Applied Mathematics and Optimization最新文献

In this work, we perform a topological asymptotic analysis of a particular case of thermo-electro-mechanical problem known as the coupled Joule-heating with thermal expansion problem. The objective is to obtain a closed formula of the associated first-order topological derivative. This result is useful in topology optimization since it can be used to obtain optimum designs of thermo-electro-mechanical devices, such as Micro-Electro-Mechanical Systems (MEMS). The topological derivative is obtained by means of a Lagrangian technique for a particular class of cost functionals considering regular and circular perturbations of the material properties distribution. A numerical procedure for validating the analytical expression of the obtained topological derivative is performed. Good concordance between the numerical approximation and the analytical expression has been obtained. Finally, we provide a full mathematical justification for the derived expressions and develop precise estimates for the remainder of the topological asymptotic expansion.

The present work investigates the Neumann initial boundary value problem for the following three-species predator–prey system

under no-flux boundary conditions for (u, v_1, v_2) in a bounded domain (Omega subset mathbb {R}^{N}(Nge 1)) with smooth boundary, where (chi ), (eta ), (beta ), (theta ), (alpha ), r, p are non-negative constants. For any choice of the initial datum, it is proved in this paper that the corresponding problem permits at least one global bounded weak solution provided that one of the following conditions holds:

In this paper, we consider the stability and regularity of a coupled plates transmission system with fractional rotational force and fractional damping. The rotational force and damping involve spectral fractional Laplacian operator, whose powers are in (0, 1] and [0, 2] respectively. We use the frequency domain method and multiplier technique to obtain the stability of the system. Here, we are interested in the stability of the coupled plates system when fractional damping acting on two plate equations simultaneously or only on one plate equation. It is find that the system decays to zero exponentially or polynomially, in which the fractional rotational force and the wave velocities also play important roles. We prove that the decay rates of polynomials obtained are all optimal. The obtained stability results indicate that the presence of higher fractional inertia term has a negative effect on the stability of the system, while the presence of higher fractional damping has a positive effect on the stability of the system. For the order of fractional rotational force is fixed, when fractional damping of the same order is added to equation with fractional inertia term instead of added to equation without fractional inertia term, the system exhibits better stability, which give us the control methods for designing stabilizers of plate coupled systems, and provide a theoretical basis for the design of stabilizers. In addition, we obtain the regularity results when fractional damping acting on two plate equations simultaneously, including the lacks of analytic, the lacks of Gevrey class, analytic, Gevrey class of the corresponding semigroup, and give the orders of Gevrey class. This paper extends the results of previous studies. Transmission systems for coupled plates with fractional damping arise in the fields of physics, mechanics and electronic circuit, etc. So the obtained results have important theoretical and practical significance.

This paper considers the following Keller-Segel-type fully parabolic system

under no-flux boundary conditions in a smoothly bounded domain (Omega subset mathbb {R}^n), (nge 1), where the parameters r, (mu ), (d_v), (d_w) are positive constants and (alpha >1). If the motility function enjoys (phi in C^3((0,infty ))) with (phi (s)>0) for all (s>0), it is shown that the system admits a global classical solution for any appropriately regular initial value when (alpha >max bigl {frac{n+2}{4},1bigr }). Additionally, if we exclude the singular at (s=0), i.e., (phi in C^3([0,infty ))), (phi >0) on ([0,infty )), then the smooth classical solution is globally bounded when any of the following conditions are met: (i) (nle 5), (alpha >1); (ii) (nge 6), (alpha >2); (iii) (nge 6), (alpha =2) and (mu >mu _*), where (mu _*) is a positive constant independent of t, and further, such bounded solution will be stable at the constant (bigl ((frac{r}{mu })^frac{1}{alpha -1}, 0, (frac{r}{mu })^frac{1}{alpha -1}bigr )) with exponential decay rate. Finally, in the case of (nge 6) and (1<alpha le 2) we also showed that the system has at least one global weak solution which will become smooth after some waiting time.

We perform a detailed study of a simple mathematical model addressing the problem of optimally regulating a process subject to periodic external forcing, which is interesting both in view of its direct applications and as a prototype for more general problems. In this model one must determine an optimal time-periodic ‘effort’ profile, and the natural setting for the problem is in a space of periodic non-negative measures. We prove that there exists a unique solution for the problem in the space of measures, and then turn to characterizing this solution. Under some regularity conditions on the problem’s data, we prove that its solution is an absolutely continuous measure, and provide an explicit formula for the measure’s density. On the other hand, when the problem’s data is discontinuous, the solution measure can also include atomic components, representing a concentrated effort made at specific time points. Complementing our analytical results, we carry out numerical computations to obtain solutions of the problem in various instances, which enable us to examine the interesting ways in which the solution’s structure varies as the problem’s data is varied.

An abstract nonautonomous parabolic linear-quadratic regulator problem with very general final cost operator (P_{T}) is considered, subject to the same assumptions under which a classical solution of the associated differential Riccati equation was shown to exist, in two papers appeared in 1999 and 2000, by Terreni and the first named author. We prove an optimal uniqueness result for the integral Riccati equation in a wide and natural class, filling a gap existing in the autonomous case, too. In addition, we give a regularity result for the optimal state.

This paper focuses on the stabilization of a transmission model with variable coefficients. The transmission model is coupled by wave equation and plate equation in different domains through a common boundary, in which the memory damping and the time-varying delay are pasted into the edge of the wave equation. Applying the Riemannian geometry method, convex analysis, compactness–uniqueness argument and a suitable assumption of the time-varying delay, we establish the energy decay rate which is driven by the solution of an ODE under a wider assumption of the memory kernel function and some conditions on the coefficient matrix.

We study a variational model in nonlinear elasticity allowing for cavitation which penalizes both the volume and the perimeter of the cavities. Specifically, we investigate the approximation of the energy (in the sense of (Gamma )-convergence) by means of functionals defined on perforated domains. Perforations are introduced at flaw points where singularities are expected and, hence, the corresponding deformations do not exhibit cavitation. Notably, those points are not prescribed but rather selected by the variational principle. Our analysis is motivated by the numerical simulation of cavitation and extends previous results on models which solely accounted for elastic energy without contributions related to the formation of cavities.

We are concerned with the asymptotic behavior of wave equations with dynamic boundary conditions, subject to internal memory damping. Instead of the assumption that the memory kernel is non-negative and monotonically decreasing in previous articles, here we assume the primitive function of the memory kernel is a generalized positive definite kernel (GPDK), which can be sign-varying. Under some appropriate hypotheses, we establish the stabilization results of the system by utilizing the property of the memory damping and constructing auxiliary system. This is the first work considering wave equations with GPD-type memory kernel and dynamic boundary conditions.

In this paper, for the Hamilton–Jacobi–Bellman equation with an infinite horizon and state constraints, we construct a suitably regular representation. This allows us to reduce the problem of existence and uniqueness of solutions to the Frankowska and Basco theorem from Basco and Frankowska (Nonlinear Differ Equ Appl 26:1–24, 2019). Furthermore, we demonstrate that our representations are stable. The obtained results are illustrated with examples.