Applied Mathematics and Optimization最新文献

The global existence of a weak solution of a mixed boundary value problem for the stationary mass transfer equations with variable coefficients is proved. The maximum and minimum principle for the substance concentration is established. The solvability of a multiplicative control problem for the considered model is proved.

This paper is concerned with the decay estimate of solutions to the semilinear wave equation subject to two localized dampings in a bounded domain. The first one is of the nonlinear Kelvin–Voigt type which is distributed around a neighborhood of the boundary and the second is a frictional damping depending in the first one. We show uniform decay rate results of the corresponding energy for all initial data taken in bounded sets of finite energy phase-space. The proof is based on obtaining an observability inequality which combines unique continuation properties and the tools of the Microlocal Analysis Theory

This paper is concerned with a linear quadratic (LQ) nonzero-sum stochastic differential game for regime switching diffusions with mean-field interactions. The salient features of this paper include that the concept of strategies is first adopted in the LQ nonzero-sum game and conditional mean-field terms appear in the state equation and cost functionals. First, a candidate optimal feedback control-strategy pair for the two players is formally constructed based on solutions of four coupled Riccati equations. Then, we verify that the formal optimal pair is indeed a Nash equilibrium for the game by a delicate multi-step completion of squares. The four Riccati equations introduced in this paper are new in the literature. Uniqueness of solutions to the Riccati equations for the general case and existence of solutions for a special case are obtained. Finally, a numerical example is reported to demonstrate the theoretical results.

This paper is concerned with the existence and computation of an equilibrium for a non-stationary average stochastic zero-sum game with Borel spaces, in which the payoff functions and transition probabilities are allowed to change over time. First, we present an extension of the span-fixed point theorem for an operator to a sequence of time-dependent operators. Second, we find a new set of conditions, which is the generalization of the ergodicity ones in the existing literature. Using the extension of the span-fixed point theorem and the novel conditions, we prove the existence of a solution to the average-reward game equations (ARGEs). Third, by the ARGEs we establish the existence of the value and the equilibrium for this game. Moreover,by constructing an approximation sequence of the solution to the ARGEs, we provide a rolling horizon algorithm for computing the value and ( varepsilon )-equilibria, and also prove the convergence of the algorithm. Finally, we illustrate the conditions and results in this paper by several energy management models.

In this paper, we consider a two-layered beam system with an interfacial slip, stabilized only by one viscoelastic vs. frictional damping acting on a small portion of the beam. We show that the local damping is enough to induce the whole dissipation mechanism, and give a general and explicit energy decay rate only under basic conditions on the damping. Meanwhile, we obtain optimal decay rates, when the frictional damping is near linear or polynomial, and the behavior of the memory kernel at infinity is either unquantified or quantified in a quite general way, by means of quantifying the effectiveness of each type of the damping. In order to handle the difficulty caused by the local feature of the damping, we manage to find fitting weighted functions to process region segmentation, as well as to construct appropriate auxiliary functionals. Our results improve and generalize the existing related results for the system to a large extent, and they are novel even for the classical Timoshenko beam system (without slip).

In this paper, we reconstruct a singular time dependent source function of a fractional subdiffusion problem using observational data obtained from a single point of the boundary and inside of the domain. Specifically, the singular function under consideration is represented by the Dirac delta function which makes the analysis interesting as the temporal component of unknown source belongs to a Sobolev space of negative order. We establish the uniqueness of the examined inverse problem in both scenarios. In addition, we analyze local stability of the solution of our inverse problem. To numerically reconstruct a point-wise source, we use the techniques of topological derivatives by converting the inverse source problem in an optimization one. More precisely, we develop a second-order non-iterative reconstruction algorithm to achieve our goal. The efficacy of the proposed approach is substantiated through diverse numerical examples.

In this paper we demonstrate both theoretically as well as numerically that neural networks can detect model-free static arbitrage opportunities whenever the market admits some. Due to the use of neural networks, our method can be applied to financial markets with a high number of traded securities and ensures almost immediate execution of the corresponding trading strategies. To demonstrate its tractability, effectiveness, and robustness we provide examples using real financial data. From a technical point of view, we prove that a single neural network can approximately solve a class of convex semi-infinite programs, which is the key result in order to derive our theoretical results that neural networks can detect model-free static arbitrage strategies whenever the financial market admits such opportunities.

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.