Applied Mathematics and Optimization最新文献

In this paper, we consider an optimal control problem of an ordinary differential inclusion governed by the hypergraph Laplacian, which is defined as a subdifferential of a convex function and then is a set-valued operator. We can assure the existence of optimal control for a suitable cost function by using methods of a priori estimates established in the previous studies. However, due to the multivaluedness of the hypergraph Laplacian, it seems to be difficult to derive the necessary optimality condition for this problem. To cope with this difficulty, we introduce an approximation operator based on the approximation method of the hypergraph, so-called “clique expansion.” We first consider the optimality condition of the approximation problem with the clique expansion of the hypergraph Laplacian and next discuss the convergence to the original problem. In appendix, we state some basic properties of the clique expansion of the hypergraph Laplacian for future works.

The paper is concerned with the exact controllability of the problems described by an evolution of hemivariational inequalities within the framework of reflexive state spaces and uniformly convex control spaces, where the controls are drawn from the space (L^p(I, U)), (~1<p<infty ). We first introduce an appropriate definition for solutions to the hemivariational inequality problem, as this has not been previously established in the literature. Using this solution framework, we demonstrate that the solutions of the associated differential inclusion problem involving the Clarke subdifferential operator also serve as solutions to the original problem. Consequently, we establish the exact controllability of the original problem through the exact controllability of the corresponding differential inclusion problem. This work presents a novel approach by assuming that the control space U is a uniformly convex Banach space, which helps resolve challenges related to convexity in constructing the necessary control–a difficulty that does not arise when U is a separable Hilbert space.

We will consider the use of mixed finite element method for the semi-discretization of the Schrödinger integro-differential systems. We investigate the problem of boundary observability for the semi-discrete models. We show that the semi-discretization based on a mixed finite element method with two different basis functions for the position and its differentiation with respect to time of Schrödinger type integro-differential equation, are uniformly observability as the discretization parameter h goes to zero. Some numerical results are given to illustrate our theoretical finds.

In this study, we consider a relaxed Dirichlet problem in linear elasticity, which is a generalized Dirichlet problem for homogeneous linear elastic materials involving a potential in the form of a symmetric and positive semi-definite matrix of Borel measures that do not charge polar sets. We present an explicit approximation procedure by means of sequences of classical Dirichlet problems in strongly perturbed domains. Then, we give a characterization of these measures, when the components of the data are nonnegative, in terms of solutions in closed convex sets of particular relaxed Dirichlet problems. Finally, we give some applications to the shape optimization for Dirichlet problems in the linear elasticity framework.

This paper considers numerical approximations of a Timoshenko beam under boundary control. The continuous system under boundary feedback is known to be exponentially stable. Firstly, the continuous system is transformed into an equivalent first-order port-Hamiltonian formulation. A basically order reduction finite difference scheme is applied to derive a family of semi-discretized systems. Secondly, a completely new method which is based on a mixed discrete observability inequality involving final state observability and exact observability is developed to prove the uniform exponential stability of the discrete systems. More interestingly, the proof for the stability of discrete systems is almost parallel to that of the continuous counterpart. Thirdly, the solutions of the semi-discretized systems are shown to be strongly convergent to the solution of the original system through Trotter-Kato theorem. Finally, both exact controllability of continuous system and the discrete systems are proved in light of Russell’s “controllability via stability” principle and the explicit controls are derived. Moreover, the discrete controls are shown in first time to be convergent to the continuous control by proposed approach.

We develop a fitted value iteration (FVI) method to compute bicausal optimal transport (OT) where couplings have an adapted structure. Based on the dynamic programming formulation, FVI adopts a function class to approximate the value functions in bicausal OT. Under the concentrability condition and approximate completeness assumption, we prove the sample complexity using (local) Rademacher complexity. Furthermore, we demonstrate that multilayer neural networks with appropriate structures satisfy the crucial assumptions required in sample complexity proofs. Numerical experiments reveal that FVI outperforms linear programming and adapted Sinkhorn methods in scalability as the time horizon increases, while still maintaining acceptable accuracy.

In this paper, we study the relationship between maximum principle (MP) and dynamic programming principle (DPP) for forward–backward control system under consistent convex expectation dominated by G -expectation. Under the smooth assumptions for the value function, we get the relationship between MP and DPP under a reference probability by establishing a useful estimate. If the value function is not smooth, then we obtain the first-order sub-jet and super-jet of the value function at any t. However, the processing method in this case is much more difficult than that when t equals 0.

This work focuses on numerically solving a shape identification problem related to advection–diffusion processes with space-dependent coefficients using shape optimization techniques. Two boundary-type cost functionals are considered, and their corresponding variations with respect to shapes are derived using the adjoint method, employing the chain rule approach. This involves firstly utilizing the material derivative of the state system and secondly using its shape derivative. Subsequently, an alternating direction method of multipliers (ADMM) combined with the Sobolev-gradient-descent algorithm is applied to stably solve the shape reconstruction problem. Numerical experiments in two and three dimensions are conducted to demonstrate the feasibility of the methods.

This paper investigates the existence and concentration behavior of nodal solutions for the planar Schrödinger-Poisson system with subcritical exponential growth

where (varepsilon, mu >0) are parameters, V and f are continuous functions. Under suitable assumptions, the existence of nodal solutions is established, using variational methods. Furthermore, we prove that the nodal solutions of (({mathcal {S}})) concentrate around the minimum point of V and exhibit exponential decay at infinity.

In this work, the multi-parameter robustness of pullback random attractors for Lamé systems is investigated. More precisely, this robustness reveals how smooth is the transition from the non-autonomous stochastic setting to the autonomous deterministic one. This is achieved by establishing the upper semicontinuity for a three-parameter family of pullback random attractors.