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This paper addresses optimal control problems with loss control regions. In that context the state space is partitioned into disjoint subsets, referred to as regions , which are classified into two types: control regions and loss control regions. When the state belongs to a control region, the control is permanent (i.e. the control value is authorized to be modified at any time). On the contrary, when the state belongs to a loss control region, the control must remain constant as long as the state belongs to this region. The objective of this paper is twofold. First, we reformulate the above setting into a hybrid optimal control problem (with spatially heterogeneous dynamics) involving moreover a regionally switching parameter, and we prove a corresponding hybrid maximum principle: hence first-order necessary optimality conditions in a Pontryagin form are obtained. Second, this paper proposes a two-steps numerical scheme to solve optimal control problems with loss control regions. It is based on a direct numerical method (applied to a regularized problem) which initializes an indirect numerical method (applied to the original problem and based on the aforementioned necessary optimality conditions). This numerical approach is applied to several illustrative examples. [ABSTRACT FROM AUTHOR]

The authors give a stochastic maximum principle for square-integrable optimal control of linear stochastic systems. The control domain is not necessarily convex and the cost functional can have a quadratic growth. In particular, they give a stochastic maximum principle for the linear quadratic optimal control problem. [ABSTRACT FROM AUTHOR]

The propagation of rumors indisputably inflicts profound negative impacts on society and individuals. This article introduces a new unaware ignorants-aware ignorants-spreaders-recovereds rumor spreading model that combines individual vigilance self-awareness with nonlinear spreading rate. Initially, the positivity of the system solutions and the existence of its positive invariant set are rigorously proved, and the rumor propagation threshold is solved using the next-generation matrix method. Next, a comprehensive analysis is conducted on the existence of equilibrium points of the system and the occurrence of backward bifurcation. Afterward, the stability of the system is validated at both the rumor-free equilibrium and the rumor equilibrium, employing the Jacobian matrix approach as well as the Lyapunov stability theory. To enhance the efficacy of rumor propagation management, a targeted optimal control strategy is formulated, drawing upon the Pontryagin's Maximum principle as a guiding framework. Finally, through sensitivity analyses, numerical simulations, and tests of real cases, we verify the reliability of the theoretical results and further consolidate the solid foundation of the above theoretical arguments. [ABSTRACT FROM AUTHOR]

In this work, we investigate a linear second-order numerical method for the Allen-Cahn equation with general mobility. The proposed scheme is a combination of the two-step first- and second-order backward differentiation formulas for time approximation and the central finite difference for spatial discretization, two additional stabilizing terms are also included. The discrete maximum bound principle of the numerical scheme is rigorously proved under mild constraints on the adjacent time-step ratio and the two stabilization parameters. Furthermore, the error estimates in H 1 -norm for the case of constant mobility and L ∞ -norm for the general mobility case, as well as the energy stability for both cases are obtained. Finally, we present extensive numerical experiments to validate the theoretical results, and develop an adaptive time-stepping strategy to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we develop a comprehensive study on principal eigenvalues and both the (weak and strong) maximum and comparison principles related to an important class of nonlinear systems involving fractional m-Laplacian operators. Explicit lower bounds for principal eigenvalues of this system in terms of the diameter of bounded domain Ω ⊂ R N are also proved. As application, we measure explicitly how small has to be diam (Ω) so that weak and strong maximum principles associated to this problem hold in Ω . [ABSTRACT FROM AUTHOR]

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. [Izv. Akad. Nauk SSSR Mekh. Ghidk. Gaza 2, 103–108 (1988)] in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region from the top to the interface in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle and the comparison theorem. Moreover, the existence of its steady solution and its stability are discussed. [ABSTRACT FROM AUTHOR]

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. in 1988 in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region below an interface to the bottom in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle. Moreover, the existence of its steady solution and its stability are shown. [ABSTRACT FROM AUTHOR]

In this research, we study certain characteristics of the solution for an equation involving uniformly elliptic nonlocal operator by establishing various maximum principles in bounded and unbounded regions. Additionally, we don't need the conditions that the narrow region principle's bound assumption and decay in the unbounded domain. In order to get the monotonicity of the solution, we use several maximum principles and averaging effects to overcome the challenges that generated by the uniformly elliptic nonlocal operator and fractional‐order variable t$$ t $$. Of course, while investigating the qualitative characteristics of the equation, we also provide some estimates of auxiliary function. For the study of additional nonlocal operators, we believe the method of uniformly elliptic nonlocal operators will also be helpful. [ABSTRACT FROM AUTHOR]

In this paper, we consider self-similar solutions of the r -mean curvature flow into the (n + 1) -dimensional Euclidean space. By employing some general maximum principles as the main tool, we characterize some self-similar solutions of the r -mean curvature flow under some suitable geometric constraints. [ABSTRACT FROM AUTHOR]

This work is dedicated to the study of existence of solutions for a Schrödinger–Poisson type system involving possible different integrodifferential operators in the presence of a nonnegative but not bounded away from zero potential. We consider a pure power nonlinearity u ↦ | u | p - 2 u , with 4 - β < p < 6 / (3 - 2 α) , where α and β are positive parameters described in each equation. Our argument is variational in nature and involves the use of a deformation type lemma on a minimization over a Nehari–Pohozaev manifold. Some regularity results are provided and a maximum type principle for integrodifferential operators is proved. [ABSTRACT FROM AUTHOR]

In this paper we state some sharp maximum principle, i.e. we characterize the geometry of the sets of minima for supersolutions of equations involving the $k$ -th fractional truncated Laplacian or the $k$ -th fractional eigenvalue which are fully nonlinear integral operators whose nonlocality is somehow $k$ -dimensional. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the tempered fractional parabolic problem $$ \begin{align*}\frac{\partial u}{\partial t}(x, t)-\left(\Delta+\lambda\right)^{\frac{\alpha}{2}} u(x, t)=f(u(x, t)), \end{align*} $$ where $-\left (\Delta +\lambda \right)^{\frac {\alpha }{2}}$ is a tempered fractional operator with $\alpha \in (0,2)$ and $\lambda $ is a sufficiently small positive constant. We first establish maximum principle principles for problems involving tempered fractional parabolic operators. And then, we develop the direct sliding methods for the tempered fractional parabolic problem, and discuss how they can be used to establish monotonicity results of solutions to the tempered fractional parabolic problem in various domains. We believe that our theory and methods can be conveniently applied to study parabolic problems involving other nonlocal operators. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in R2$\mathbb {R}^2$. We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system. [ABSTRACT FROM AUTHOR]

This article investigates the finite horizon linear quadratic gaussian (LQG) control problem for networked control systems with two‐way Markovian packet dropouts and input delay, where packet dropouts occur from the sensor to the estimator and from the controller to the actuator, and delay occurs from the controller to the actuator. The novelty of this work lies in providing a complete solution for the optimal control problem subject to two‐way Markovian packet dropouts and input delay. The contributions of this article are as follows: first, by applying the maximum principle to the discrete‐time linear systems and the quadratic cost function involving input delay and Markovian packet dropouts, a solution to the forward and backward stochastic difference equations (FBSDEs) is derived. Second, a sufficient and necessary condition for the optimal control problem is obtained by decoupling the coupled Riccati equations. Finally, the explicit solution of the optimal controller is presented based on the complete square method. Numerical examples are provided to demonstrate the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following fractional $ p $-Laplacian equations with the Hardy potential:$ \mathcal{L}^{\alpha,p}_{\mu}u(x) = f(x,u(x),\nabla u(x)), $where $ \mathcal{L}^{\alpha,p}_{\mu} = (-\Delta)^{\alpha}_{p}+\frac{\mu}{|x|^{\alpha p}} $. Under some mild conditions on the nonlinearity $ f(x,u,\nabla u) $, we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional $ p $-Laplacian with the Hardy potential in bounded domains. Moreover, we also prove a maximum principle along a sequence of approximate maximum points in unbounded domains. Our results are extensions of Chen and Li [4] and Dai, Liu and Wang [16]. [ABSTRACT FROM AUTHOR]

The Allen-Cahn (AC) model is a mathematical equation that represents the phase separation process. The AC equation has numerous applications in various disciplines, such as image processing, physics, and biology. It models phase transitions, such as solidification and grain growth in materials, pattern formation in chemical reactions, and domain coarsening in biological systems like lipid membranes. Numerical methods are crucial for solving the AC equation due to its complexity and nonlinear nature. Analytical solutions are often extremely difficult to obtain. Therefore, the development of efficient numerical techniques is indispensable for approximating solutions and studying phase transitions, material behavior, and pattern formation accurately. We investigate the stability of an explicit finite difference method (FDM) used to numerically solve the two-dimensional (2D) AC model with a high-order polynomial potential, which was recently proposed to preserve a more intricate structure of interfaces. To demonstrate the precision and optimal estimate of our stability constraints, we conduct various computational tests using the derived time step formulas that ensure the maximum principle. [ABSTRACT FROM AUTHOR]

This paper mainly deals with the monotonicity and Liouville type theorems of nonnegative solutions for a class of fully nonlinear fractional equations in half-space. Compared with serial works in [9,14,20], the restrictions on the monotonicity of nonlinear terms and the decay assumption on solutions are not necessary here. Moreover, the monotonicity result of nonnegative bounded solutions for corresponding cooperative systems is also derived. Specifically, we establish some new types of maximum principles for fully nonlinear fractional equations and systems in unbounded domains, which will play crucial roles in carrying on the method of moving planes. [ABSTRACT FROM AUTHOR]

In this paper, based on the Lagrange multiplier method, we construct a maximum principle preserving scheme for the time-fractional Allen-Cahn equation of 2- α (0 < α < 1) order. The correction energy of this scheme is increased by a term compared to the original energy, which is O (τ α). We prove that our scheme is unconditionally stable related to the corrected energy and verify the convergence, maximum principle, and energy conservation properties of the algorithm through numerical examples. We also find that the larger the α , the faster the evolution of the time-fractional Allen-Cahn equation. [ABSTRACT FROM AUTHOR]

In this paper, we develop a convex splitting algorithm for the time-dependent Ginzburg-Landau equation, which can preserve both the energy stability and maximum bound principle. The basic idea of the convex splitting method is to decompose the energy functional into the convex part and the concave part. The term corresponding to the convex part of the equation is implicitly treated, and the concave part is explicitly processed. The backward Euler time discretizing method is chosen for the time-dependent Ginzburg-Landau equation. The theoretical analysis proves that the convex splitting method can preserve the maximum bound principle and energy stability. The numerical results show that the numerical algorithm is stable. [ABSTRACT FROM AUTHOR]

In this paper, we prove new Liouville-type properties of fully nonlinear subelliptic equations modelled on the Heisenberg group. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential equations. [ABSTRACT FROM AUTHOR]

We investigate the maximum principle for the weak solutions to the Cauchy problem for the hyperbolic fourth‐order linear equations with constant complex coefficients in the plane bounded domain. [ABSTRACT FROM AUTHOR]

Summary: The existence, solution structure, and equivalence of sparse control for linear time‐varying systems are investigated. We proved the existence of the optimal solution of sparse control for the linear time‐varying systems. By using the non‐smooth maximum principle, we are given the necessary conditions for the sparse control optimality for linear time‐varying systems. We proved that the optimal solution of sparse control taking values of ±1$$ \pm 1 $$ and 0. Under the normality of the linear time‐varying systems, we proved an equivalence theorem between the sparse control and L1$$ {L}^1 $$ optimal control. Numerical examples are included to illustrate the effectiveness of the sparse control for linear time‐varying systems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present some applications of the Neumann Laplacian in image processing, along with the necessary mathematical background. We prove weak and strong versions of the maximum principle for weak solutions of elliptic and parabolic problems and apply them to a Fisher K.P.P.-type equation. The original contribution lies in the application of this equation in image processing, where various diffusion-like effects can be achieved. Additionally, a review of the basics of linear and nonlinear PDEs with Neumann boundary conditions is provided, along with updated bibliography and recent qualitative results. There are also some new theoretical results developed in this work. [ABSTRACT FROM AUTHOR]

Landscape functions are a popular tool used to provide upper bounds for eigenvectors of Schrödinger operators on domains. We review some known results obtained in the last 10 years, unify several approaches used to achieve such bounds, and extend their scope to a large class of linear and nonlinear operators. We also use landscape functions to derive lower estimates on the principal eigenvalue—much in the spirit of earlier results by Donsker–Varadhan and Bañuelos–Carrol—as well as upper bounds on heat kernels. Our methods solely rely on order properties of operators: We devote special attention to the case where the relevant operators enjoy various forms of elliptic or parabolic maximum principle. Additionally, we illustrate our findings with several examples, including p$p$‐Laplacians on domains and graphs as well as Schrödinger operators with magnetic and electric potential, also by means of elementary numerical experiments. [ABSTRACT FROM AUTHOR]

The G-Brownian-motion-driven stochastic differential equations (G-SDEs) as well as the G-expectation, which were seminally proposed by Peng and his colleagues, have been extensively applied to describing a particular kind of uncertainty arising in real-world systems modeling. Mathematically depicting long-time and limit behaviors of the solution produced by G-SDEs is beneficial to understanding the mechanisms of system's evolution. Here, we develop a new G-semimartingale convergence theorem and further establish a new invariance principle for investigating the long-time behaviors emergent in G-SDEs. We also validate the uniqueness and the global existence of the solution of G-SDEs whose vector fields are only locally Lipschitzian with a linear upper bound. To demonstrate the broad applicability of our analytically established results, we investigate its application to achieving G-stochastic control in a few representative dynamical systems. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the symmetry and monotonicity of positive solutions for linear coupling elliptic systems in a ball in $ \mathbf{R}^N $ R N . Using the Alexandrov–Serrin method of moving planes combined with the strong maximum principle, we prove that the solutions of elliptic systems with linear couplings in a ball are symmetric w.r.t. 0 and radially decreasing. For our problems, the tangential gradient of solutions and the coupling conditions play important roles in using the moving plane method. Our results on the symmetry of solutions are further research based on the existence of solutions in [1]. [ABSTRACT FROM AUTHOR]

Random environment and human activities have important effects on the survival of listeria. In this paper, treating infected people and removing bacteria from the environment as control strategies, we developed a listeriosis model that considers random noise and spatial diffusion. By constructing a Lyapunov function, we demonstrated the existence and uniqueness of the global positive solution of the model. However, it was a challenging task to realize the optimal control of the model by solving the Pontryagin random maximum principle with the lowest control cost. Therefore, our study on near-optimal controls is of great significance for controlling the spread of listeriosis. Initially, we gave some adjoint equations and a priori estimates. Subsequently, the Pontryagin random maximum principle was utilized to establish the sufficient and necessary conditions for achieving near-optimal controls. Ultimately, the theoretical findings are corroborated through numerical analysis. [ABSTRACT FROM AUTHOR]

Nowadays, rumors spread more rapidly than before, leading to more panic and instability in society. Therefore, it is essential to seek out propagation law in order to prevent rumors from spreading further and avoid unnecessary harm. There is a connection between rumor models and symmetry. The consistency of a system or model is referred to as the level of symmetry under certain transformations. For this purpose, we propose a fractional-order Ignorant–Latent–Spreader–Remover (ILSR) rumor propagation model that incorporates media reports and a nonlinear inhibition mechanism. Firstly, the boundedness and non-negativeness of the solutions are derived under fractional differential equations. Secondly, the threshold is used to evaluate and illustrate the stability both locally and globally. Finally, by utilizing Pontryagin's maximum principle, we obtain the necessary conditions for the optimal control in the fractional-order rumor propagation model, and we also obtain the associated optimal solutions. Furthermore, the numerical results indicate that media reports can decrease the spread of rumors in different dynamic regions, but they cannot completely prevent rumor dissemination. The results are also exhibited and corroborated by replicating the model with specific hypothetical parameter values. It can be inferred that fractional order yields more favorable outcomes when rumor permanence in the population is higher. The presented method facilitates the acquisition of profound insights into the dissemination dynamics and subsequent consequences of rumors within a societal network. [ABSTRACT FROM AUTHOR]

In this paper, we first study the dual fractional parabolic equation ∂ t α u (x , t) + (− Δ) s u (x , t) = f (u (x , t)) in  B 1 (0) × ℝ , subjected to the vanishing exterior condition. We show that for each t ∈ ℝ , the positive bounded solution u (⋅ , t) must be radially symmetric and strictly decreasing about the origin in the unit ball in ℝ n . To overcome the challenges caused by the dual nonlocality of the operator ∂ t α + (− Δ) s , some novel techniques were introduced. Then we establish the Liouville theorem for the homogeneous equation in the whole space ∂ t α u (x , t) + (− Δ) s u (x , t) = 0 in  ℝ n × ℝ. We first prove a maximum principle in unbounded domains for antisymmetric functions to deduce that u (x , t) must be constant with respect to x. Then it suffices for us to establish the Liouville theorem for the Marchaud fractional equation ∂ t α u (t) = 0 in  ℝ. To circumvent the difficulties arising from the nonlocal and one-sided nature of the operator ∂ t α , we bring in some new ideas and simpler approaches. Instead of disturbing the antisymmetric function, we employ a perturbation technique directly on the solution u (t) itself. This method provides a more concise and intuitive way to establish the Liouville theorem for one-sided operators ∂ t α , including even more general Marchaud time derivatives. [ABSTRACT FROM AUTHOR]

The aim of this study is to parametrically analyze the vibration characteristics of weaving needles during the feeding motion, providing effective references for the selection of parameters in the three-dimensional weaving process. To control the degree of carbon fiber wear during the manufacturing process and maximize its maximum forming height, we have established a systematic dynamic model based on Hamilton's principle. The state equation was solved using the Runge-Kutta method. The influence of different weaving parameters on the weaving needle's free-end vibration response and equilibrium position of was studied. The results indicate that the weaving velocity and height of the cross-section have a significant impact on the vibration amplitude of the free end of the weaving needle and the deflection of the equilibrium position. Moreover, the increase in deflection exhibits a certain inhibitory effect on the amplitude. [ABSTRACT FROM AUTHOR]

In this paper, we first derive the existence and uniqueness theorems for solutions to a class of generalized mean-field delay stochastic differential equations and mean-field anticipated backward stochastic differential equations (MFABSDEs). Then we study the stochastic maximum principle for generalized mean-field delay control problem. Since the state equation is distribution-depending, we define the adjoint equation as a MFABSDE in which all the derivatives of the coefficients are in Lions' sense. We also provide a sufficient condition for the optimality of the control. [ABSTRACT FROM AUTHOR]

In this paper, a finite volume (TPFA) method is employed to simulate a degenerate breast cancer model that captures the progressive mutations from a normal breast stem cell to a tumor cell. The model incorporates a degenerate parabolic equation to represent the interaction between solid tumor growth and its environment, which involves the release of degradative enzymes governed by a partial differential equation. The discrete maximum principle is verified to be satisfied by the proposed finite volume scheme, and the convergence of the existing discrete solutions to a weak solution is proven. Additionally, the development of breast cancer is demonstrated through a numerical experiment. [ABSTRACT FROM AUTHOR]

A nonlinear vertex-centered finite volume scheme that preserves the discrete maximum principle (DMP) is proposed for anisotropic diffusion equations on polygonal meshes. The new scheme is constructed from a vertex-centered linearity-preserving scheme combined with a nonlinear correction technique. The key ingredient is the introduction of the correction coefficient and the limiter. Particularly, the choice of the limiter is discussed and two new and effective limiters are proposed. Different from most existing nonlinear DMP-preserving schemes, the new nonlinear scheme does not rely on the nonlinear combination of linear flux approximations and requires no interpolation of the auxiliary unknowns. More important is that, it can be proved theoretically that the new scheme is symmetric and coercive on general polygonal meshes with star-shaped cells. Moreover, other theoretical results such as the DMP property and stability of the correction scheme are also presented. Numerical experiments demonstrate the accuracy, efficiency, and DMP property of the new scheme on general polygonal meshes with heterogeneous and anisotropic diffusion tensors. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the monotonicity and one-dimensional symmetry of solutions for parabolic equations related to the weighted fractional Laplacian on the whole space. We first establish a generalized weighted average inequality and the maximum principle in unbounded domains, and then carry out the sliding method to obtain monotonicity and one-dimensional symmetry of solutions. [ABSTRACT FROM AUTHOR]

Sufficient conditions are obtained for a signed maximum principle for boundary value problems for Riemann–Liouville fractional differential equations with analogues of Neumann or periodic boundary conditions in neighborhoods of simple eigenvalues. The primary objective is to exhibit four specific boundary value problems for which the sufficient conditions can be verified. To show an application of the signed maximum principle, a method of upper and lower solutions coupled with monotone methods is developed to obtain sufficient conditions for the existence of a maximal solution and a minimal solution of a nonlinear boundary value problem. A specific example is provided to show that sufficient conditions for the nonlinear problem can be realized. [ABSTRACT FROM AUTHOR]

In this paper, we study the modern mathematical theory of the optimal control problem associated with the fractional Roesser model and described by Caputo partial derivatives, where the functional is given by the Riemann-Liouville fractional integral. In the formulated problem, a new version of the increment method is applied, which uses the concept of an adjoint integral equation. Using the Banach fixed point principle, we prove the existence and uniqueness of a solution to the adjoint problem. Then the necessary and sufficient optimality condition is derived in the form of the Pontryagin's maximum principle. Finally, the result obtained is illustrated by a concrete example. [ABSTRACT FROM AUTHOR]

We investigate optimal control problems with L 0 constraints, which restrict the measure of the support of the controls. We prove necessary optimality conditions of Pontryagin maximum principle type. Here, a special control perturbation is used that respects the L 0 constraint. First, the maximum principle is obtained in integral form, which is then turned into a pointwise form. In addition, an optimization algorithm of proximal gradient type is analyzed. Under some assumptions, the sequence of iterates contains strongly converging subsequences, whose limits are feasible and satisfy a subset of the necessary optimality conditions. [ABSTRACT FROM AUTHOR]

We study a repulsion-diffusion equation with immigration and linear diffusion, whose asymptotic behaviour is related to stability of long-term dynamics in spatial population models and other branching particle systems. We prove well-posedness and find sharp conditions on the repulsion under which a form of the maximum principle and a strong notion of global boundedness of solutions hold. The critical asymptotic strength of the repulsion is $ |x|^{1-d} $, that of the Newtonian potential. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of one-dimensional hyperbolic delay differential equations (HDDEs) and their corresponding initial conditions. HDDEs are a class of differential equations that involve a delay term, which represents the effect of past states on the present state. The delay term poses a challenge for the application of standard numerical methods, which usually require the evaluation of the differential equation at the current step. To overcome this challenge, various numerical methods and analytical techniques have been developed specifically for solving a system of first-order HDDEs. In this study, we investigate these challenges and present some analytical results, such as the maximum principle and stability conditions. Moreover, we examine the propagation of discontinuities in the solution, which provides a comprehensive framework for understanding its behavior. To solve this problem, we employ the method of lines, which is a technique that converts a partial differential equation into a system of ordinary differential equations (ODEs). We then use the Runge–Kutta method, which is a numerical scheme that solves ODEs with high accuracy and stability. We prove the stability and convergence of our method, and we show that the error of our solution is of the order O (Δ t + h ¯ 4) , where Δ t is the time step and h ¯ is the average spatial step. We also conduct numerical experiments to validate and evaluate the performance of our method. [ABSTRACT FROM AUTHOR]

In the class of linear optimal control problems, a new approach to finding a solution to the boundary value problem of the maximum principle is proposed. The approach is based on the maximum principle expressed as fixed point problems, which makes it possible to construct new methods for finding extremal controls and determine solutions to the boundary value problem of the maximum principle. The efficiency of fixed point methods is illustrated by a model example. [ABSTRACT FROM AUTHOR]

We consider three optimal control problems (for linear, bilinear, and quadratic functionals) for a special bilinear system with a matrix of rank 1. For the first problem, we obtain two versions of conditions for initial data of the system and functional such that the maximum principle becomes a sufficient optimality condition. In this case, the problem becomes very simple: the optimal control is determined in the integration process of the phase or adjoint system (one Cauchy problem). Next, the optimization problem for a bilinear functional is considered. Sufficient optimality conditions for the boundary controls without switching points are obtained. These conditions are represented as inequalities for functions of one variable (time). The optimal control problem with the quadratic functional is reduced to the bilinear case on the basis of special increment formula. [ABSTRACT FROM AUTHOR]

In this article, we present a hybrid numerical scheme for solving the Allen–Cahn (AC) equation on a nonuniform mesh. The AC equation represents a model for antiphase domain coarsening in a binary mixture. To solve the AC equation on nonuniform grids, the AC equation is split into linear and nonlinear terms applying the operator splitting method. As the first step, the nonlinear term is solved using the separation of variables. Next, the diffusion term is decomposed into linear operators in each space direction. In each direction, we sequentially solve each diffusion equation by applying the implicit Euler method on a nonuniform grid to update the numerical solution. Because the implicit Euler method and analytic solution do not depend on the time step size, the proposed hybrid numerical method for the AC equation on a nonuniform mesh is unconditionally stable. In addition, we prove the proposed scheme satisfies the maximum principle. To verify the superior performance of the proposed method, we conduct numerical simulations such as motion by mean curvature, total energy non-increasing property, the maximum principle and unconditional stability. [ABSTRACT FROM AUTHOR]

In this paper, by using critical point theory, we study the infinitely many solutions of partial discrete Kirchhoff type problems. Moreover, we acquire some sufficient conditions for the existence of positive solutions to the boundary value problems by using the strong maximum principle. As far as we know, this is the research of partial discrete Kirchhoff type problems for the first time. Finally, our major results are explained with four examples. [ABSTRACT FROM AUTHOR]

In this study, we aimed to explore the dynamics of rail passengers' negative attitudes that can be influenced by safety concerns and unreliable train operations. We mainly formulated and analyzed a mathematical model of fractional order and derived an optimal control problem considering the Caputo fractional order derivative. In the analysis part of the model, we proved that the solutions of the model for the dynamical system are non-negative and bounded, and determined the passengers' negative attitude-free and negative attitude persistence equilibrium points of the model. Both the local and global stabilities of these equilibrium points were examined. Furthermore, we verified the conditions necessary for the existence of optimal control strategies. We then proceeded to analyze the proposed control strategies, which aim to prevent negative attitudes and improve the attitudes of passengers who have already developed negative attitudes. Finally, we conducted numerical simulations to examine the effects of these control strategies. The results revealed that protecting passengers from developing negative attitudes and improving the attitudes of those who have already developed such attitudes are crucial for improving the overall attitude of railway passengers. These measures can effectively address any negative experiences caused by safety concerns and unreliable train operations. [ABSTRACT FROM AUTHOR]

We present and evaluate several explicit, large time-stepping algorithms for the Allen-Cahn equation. Our approach incorporates a stabilization technique and uses Taylor series approximations for exponential functions to develop a family of up to third-order parametric Runge–Kutta schemes that maintain fixed-points and maximum principle for any time step τ > 0 . We also introduce a new relaxation technique that eliminates time delay caused by stabilization. To further decrease the stabilization parameter, we utilize an integrating factor with respect to the stiff linear operator and develop a parametric relaxation integrating factor Runge–Kutta (pRIFRK) framework. Compared to existing maximum-principle-preserving (MPP) schemes, the proposed parametric relaxation approaches are free from limiters, cut-off post-processing, exponential decay, or time delay. Linear stability analysis determines that the parametric approaches are A-stable when appropriate parameters are used. In addition, we provide error estimates in the l ∞ -norm with the help of the MPP property. We demonstrate the high-order temporal accuracy, maximum-principle-preservation, energy stability, and delay-free properties of the proposed schemes through a set of experiments on 1D, 2D, and 3D problems. [ABSTRACT FROM AUTHOR]

The geometric approach to the study of the Herglotz problem developed in Massa and Pagani [J. Math. Phys. 64, 102902 (2023)] is extended to the case in which the evolution of the system is subject to a set of non-holonomic constraints. The original setup is suitably adapted to the case in study. Various aspects of the problem are considered: the direct derivation of the evolution equations; the super-lagrangian approach; the resulting super-Hamiltonian and its relation with Pontryagin's maximum principle; the abnormality index of the extremals; the invariance properties of the theory and the consequent existence of Herglotz Lagrangians gauge equivalent to ordinary ones. [ABSTRACT FROM AUTHOR]

Assuming Perelman's estimates, we give a new proof of uniform L^\infty estimate along normalized Kähler-Ricci flow on Fano manifolds with Kähler-Einstein metrics, using Chen-Cheng's auxiliary Monge-Ampère equation and the Alexandrov-Bakelman-Pucci maximum principle. This proof does not use pluripotential theory. [ABSTRACT FROM AUTHOR]

We consider generalized solutions of the Perona--Malik equation in dimension one, defined as all possible limits of solutions to the semidiscrete approximation in which derivatives with respect to the space variable are replaced by difference quotients. Our first result is a pathological example in which the initial data converge strictly as bounded variation functions, but strict convergence is not preserved for all positive times and, in particular, many basic quantities, such as the supremum or the total variation, do not pass to the limit. Nevertheless, in our second result we show that all our generalized solutions satisfy some of the properties of classical smooth solutions, namely, the maximum principle and the monotonicity of the total variation. The verification of the counterexample relies on a comparison result with suitable sub-/super-solutions. The monotonicity results are proved for a more general class of evolution curves, that we call uv-evolutions. [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach for analyzing efficient numerical solution of a class of singularly perturbed parabolic convection-diffusion PDEs having a finite jump discontinuity mostly in the source function at the interior of the spatial domain. These PDEs often appear in mathematical modeling of the semiconductor devices; and solutions of such problems usually exhibit a weak interior layer at one side of the point of discontinuity along with a boundary layer at one side of the spatial domain. The prime objectives of this paper are to overcome the theoretical challenge related to the monotonocity of the finite difference operator which utilizes one-sided second-order difference operators at the interface point; and also to establish higher-order numerical approximation in space, regardless of smaller and larger values of the parameter ε. Proving discrete maximum principle of the proposed difference operator is found to be challenging task on the standard Shishkin mesh adapted to both boundary and weak interior layers. We overcome this difficulty by constructing a special non-uniform mesh, called modified layer-adapted mesh; and hereby establish the stability as well as the parameter-uniform error estimate in the discrete supremum norm. Finally, the theoretical estimate is verified with numerical results for test examples with and without the exact solution. Moreover, numerical results are obtained for the semilinear PDEs and also compared with the implicit upwind scheme to exhibit the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]