Applied Mathematics and Optimization最新文献

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.

This paper studies the optimum design of beam networks modeled with Timoshenko beams. To account for multiple load cases, an auxiliary optimal control problem is introduced. Optimal distributed control problems for Timoshenko beam networks are solved through the associated optimality system, where the shape functional of the network is defined by the optimal value of the control cost. For control problems exhibiting the turnpike property, the optimum network design is carried out using the steady-state beam model and the corresponding steady-state control problem. A domain decomposition method is adopted to handle topological changes, while the Steklov–Poincaré operator is used to reformulate the beam network model as an interface problem on subdomain boundaries. This approach is applicable under additional assumptions on the network loading. Consequently, the topological derivative of the Steklov–Poincaré operator is incorporated into the optimality system of the control problem, enabling sensitivity analysis with respect to topological changes. The topological derivative of the cost functional with respect to the size of small cycles is derived and computed. Finally, numerical experiments are presented to illustrate and corroborate the analytical results.

We consider an optimal control problem for the Navier–Stokes system with Tresca boundary conditions. With such boundary conditions, the weak formulation of the system is a variational inequality. We approximate this system and the optimal control problem by regularizing the boundary conditions leading to a variational equality. We show that for the approximate system, there exists an optimal control and we derive the first optimality condition by using an adjoint system. We also prove that the approximate optimal controls converge towards an optimal control for the Navier–Stokes system with Tresca boundary conditions. Finally we show that as the threshold of the Tresca law goes to infinity, the corresponding optimal controls converge towards an optimal control for the Navier–Stokes system with the Dirichlet boundary condition.

The goal of this article is to study a control problem for the Benney–Lin equation with multiple objectives, by means of localized interior controls. The primary objective is to steer the solution to a given control-free trajectory, along with a secondary goal of solving a non-cooperative/competitive optimization problem associated with the solution of underlying control system. To study such multi-objective hierarchical control problem, we employ a well-known Stackelberg–Nash strategy. More precisely, assuming the existence of a control (referred to as leader) responsible for driving the solution to a free trajectory, we characterize the other two controls (referred to as followers) which solve the non-cooperative optimization problem under study. The characterization of the followers is influenced by the choice of leader, leading to a coupled optimality system. Consequently, this multi-objective control problem for the Benney–Lin equation simplifies to a single-objective control problem for the optimality system.