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This paper presents an investigation into the stability and control aspects of delayed partial differential equation (PDE) systems utilizing the Lyapunov method. PDEs serve as powerful mathematical tools for modeling diverse and intricate systems such as heat transfer processes, chemical reactors, flexible arms, and population dynamics. However, the presence of delays within the feedback loop of such systems can introduce significant challenges, as even minor delays can potentially trigger system instability. To address this issue, the Lyapunov method, renowned for its efficacy in stability analysis, is employed to assess the exponential stability of a specific cohort of delayed PDE systems. By adopting Dirichlet boundary conditions and incorporating delay-dependent techniques such as the Galerkin method and Halanay inequality, the inherent stability properties of these systems are rigorously examined. Notably, the utilization of Dirichlet boundary conditions in this study allows for simplified analysis, and it is worth mentioning that the stability analysis outcomes under Neumann conditions and combined boundary conditions align with those of the Dirichlet boundary conditions discussed herein. Furthermore, this research endeavor delves into the implications of the obtained results in terms of control considerations and convergence rates. The integration of the Galerkin method aids in approximating the behavior of dominant modes within the system, thereby enabling a more comprehensive understanding of stability and control. The exploration of convergence rates provides valuable insights into the speed at which stability is achieved in practice, thus enhancing the practical applicability of the findings. The outcomes of this study contribute significantly to the broader comprehension and effective control of delayed PDE systems. The elucidation of stability behaviors not only provides a comprehensive understanding of the impact of delays but also offers practical insights for the design and implementation of control strategies in various domains. Ultimately, this research strives to enhance the stability and reliability of complex systems represented by PDEs, thereby facilitating their effective utilization across numerous scientific and engineering applications.

In this paper, using the notion of ( I q )–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on the entire interval [ a , b ] have been established for four different types of boundary conditions for a singularly perturbed differential equation ε y ″ = f (x , y , y ′) ,   a ≤ x ≤ b . In the second part of the paper, by employing the Peano phenomenon, we analyzed the structure of the solutions of the reduced problem f (x , y , y ′) = 0 . [ABSTRACT FROM AUTHOR]

In this document we will study and solve the nonlinear partial differential equation, with initial conditions for vehicle entry that serves to model the dynamics of traffic flow. To find a numerical solution to the dynamics that govern the behavior of traffic flow, the Finite Element Method in a spatial dimension was used. In accordance with the temporal dynamics, simulations were developed to know the flow in terms of time. The numerical solution is interesting for predicting the number of vehicles at the entrance to a high-flow road. Some theorems are enunciated that guarantee the existence of the solution and the uniqueness is given by the boundary conditions. [ABSTRACT FROM AUTHOR]

Several important problems in Mechanics can be efficiently solved using Raviart–Thomas mixed finite element methods. Whenever the domain of interest has a curved boundary the methods of this family for N-simplexes are the natural choice. But in this case the question arises on the best way to prescribe normal flux conditions across the boundary, if any. It is generally acknowledged that the normal component of the flux variable should preferably not take up corresponding prescribed values at nodes shifted to the boundary of the approximating polytope in the underlying normal direction. This is because an accuracy downgrade is to be expected, as shown in Bertrand and Starke (2016). In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this work an alternative with straight-edged triangles for two-dimensional problems is examined. The key feature of this approach is a Petrov–Galerkin formulation, in which the test-flux space is a little different from the shape-flux space. Based on previous author’s experience with this technique, as applied to Lagrange finite elements, it would lead to an overall accuracy improvement here as well. The experimentation reported hereafter provides examples and counterexamples confirming or not such an expectation, depending on the unknown field of the mixed problem at hand.

This manuscript explores the flow features of the Modified Cross Model in a channel with symmetric trapezoidal cavities in the presence of a circular obstacle. The non-dimensional governing equations and model for different parameters are evaluated via a Galerkin Finite Element Method The system of non-linear algebraic equations is computed by adopting the Newton method. A space involving the quadratic polynomials (P2) has been selected to compute for the velocity profile while the pressure profile is approximated by a linear (P1) finite element space of functions. Simulations are performed for a wide range of physical parameters such as modified parameter (from 0.0 to 0.5), power-law index (from 0.5 to 1.5), relaxation parameter (from 1 to 3), and Reynolds number (from 10 to 40). For the case of a modified parameter (b) and relaxation parameter (λ), it is observed that the drag coefficient (CD) shows an increasing trend while the lift coefficient (CL) is changing sign at lower values of (λ), and then becomes positive at λ=3.

In this paper, using the notion of (Iq)–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on the entire interval [a,b] have been established for four different types of boundary conditions for a singularly perturbed differential equation εy″=f(x,y,y′), a≤x≤b. In the second part of the paper, by employing the Peano phenomenon, we analyzed the structure of the solutions of the reduced problem f(x,y,y′)=0.

This work reviews some basic features on the second (first nontrivial) eigenvalue λ2 to the Neumann problem ... where Ω is a bounded Lipschitz domain of RN, is the outer unit normal, and ∆pu = div(∇) is the p-Laplacian operator. We are mainly concerned with the variational characterization of &#955;2 and place emphasis on the range 1 < p < 2, where the nonlinearity |∆| becomes non smooth. We also address the corresponding result for the p-Laplacian in graphs. [ABSTRACT FROM AUTHOR]

In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions.

Spectral problems are considered which appear in description of small transversal vibrations of Stieltjes strings. It is shown that the eigenvalues of the Neumann-Neumann problem, i.e. the problem with the Neumann conditions at both ends of the string interlace with the union of the spectra of the Neumann-Dirichlet problems, i.e. problems with the Neumann condition at one end and Dirichlet condition at the other end on two parts of the string. It is shown that the spectrum of Neumann-Neumann problem on the whole string, the spectrum of Neumann-Dirichlet problem on the left part of the string, all but one eigenvalues of the Neumann-Dirichlet problem on the right part of the string and total masses of the parts uniquely determine the masses and the intervals between them.

We study the dynamics of a Neumann boundary value problem arising in fluid dynamics. We prove the nonexistence, existence and uniqueness of positive solutions under suitable conditions. At the same time, under stricter conditions, we also obtain the dynamic properties of the Neumann boundary value problem, such as the stability and instability of positive solutions. The methods of proof mainly involve the upper and lower solutions method, eigenvalue theory and some analysis techniques. [ABSTRACT FROM AUTHOR]

In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract We show uniqueness for overdetermined elliptic problems defined on topological disks Ω with C 2 boundary, i.e. , positive solutions u to Δ u + f (u) = 0 in Ω ⊂ (M 2 , g) so that u = 0 and ∂ u ∂ η → = c t e along ∂Ω, η → the unit outward normal along ∂Ω under the assumption of the existence of a candidate family. To do so, we adapt the Gálvez–Mira generalized Hopf-type Theorem [19] to the realm of overdetermined elliptic problem. When (M 2 , g) is the standard sphere S 2 and f is a C 1 function so that f (x) > 0 and f (x) ≥ x f ′ (x) for any x ∈ R + ⁎ , we construct such candidate family considering rotationally symmetric solutions. This proves the Berestycki–Caffarelli–Nirenberg conjecture in S 2 for this choice of f. More precisely, this shows that if u is a positive solution to Δ u + f (u) = 0 on a topological disk Ω ⊂ S 2 with C 2 boundary so that u = 0 and ∂ u ∂ η → = c t e along ∂Ω, then Ω must be a geodesic disk and u is rotationally symmetric. In particular, this gives a positive answer to the Schiffer conjecture D (cf. [33,35]) for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains (cf. [28] , also called Serrin Problem) in S 2. [ABSTRACT FROM AUTHOR]

We investigate the geometry and topology of extremal domains in a Hadamard manifold, i.e., domains that support a positive solution to an overdetermined elliptic problem (OEP). First, we study narrow properties of such domains and characterize the boundary at infinity. We give an upper bound for the Hausdorff dimension of its boundary at infinity and how the domain behaves at infinity. This shows interesting relations with the Singular Yamabe Problem. Later, we focus on extremal domains in the Hyperbolic Space. Symmetry and boundedness properties will be shown. In a certain sense, we extend Levitt–Rosenberg's Theorem to OEPs, which suggests a strong relation with constant mean curvature hypersurfaces. In particular, we are able to prove the Berestycki–Caffarelli–Nirenberg Conjecture under certain assumptions either on the boundary at infinity of the extremal domain or on the OEP itself. Also a height estimate for solutions on extremal domains in a Hyperbolic Space will be given. [ABSTRACT FROM AUTHOR]

A time-optimal control problem for linear infinite order distributed parabolic systems involving constant time lags appear both in the state equation and in the boundary condition is presented. Some particular properties of the optimal control are discussed.

We study a Neumann boundary-value problem on the half line for a second order equation, in which the nonlinearity depends on the (unknown) Dirichlet boundary data of the solution. An existence result is obtained by an adapted version of the method of upper and lower solutions, together with a diagonal argument.

Abstract: In this paper, the optimal boundary control problem for (n × n) infinite order distributed parabolic systems, with boundary conditions involving multiple time-varying lags is considered. Constraints on controls are imposed. Necessary and sufficient optimality conditions for the Neumann problem with the quadratic performance functional are derived. [Copyright &y& Elsevier]

In this paper, by using the theorems of [Lions (1971) and Lions&Magenes (1972) ], the optimal control problem for distributed hyperbolic systems, involving second order operator with an infinite number of variables, in which constant lags appear both in the state equations and in the boundary conditions is considered. The optimality conditions for Neumann boundary conditions are obtained and the set of inequalities that characterize these conditions is formulated. Also, several mathematical examples for derived optimality conditions are presented. Finally, we consider an optimal distributed control problem for (nxn)-infinite variables hyperbolic systems. [ABSTRACT FROM AUTHOR]

Abstract: For a non-local reaction–diffusion problem with either homogeneous Dirichlet or homogeneous Neumann boundary conditions, the questions of blow-up are investigated. Specifically, if the solutions blow up, lower bounds for the time of blow-up are derived. [Copyright &y& Elsevier]

Abstract: We show the existence of positive solution for the following class of singular Neumann problem in with on , where , is a positive parameter, , , , and are radially symmetric nonnegative functions. Using variational methods and sub- and supersolutions, we obtain a solution for an approximated problem involving mixed boundary conditions. The limit of the approximated solutions, is a positive solution. [Copyright &y& Elsevier]

We study a Neumann boundary-value problem on the half line for a second order equation, in which the nonlinearity depends on the (unknown) Dirichlet boundary data of the solution. An existence result is obtained by an adapted version of the method of upper and lower solutions, together with a diagonal argument. [ABSTRACT FROM AUTHOR]

Abstract: This work deals with the existence and symmetry of positive solutions for a Neumann boundary value problem. It is a generalization of the work of Pedro J. Torres. The main result is the uniqueness of positive solutions, which is proved by an analytical method, for a given interval of the positive parameter . [Copyright &y& Elsevier]