This paper proposes a dynamic programming approach targeted to solve a natural resource problem of water storage management for irrigation in an environmentally and socially sustainable way. The problem we examine in our formulation, which deals with the control of water storage in tanks, incorporates assumptions that are milder compared to those previously discussed in the literature concerning similar approaches. Specifically, we consider a time periodic optimal control problem whose performance functional to be maximized merely satisfies a contraction assumption (in the sense of Boyd and Wong) weaker than the one usually considered in the pertinent literature. By using an appropriate fixed point theorem, a time periodic value function can be constructed to enable the definition of optimal feedback control strategies. After showing the insufficiency of Banach contraction frameworks, and proving the underlying auxiliary mathematical results, we show our main result under conditions that are weaker than the previous related ones. Simulation results illustrate the performance of our approach.

This paper discusses the Maxwell system of electrodynamics in the context of distributions. It is used to establish boundary conditions for fields at the interface when the charge and current densities are concentrated measures on the interface. The paper derives the generalized Snell's law from this analysis.

We consider an alternative approach to investigate three-dimensional exterior mixed boundary value problems (BVP) for the steady state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, Ω − ⊂ R 3 , has a compact boundary surface S = ∂ Ω − , which is divided into two disjoint parts, the Dirichlet part S D and the Neumann part S N , where the displacement vector (the Dirichlet type condition) and the stress vector (the Neumann type condition) are prescribed respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov-Poincaré type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right-hand sides of the resulting boundary integral equations system are vector-functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector-functions belonging to the Sobolev space [ W 2 , loc 1 ( Ω − ) ] 3 and satisfying the Sommerfeld-Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the L p -based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses C α -Hölder continuity property in the closed domain Ω − ‾ with α = 1 2 − ε , where ε>0 is an arbitrarily small number.

The Moore-Gibson-Thompson equation with a viscoelastic memory and a forcing is considered. The existence and uniqueness of a local solution is obtained via the Faedo-Galerkin's method. Furthermore, blowing-up solutions with or without a positive initial energy exist due to the nonlinear forcing.

The investigation of linear periodic systems is a prominent subject of research within the domain of linear systems theory. In this context, the state-space realization of such systems is of particular significance. In fact, after successfully treating the realization problem for time-invariant systems, various researchers have directed their attention toward investigating the case of linear periodic systems. Meanwhile, in the late 1980s, Jan C. Willems proposed an approach that broadened the range of systems studied, now referred to as the behavioral approach. This approach views the behavior of a system as its fundamental element, including all signals that adhere to the system laws (also called system trajectories). More recently, the behavioral framework has also been extended separately to periodic and quaternionic behavioral systems. Our work consists of considering linear periodic input/output quaternionic behavioral systems and using recent developments in order to obtain quasi-minimal and uniform state-space realizations.

In this paper, Hopf and zero-Hopf bifurcations are investigated for a class of three-dimensional cubic Kolmogorov systems with one positive equilibrium. Firstly, by computing the singular point quantities and figuring out center conditions, we determined that the highest order of the positive equilibrium is eight as a fine focus, which yields that there exist at most seven small amplitude limit cycles restricted to one center manifold and Hopf cyclicity 8 at the positive equilibrium. Secondly, by using the normal form algorithm, we discuss the existence of stable periodic solution via zero-Hopf bifurcation around the positive equilibrium. At the same time, the relevance between zero-Hopf bifurcation and Hopf bifurcation is analyzed via its special case, which is rarely considered. Finally, some related illustrations are given by means of numerical simulation.

In this paper, we study the uniqueness of nodal radial solutions to nonlinear elliptic equations in the unit ball in R 3 . Under suitable conditions, we prove that, for any given positive integer k, the problem we considered has at most one solution possessing exactly k−1 nodes. Together with the results presented by Nagasaki [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (2): 211–232, 1989] and Tanaka [Proc. Roy. Soc. Edinburgh Sect. A. 138 (6): 1331–1343, 2008], we can prove that more types of nonlinear elliptic equations have the uniqueness of nodal radial solutions.

This manuscript is devoted to a derivative-free parametric iterative step to obtain roots simultaneously for both nonlinear systems and equations. We prove that when it is added to an arbitrary scheme, it doubles the convergence order of the original procedure and defines a new algorithm that obtains several solutions simultaneously. Different numerical experiments are carried out to check the behaviour of the iterative methods by changing the value of the parameter and the initial guesses. Also, it is perform a numerical example where the dynamical planes are carried out to see and compare the basins of attraction.

The construction of (hierarchical) curl-conforming basis functions has been a hot topic in the last decades in the finite element community. Especially, functions applied to simplices have been quite popular after the work by Nédélec in 1980. Many mixed-order and full-order families have been provided in the last years, but sometimes it is difficult to assess if they belong to the original space proposed by Nédélec (especially when orthogonalization procedures are applied). Here, a tool to determine if a family of basis functions belongs to the Nédélec space is provided. Since affine coordinates are the most frequent choice for simplices, particularities about its use with this kind of coordinates are detailed. A detailed survey of existing families is provided, and the practical application of the tool to a representative set of these families is discussed. The tool is also available for the community in a public repository.

In this paper, we investigate the the existence and stability of non-trivial steady state solutions of a class of chemotaxis models with zero-flux boundary conditions and Dirichlet boundary conditions on one-dimensional bounded interval. By using upper-lower solution and the monotone iteration scheme method, we get the existence of the steady-state solution of the chemotaxis model. Moreover, by adopting the “inverse derivative” technique and the weighted energy method to obtain the stability of the steady-state solution of this chemotaxis model.

We present some existence and localization results for periodic solutions of impulsive first-order coupled non-linear systems of two equations, without requiring periodicity for the nonlinearities. The arguments are based on Schauder's Fixed Point Theorem together with the upper and lower solution method, where the upper and lower solutions are not necessarily well-ordered. In addition, results on equi-regulated functions are required for the impulsive analysis. An application to a Wilson-Cowan system of two strongly coupled neurons illustrates one of the main results.

This paper is devoted to analysing a kind of fractional neutral stochastic system (FNSS). Firstly, by introducing the notion of newly defined two-parameter Mittag-Leffler matrix function, we derive the solution of the corresponding linear stochastic system. Subsequently, for the linear case, by virtue of the Grammian matrix, we give a suffcient and necessary condition to guarantee the relatively exact controllability for the addressed case. Furthermore, for the nonlinear one, the relatively exact controllability is obtained by fixed point and explore it via Banach contraction principle. Finally, two examples are provided to intensify our theoretical conclusions.

This paper is concerned with high moment and pathwise error estimates for both velocity and pressure approximations of the Euler-Maruyama scheme for time discretization and its two fully discrete mixed finite element discretizations. Optimal rates of convergence are established for all pth moment errors for p≥2 using a novel bootstrap technique. The almost optimal rates of convergence are then obtained using Kolmogorov’s theorem based on the high moment error estimates. Unlike for the velocity error estimate, the high moment and pathwise error estimates for the pressure approximation are proved in a time-averaged norm. In addition, the impact of noise types on the rates of convergence for both velocity and pressure approximations is also addressed. Finally, numerical experiments are also provided to validate the proven theoretical results and to examine the dependence/growth of the error constants on the moment order p.

In this paper we study the spectral properties of a family of discrete one-dimensional quasi-periodic Schrödinger operators (depending on a phase theta). In large disorder, under some suitable conditions on v and a diophantine rotation number, we prove using basically K.A.M theory that the spectrum of this operator is pure point for all θ∈[0 ,2 π) with exponential decaying eigenfunctions.

In this paper, we use the generalized notions of Riemann-Liouville (fractional calculus with respect to a regular function σ) to extend the definitions of fractional integration and derivative from the functional sense to the distributional sense. First, we give some properties of fractional integral and derivative for the functions infinitely differentiable with compact support. Then, we define the weak derivative, as well as the integral and derivative of a distribution with compact support, the integral and derivative of a distribution using the convolution product. Then, we generalize those concepts from the unidimensional to the multidimensional case. Finally, we propose the definitions of some usual differential operators.

One-hidden-layer feedforward neural networks are described as functions having many real-valued parameters. The larger the number of parameters is, neural networks can approximate various functions (universal approximation property). The essential optimal order of approximation bounds is already derived in 1996. We focused on the numerical experiment that indicates the neural networks whose parameters have stochastic perturbations gain better performance than ordinary neural networks, and explored the approimation property of neural networks with stochastic perturbations. In this paper, we derived the quantitative order of variance of stochastic perturbations to achieve the essential approximation order.

The aim of this paper is to decompose the displacements of thin-walled beams with rectangular cross-section. The decomposition is accompanied by estimates of all its terms with respect to the norm of the strain tensor. Korn's inequality is also given.

Atherosclerosis is a chronic inflammatory disease that poses a serious threat to human health. It starts with the buildup of plaque in the artery wall, which results from the accumulation of pro-inflammatory factors and other substances. In this paper, we propose a mathematical model of early atherosclerosis with a free boundary and time delay. The time delay represents the transformation of macrophages into foam cells. We obtain an explicit solution and analyze the stability of the model and the effect of the time delay on plaque size. We show that for non-radial symmetric perturbations, when $n = 0$ or $1$, the steady-state solution $(M_*,p_*,r_*)$ is linearly stable; when $n \ge 2$, there exists a critical parameter $L_*$ such that the steady-state solution is linearly stable for $L < L_*$ and unstable for $L > L_*$. Moreover, we find that smaller plaque are associated with the presence of time delay.