Applied Mathematics and Optimization最新文献

This note addresses the well-posedness of weak solutions for a general linear evolution problem on a separable Hilbert space. For this classical problem there is a well known challenge of obtaining a priori estimates, as a constructed weak solution may not be regular enough to be utilized as a test function. This issue presents an obstacle for obtaining uniqueness and continuous dependence of solutions. Utilizing a generic weak formulation (involving the adjoint of the system’s evolution operator), the classical reference (Ball in Proceedings of the American Mathematical Society 63:370-373, 1977) provides a characterization which makes equivalent well-posedness of weak solutions and generation of a (C_0)-semigroup. On the other hand, the approach in (Ball in Proceedings of the American Mathematical Society 63:370-373, 1977) does not take into account any underlying energy estimate, and requires a characterization of the adjoint operator, the latter often posing a non-trivial task. We propose an alternative approach, when the problem is posed on a Hilbert space and admits an underlying “formal" energy estimate. For such a Cauchy problem, we provide a general notion of weak solution and through a straightforward observation, obtain that arbitrary weak solutions have additional time regularity and obey an a priori estimate. This yields weak well-posedness. Our result rests upon a central hypothesis asserting the existence of a “good" Galerkin basis for the construction of a weak solution. A posteriori, a (C_0)-semigroup may be obtained for weak solutions, and by uniqueness, weak and semigroup solutions are equivalent.

We consider a singular control problem that aims to maximize the expected cumulative rewards, where the instantaneous returns depend on the state of a controlled process. The contributions of this paper are twofold. Firstly, to establish sufficient conditions for determining the optimality of the one-barrier strategy when the uncontrolled process X follows a spectrally negative Lévy process with a Lévy measure defined by a completely monotone density. Secondly, to verify the optimality of the ((2n+1))-barrier strategy when X is a Brownian motion with a drift. Additionally, we provide an algorithm to compute the barrier values in the latter case.

Archetypal analysis is an unsupervised machine learning method that summarizes data using a convex polytope. In its original formulation, for fixed k, the method finds a convex polytope with k vertices, called archetype points, such that the polytope is contained in the convex hull of the data and the mean squared Euclidean distance between the data and the polytope is minimal. In the present work, we consider an alternative formulation of archetypal analysis based on the Wasserstein metric, which we call Wasserstein archetypal analysis (WAA). In one dimension, there exists a unique solution of WAA and, in two dimensions, we prove the existence of a solution, as long as the data distribution is absolutely continuous with respect to the Lebesgue measure. We discuss obstacles to extending our result to higher dimensions and general data distributions. We then introduce an appropriate regularization of the problem, via a Rényi entropy, which allows us to obtain the existence of solutions of the regularized problem for general data distributions, in arbitrary dimensions. We prove a consistency result for the regularized problem, ensuring that if the data are iid samples from a probability measure, then as the number of samples is increased, a subsequence of the archetype points converges to the archetype points for the limiting data distribution, almost surely. Finally, we develop and implement a gradient-based computational approach for the two-dimensional problem, based on the semi-discrete formulation of the Wasserstein metric. Detailed numerical experiments are provided to support our theoretical findings.

Singular fourth-order Abreu equations have been used to approximate minimizers of convex functionals subject to a convexity constraint in dimensions higher than or equal to two. For Abreu type equations, they often exhibit different solvability phenomena in dimension one and dimensions at least two. We prove the analogues of these results for the variational problem and singular Abreu equations in dimension one, and use the approximation scheme to obtain a characterization of limiting minimizers to the one-dimensional variational problem.

It was proved in Iwabuchi and Ogawa (J Elliptic Parabol Equ 7(2):571–587, 2021) that the Cauchy problem for the full compressible Navier–Stokes equations of the ideal gas is ill-posed in (dot{B}_{p, q}^{2 / p}(mathbb {R}^2) times dot{B}_{p, q}^{2 / p-1}(mathbb {R}^2) times dot{B}_{p, q}^{2 / p-2}(mathbb {R}^2) ) with (1le ple infty ) and (1le q<infty ). In this paper, we aim to solve the end-point case left in [17] and prove that the Cauchy problem is ill-posed in (dot{B}_{p, infty }^{d / p}(mathbb {R}^d) times dot{B}_{p, infty }^{d / p-1}(mathbb {R}^d) times dot{B}_{p, infty }^{d / p-2}(mathbb {R}^d)) with (1le ple infty ) by constructing a sequence of initial data for showing discontinuity of the solution map at zero. As a by-product, we demonstrate that the Cauchy problem for the incompressible Navier–Stokes equations is also ill-posed in (dot{B}_{p,infty }^{d/p-1}(mathbb {R}^d)), which is an interesting open problem in itself.

This paper is concerned with a state constrained optimal control problem governed by a Moreau’s sweeping process with a controlled drift. The focus of this work is on the Bellman approach for an infinite horizon problem. In particular, we focus on the regularity of the value function and on the Hamilton–Jacobi–Bellman equation it satisfies. We discuss a uniqueness result and we make a comparison with standard state constrained optimal control problems to highlight a regularizing effect that the sweeping process induces on the value function.

This article is intended to present a qualitative and numerical analysis of well-posedness and boundary stabilization problems of the well-known Korteweg-de Vries-Burgers equation. Assuming that the boundary control is of memory type, the history approach is adopted in order to deal with the memory term. Under sufficient conditions on the physical parameters of the system and the memory kernel of the control, the system is shown to be well-posed by combining the semigroups approach of linear operators and the fixed point theory. Then, energy decay estimates are provided by applying the multiplier method. An application to the Kuramoto-Sivashinsky equation will be also given. Lastly, we present a numerical analysis based on a finite difference method and provide numerical examples illustrating our theoretical results.

We study an optimal control problem for a coupled Schrödinger-elliptic evolution system that describes the propagation of a laser beam in nematic liquid crystals. We consider a bilinear control related to an electric field depending on the optical axis acting on the sample. This problem arises from the study of an optimal way to transform the input signal into a target signal by modifying a system parameter related to the bias electric field. We prove well-posedness, existence and first order necessary conditions for an optimal solution.

This paper deals with the local null controllability of the complete Boussinesq system (where quadratic viscous terms are kept in the right hand side of the heat equation) with distributed controls supported in small sets.