Applied Mathematics and Optimization最新文献

Diffeological spaces firstly introduced by J. M. Souriau in the 1980 s are a natural generalization of smooth manifolds but optimization techniques are only known on manifolds so far. Generalizing these techniques to diffeological spaces is very challenging because of several reasons. One of the main reasons is that there are various definitions of tangent spaces which do not coincide. Additionally, one needs to deal with a generalization of a Riemannian space in order to define gradients which are indispensable for optimization methods. One main aim of this paper is a suitable definition of a tangent space in view to optimization methods. Based on this definition, we present a diffeological Riemannian space and a diffeological gradient, which we need for the formulation of an optimization algorithm on diffeological spaces. Moreover, in order to be able to update the iterates in an optimization algorithm on diffeological spaces, we present a diffeological retraction and the Levi-Civita connection on diffeological spaces. This paper also illustrates the novel objects by examples. Finally, we formulate the steepest descent method on diffeological spaces and apply it to an example.

In this paper, we investigate the connection between a class of doubly reflected backward stochastic differential equations, driven by a right continuous with left limits martingale M with two completely separated reflection obstacles, a stochastic Lipschitz driver f, and a generalized Dynkin game, where the game payoff is expressed in terms of a nonlinear expectation (mathcal {E}^{f,M}).

We propose three modified contact boundary conditions incorporating both the velocity and the displacement with a parameter (delta ) for the viscoelastic problem. As (delta ) approaches 0, these conditions formally reduce to the conventional Signorini, Tresca-friction, and Clarke-subdifferential type boundary conditions, respectively. Consequently, the modified conditions, as a generalization of the conventional ones, can be viewed as contact conditions in the displacement with a dynamic setting. We derive weak formulations for the viscoelastic contact model under three modified contact conditions and explore their well-posedness. Additionally, we provide bounds on the weak solutions with respect to the parameter (delta ).

In this paper, we examine a stochastic linear-quadratic control problem characterized by regime switching and Poisson jumps. All the coefficients in the problem are random processes adapted to the filtration generated by Brownian motion and Poisson random measure for each given regime. The model incorporates two distinct types of controls: the first is a conventional control that appears in the continuous diffusion component, while the second is an unconventional control, dependent on the variable z, which influences the jump size in the jump diffusion component. Both controls are constrained within general closed cones. By employing the Meyer-Itô formula in conjunction with a generalized squares completion technique, we rigorously and explicitly derive the optimal value and optimal feedback control. These depend on solutions to certain multi-dimensional fully coupled stochastic Riccati equations, which are essentially backward stochastic differential equations with jumps (BSDEJs). We establish the existence of a unique nonnegative solution to the BSDEJs. One of the major tools used in the proof is the newly established comparison theorems for multi-dimensional BSDEJs.

We consider the mixed problem for the “Kirchhoff plate equation” on an open bounded domain (Omega ) in (mathbb {R}^n, n = 1, 2, 3,, ldots ) with sufficiently smooth boundary (Gamma = partial (Omega )). Under both Dirichlet and Neumann homogeneous Boundary Conditions, the dynamical system defines a strongly continuous group of unitary operators on an appropriate function space. We then introduce a suitably devised Neumann boundary control in feedback form, as to force the new dynamic (to be well-posed and) to asymptotically decay in an optimal function space (the same space of optimal regularity and exact controllability under open-loop control, (L^2)-in time and space.) We obtain the following results: (1) uniform stabilization for (n=1); (2) polynomial/rational stability for (n = 2,3,4, ldots ); (3) and, independently, strong stabilization for any dimension n. In the present paper, we employ a frequency domain approach, based on technical PDE-estimates.

Motivated by stability issues for suspension bridges, the analysis focuses on the maximization of the torsional eigenvalues of a nonhomogeneous multi-span fish-bone plate with respect to the mass density. The incorporation of internal piers significantly impacts the spectral properties of the system. After a general spectral theorem, a characterization of the densities maximizing the first and the second torsional eigenvalue is provided, starting from the corresponding results for the nonhomogeneous Dirichlet problem. In the case where the mass of the central span is equal to its length, more explicit insight is then given, taking into account the role of the position of the piers and discussing the scenario for higher-order eigenvalues, as well.

In this article, we investigate the global existence of martingale suitable weak solutions to stochastic Ericksen–Leslie equations with additive noise in a 3D torus. The notion of suitable weak solutions has been introduced to address possible emergence of finite-time singularities, which remains a notably challenging question in the field of fluid dynamics. Weak solutions offer an approach to account for these potential singularities. A restricted class of weak solutions that exhibit a higher level of regularity which are therefore more likely to be physically meaningful, is naturally called for. Consequently, suitable weak solutions, i.e., weak solutions that satisfy a local energy inequality, become a focus of research, including investigations about how regular these solutions can be. In this article, we prove that, despite the presence of white noise, the paths of martingale suitable weak solutions of 3D stochastic Ericksen–Leslie equations exhibit singular points of one-dimensional parabolic Hausdorff measure zero. To establish this result, we have utilized two techniques, which can potentially be generalized to handle other stochastically forced complex fluid dynamics equations with a similar structure. Firstly, a local energy-preserving approximation is constructed which markedly facilitates the proof of the global existence of martingale suitable weak solutions; secondly, to demonstrate partial regularity of these solutions, a blow-up argument is formulated, which efficiently yields the desired key estimate.

We introduce a general coupled system of parabolic equations with quadratic nonlinear terms and diffusion terms defined by fractional powers of the Laplacian operator. We develop a method to establish the rigorous convergence of the fractional diffusion case to the classical diffusion case in the strong topology of Sobolev spaces, with explicit convergence rates that reveal some unexpected phenomena. These results apply to several relevant real-world models included in the general system, such as the Navier-Stokes equations, the Magneto-hydrodynamics equations, the Boussinesq system, and the Keller-Segel system. For these specific models, this fractional approach is further motivated by previous numerical and experimental studies.

In this paper, we propose and solve an optimal reinsurance and protection problem for the classical Cramér–Lundberg model in the continuous-time setting. Suppose that the insurer can purchase per-loss reinsurance and invest in the prevention fund to reduce the claim risk, and the reinsurance premium is determined via the mean-standard-deviation premium principle. Under the time-inconsistent mean-variance criterion, the equilibrium reinsurance and protection strategies, as well as the value function, are derived explicitly by solving the extended Hamilton–Jacobi–Bellman system in a game framework. With the help of Lagrange duality, we transform the original optimization problem into an auxiliary optimization problem with constraint and build the relationship between them. Moreover, the necessary condition for protection to be effective has been identified. Finally, we illustrate the influence of model parameters on the optimal results for both the light-tailed and heavy-tailed risks, and reveal the significance of the reinsurance and protection businesses.

We are interested in the problem of identifying the viscosity of a fluid based on observations. This analysis is twofold. First, a stability property of the inverse problem is proved. Secondly, we analyse the discretization of the optimization problem using reduced Hsieh-Clough-Tocher elements, and a convergence with order 3/2 of the identified viscosity with respect to the mesh size is determined. We conclude the paper with some numerical examples showing that this 3/2 order might be enhanced with correct assumptions.