Applied Mathematics and Optimization最新文献

In this article we study the behavior of the solutions for the three-phase-lag heat equation with localized dissipation on an Euler–Bernoulli beam model. We show that semigroup S(t) associated with the problem is of Gevrey class 5 for (t>0). If the coefficients satisfy (tau _alpha > k^{*}tau _q), the solutions are always exponentially stable.

This paper is devoted to studying the long and short time behavior of the solutions to a class of non-local in time subdiffusion equations. To this end, we find sharp estimates of the fundamental solutions in Lebesgue spaces using tools of the theory of Volterra equations. Our results include, as particular cases, the so-called time-fractional and the ultraslow reaction-diffusion equations, which have seen much interest during the last years, mostly due to their applications in the modeling of anomalous diffusion processes.

This paper is concerned with the growth-driven shape-programming problem, which involves determining a growth tensor that can produce a deformation on a hyperelastic body reaching a given target shape. We consider the two cases of globally compatible growth, where the growth tensor is a deformation gradient over the undeformed domain, and the incompatible one, which discards such hypothesis. We formulate the problem within the framework of optimal control theory in hyperelasticity. The Hausdorff distance is used to quantify dissimilarities between shapes; the complexity of the actuation is incorporated in the cost functional as well. Boundary conditions and external loads are allowed in the state law, thus extending previous works where the stress-free hypothesis turns out to be essential. A rigorous mathematical analysis is then carried out to prove the well-posedness of the problem. The numerical approximation is performed using gradient-based optimisation algorithms. Our main goal in this part is to show the possibility to apply inverse techniques for the numerical approximation of this problem, which allows us to address more generic situations than those covered by analytical approaches. Several numerical experiments for beam-like and shell-type geometries illustrate the performance of the proposed numerical scheme.

In the present work we investigate an optimal control problem related to the following chemotaxis-consumption model in a bounded domain (Omega subset {mathbb {R}}^3):

with (s ge 1), endowed with isolated boundary conditions and initial conditions for (u, v), being u the cell density, v the chemical concentration and f the control acting in the v-equation through the bilinear term (f ,v, 1_{Omega _c}), in a subdomain (Omega _c subset Omega ). We address the existence of optimal control restricted to a weak solution setting, where, in particular, uniqueness of state (u, v) given a control f is not clear. Then by considering weak solutions satisfying an adequate energy inequality, we prove the existence of optimal control subject to uniformly bounded controls. Finally, we discuss the relation between the considered control problem and two other related ones, where the existence of optimal solution can not be proved.

Considering irreducibility is fundamental for studying the ergodicity of stochastic dynamical systems. In this paper, we establish the irreducibility of stochastic complex Ginzburg-Laudau equations driven by pure jump noise. Our results are dimension free and the conditions placed on the driving noises are very mild. A crucial role is played by criteria developed by the authors of this paper and T. Zhang for the irreducibility of stochastic equations driven by pure jump noise. As an application, we obtain the ergodicity of stochastic complex Ginzburg-Laudau equations. We remark that our ergodicity result covers the weakly dissipative case with pure jump degenerate noise.

In this article, we study the homogenization of optimal control problems subject to second-order semi-linear elliptic PDEs with matrix coefficients in two different types of oscillating domains: a circular domain and a domain with general low-dimensional oscillations. The cost functionals considered are of general energy type with oscillating matrix coefficients, and the coefficient matrix in the cost functional is allowed to differ from the coefficient matrix in the constrained PDE. We prove well-defined limit problems for both domains and obtain explicit forms for the limiting coefficient matrices of the cost functionals and constrained PDEs. As expected, the coefficient matrix of the limit cost functional is a combination of the original cost functional’s and constrained PDE’s coefficient matrices.

In this paper we establish the convergence of a numerical scheme based, on the Finite Element Method, for a time-independent problem modelling the deformation of a linearly elastic elliptic membrane shell subjected to remaining confined in a half space. Instead of approximating the original variational inequalities governing this obstacle problem, we approximate the penalized version of the problem under consideration. A suitable coupling between the penalty parameter and the mesh size will then lead us to establish the convergence of the solution of the discrete penalized problem to the solution of the original variational inequalities. We also establish the convergence of the Brezis–Sibony scheme for the problem under consideration. Thanks to this iterative method, we can approximate the solution of the discrete penalized problem without having to resort to nonlinear optimization tools. Finally, we present numerical simulations validating our new theoretical results.

We identify most probable flows for Kunita Brownian motions, i.e. stochastic flows with Eulerian noise and deterministic drifts. Such stochastic processes appear for example in fluid dynamics and shape analysis modelling coarse scale deterministic dynamics together with fine-grained noise. We treat this infinite dimensional problem by equipping the underlying domain with a Riemannian metric originating from the noise. The resulting most probable flows are compared with the non-perturbed deterministic flow, both analytically and experimentally by integrating the equations with various choice of noise structures.

In this paper, we consider the problem about the simultaneous realization of exact boundary controllability of final state and nodal profile for general 1-D first order quasilinear hyperbolic systems. We show that by means of boundary controls, the system (hyperbolic equations together with boundary conditions) can drive any given initial data at (t=0) to any given final data at (t=T), and the solution to the system fits exactly any given nodal profile on a boundary node or an internal node for certain subinterval ([T_1,T_2]) of [0, T]. Moreover, we give an application of the main results to the system of traffic flow.

In this paper we extend dynamic programming techniques to the study of discrete-time infinite horizon optimal control problems on compact control invariant sets with state-independent best asymptotic average cost. To this end we analyse the interplay of dissipativity and optimal control, and propose novel recursive approaches for the solution of so called shifted Bellman Equations.