solve optimal control and stabilization problems. On the other side, measure differential equations allow deterministic modeling of uncertainty, finite-speed diffusion, concentration, and other phenomena. Moreover, it represents a natural generalization of Ordinary Differential Equations to measures.We establish regularity properties of measure controls to ensure existence and uniqueness of trajectories and show applications to stabilization problems.

We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the simultaneous fusion and de- noising of color images with different spatial resolution. The characteristic feature of the proposed model is that we deal with a constrained minimization problem that lives in variable Sobolev-Orlicz spaces where the variable exponent, which is associated with non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. In view of this, we discuss the consistency of the proposed model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

We consider a multi-server queueing system with a visible queue and an arrival flow that is dynamically dependent on the system’s rating. This rating reflects the level of customer satisfaction with the quality and price of the provided service. A higher rating implies a higher arrival rate, which motivates the service provider to increase the price of the service. A steady-state analysis of this system using the proposed mechanism for changing the rating and a threshold strategy for changing the price is performed. This is carried out via the consideration of a suitably constructed multidimensional Markov chain. The impact of the variation in the threshold defining the strategy for changing the price on the key performance indicators is numerically illustrated. The results can be used to make managerial decisions, leading to an increase in the effectiveness of system operations.

We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the simultaneous contrast enhancement and denoising of color images. The characteristic feature of the proposed model is that we deal with a constrained non-convex minimization problem that lives in variable Sobolev-Orlicz spaces where the variable exponent is unknown a priori and it depends on a particular function that belongs to the domain of the objective functional. In contrast to the standard approach, we do not apply any spatial regularization to the image gradient. We discuss the consistency of the variational model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

A multi-server retrial queue with a finite number of sources of requests was considered. In contrast to similar models studied in the literature, we assumed this number is not constant but changes its value in a finite range. During the stay in the system, each source generates the service requests. These requests are processed in a finite pool of servers. After service completion of a request, the source is granted the possibility to generate another request. If the source does not use this possibility during an exponentially distributed time, it is deleted from the system. If the request finds all servers busy, it can make repeated attempts to enter the service. If all servers are busy, the request may depart from the system without service. In this case, with a fixed probability, the source that generated this request is deleted from the system. Sources arrive according to a Markov arrival process. If the number of sources in the system at the arrival epoch has the maximum allowed number, the arriving source is lost. This system is a more adequate model of many real-world systems than the standard finite source queue. Analysis of the considered system required a four-dimensional continuous-time Markov chain. The generator of the chain was obtained as a block matrix with four levels of nesting. The stationary distribution of this Markov chain was found numerically as well as the values of the system's performance measures. The dependence of these measures on the maximum allowed number of sources and the number of servers was numerically clarified. An example of solving an optimization problem was presented.

This paper investigates the propagation of in-plane surface waves in a coated thermoelastic half-space. First, it investigates a special case where the surface layer is described by the Maxwell–Cattaneo thermoelastic approach, while the half-space is filled by a thermoelastic material described by the classical Fourier law for the heat flux. The contact between the layer and the half-space is assumed to be welded, i.e., the displacements and the temperature, as well as the stresses and the heat flux are continuous through the interface of the layer and the half-space. The boundary and continuity conditions of the problem are formulated and then the exact dispersion relation of the surface waves is established. An illustrative numerical simulation is presented for the case of an aluminum thermoelastic layer coating a thermoelastic copper half-space, highlighting important aspects regarding the propagation of Rayleigh waves in such structures. The exact effective boundary conditions at the interface are also established replacing the entire effect of the layer on the half-space. The general case of the problem is also investigated when both the surface layer and the half-space are described by the Maxwell–Cattaneo thermoelasticity theory. This study helps to further understand the propagation characteristics of elastic waves in layered structures with thermal effects described by the Maxwell–Cattaneo approach.

In this paper, the problem of restoration of cloud contaminated optical images is studied in the case when we have no information about brightness of such images in the damage region. We propose a new variational approach for exact restoration of optical multi-band images utilising Synthetic Aperture Radar (EOS – Spatial Data Analytics, GIS Software, Satellite Imagery – is a cloud-based platform to derive remote sensing data and analyse satellite imagery for business and science purposes) images of the same regions. We prove existence of solutions, propose an alternating minimisation method for computing them, prove convergence of this method to weak solutions of the original problem and derive optimality conditions.

We analyse a cell of Cognitive Radio Network ( $CRN$ ) as the multiline queueing system supplying service to two Markovian arrival flows of users. Primary (or licensed) users called as High Priority Users ( $HPU\text{s}$ ) have a preemptive priority over the secondary (cognitive) users called as Low Priority Users ( $LPU\text{s}$ ). The $HPU\text{s}$ are dropped upon the arrival only if all servers are occupied by $HPU\text{s}$ . If at the arrival epoch all servers are busy but some of them provide service to $LPU\text{s}$ , service of one $LPU$ is immediately interrupted and service of the $HPU$ begins in the released server. A $LPU$ is accepted only if the number of busy servers at arrival epoch is less than the defined in advance threshold $M$ . Otherwise, the $LPU$ is permanently lost or becomes a retrial user. A retrial user repeats attempts to receive service later after random time intervals. The $LPU$ whose service is interrupted is either lost or transferred to a virtual place called as orbit. The users placed in the orbit may be impatient and can renege the system. The service time follows an exponential probability distribution with the rate determined by the user’s type. After loss of a $HPU$ , admission of $LPU\text{s}$ is blocked. $LPU\text{s}$ are informed that their access is temporarily suspended and do not generate new requests until blocking expires. The purpose of the research is the optimization of threshold $M$ and admission blocking period duration. Behavior of the system is described by a multidimensional continuous-time Markov chain. Its generator, ergodicity condition and invariant distribution are derived. Expressions for performance indicators are given. Numerical results demonstrating usefulness of blocking and significance of account of correlation in arrivals are presented. E.g., in the presented example of cost criterion optimization blocking gives 18 percent profit comparing to the system without blocking.

A queuing system having two different servers is under study. Demands enter the system according to a Markov arrival process. Service times have phase-type distribution. Service of demands is possible only if the fixed number of energy units, probably different for two servers, is available in the system at the potential service beginning moment. Energy units arrive in the system also according to a Markov arrival process and are stored in a stock (battery) of a finite capacity. Leakage of energy units from the stock can occur. Demands waiting in the infinite buffer are impatient and can leave the buffer after an exponentially distributed waiting time. One server is the main one and permanently provides service when the buffer is not empty and the required number of energy units is available. The second server is the assistant server and is switched on or off depending on the availability of energy units and queue length according to the hysteresis strategy defined by two thresholds. The assistant server is switched on when the queue length is not less than the greater threshold and is switched off when the queue length becomes smaller than the smaller threshold. The use of the assistant server has to be paid. Thus, the problem of the optimal selection of the thresholds defining the control strategy naturally arises. To solve this problem, the study of the behavior of the system under any fixed values of the parameters of the control strategy is necessary. Such a study is given in this paper. Numerical results are presented. They illustrate the feasibility of computer realization of the developed algorithms for computation of the stationary distribution of the system states and the main key performance indicators as well as the result of solving one of the possible optimization tasks.

A higher integrability of the gradient of a solution to the Zaremba problem in a bounded Lipschitz plane domain is proved for an inhomogeneous p -Laplace equation.

We study a coupled two-level variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its consistence and solvability issues. At the first level, we deal with the so-called temporal interpolation problem that can be cast as a state constrained optimal control problem for anisotropic convection-diffusion equation with two types of control functions — distributed $L^2$-control and $BV$-bounded control in coefficients. At the second level, we have a constrained minimization problem with the nonstandard growth energy functional that lives in a variable Sobolev-Orlicz space. The characteristic feature of the proposed model is the fact that the variable exponent, which is associated with non-standard growth in the objective functional, is unknown a priori and it depends on the solution of the first-level optimal control problem.

In this work, we study a sparse optimal control problem involving a quasilinear parabolic equation with variable order of nonlinearity as a state equation and with a pointwise control constraints. We show that in the case if the cost functional contains the terminal term of the tracking type, the proposed optimal control problem is ill-posed, in general. In view of this, we provide a sufficiently mild relaxation of the proposed problem and establish the existence of optimal solutions for the relaxed version. Using the compensated compactness technique and the consept of variational convergence of minimization problems, we study the attainability of optimal pairs to the relaxed problem by optimal solutions of the special approximating problems. We also discuss the optimality conditions for approximating problems and provide their substantiation.