Applied Mathematics and Optimization最新文献

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.

In this paper, we investigate the following fractional nonlinear Choquard equation:

where ({{,mathrm{varepsilon },}}>0) is a small parameter, (sin (0, 1)), (Nge 2), ((-Delta )^{s}) denotes the fractional Laplacian, and (I_{alpha }) is the Riesz potential of order (alpha in ((N-4s)_{+}, N)). The potential (Vin C^{0}(mathbb {R}^N, (0, +infty ))) satisfies

for some bounded open set (Omega subset mathbb {R}^N). The function (Fin C^{1}(mathbb {R})) is a nonlinearity of Berestycki–Lions type. By employing suitable variational methods, we establish the existence of at least (textrm{cupl}(K)+1) solutions concentrating around the set (K:={xin Omega : V(x)=m_{0} }) as ({{,mathrm{varepsilon },}}rightarrow 0^{+}.)

This paper considers a semi-discrete forward stochastic parabolic operator with homogeneous Dirichlet conditions in arbitrary dimension. We show the lack of null controllability for a spatial semi-discretization of a null-controllable parabolic system from any initial datum. However, by proving a new Carleman estimate for its semi-discrete backward stochastic adjoint system, we achieve a relaxed observability inequality, which is applied to derivative (phi)-null controllability by duality arguments.

In this paper, we generalize Chae’s Liouville-type rigidity theorems for the Navier–Stokes and Euler equations to the viscous Boussinesq system on (mathbb {R}^n). By testing the momentum equation against gradients of truncated quadratic polynomials and carefully estimating boundary contributions, we prove that if the pressure satisfies either a nonnegativity condition on its spatial integral or a Hardy space assumption ((p in L^{1}(0, T; H_{q}(mathbb {R}^{n}))) for some (q in (0,1])), and if the buoyancy field satisfies the weighted integrability condition ((1+|x|^2)theta in L^1(mathbb {R}^n)) with vanishing vertical first moment, then every weak solution must have identically vanishing velocity. Consequently, the temperature remains frozen at its initial profile and the pressure reduces to a vertical potential, yielding a complete Liouville-type theorem for the Boussinesq system.

We investigate the convergence of numerical solution of Reflected Backward Stochastic Differential Equations (RBSDEs) using the penalization approach in a general non-Markovian framework. We prove the convergence between the continuous penalized solution and the reflected one, in full generality, at order 1/2 as a function of the penalty parameter; the convergence order becomes 1 when the increasing process of the RBSDE has a bounded density, which is a mild condition in practice. The convergence is analyzed in a.s.-sense and (mathbb {L}^p)-sense ((pge 2)). To achieve these new results, we have developed a refined analysis of the behavior of the process close to the barrier. Then we propose an implicit scheme for computing the discrete solution of the penalized equation and we derive that the global convergence order is 3/8 as a function of time discretization under mild regularity assumptions. This convergence rate is verified in the case of American put options and some numerical tests illustrate these results.

This paper is concerned with a kind of partially observed nonzero-sum differential game of mean-field backward doubly stochastic differential equations, in which the coefficient contains not only the state process but also its marginal distribution. Moreover, the cost functional is also of mean-field type. A necessary condition in the form of maximum principle with Pontryagin s type for open-loop Nash equilibrium point of this type of partially observed game, and a verification theorem which is a sufficient condition for Nash equilibrium point are established. The theoretical results are applied to study a partially observed linear-quadratic game.

In this paper, we explore a static setting for the assessment of risk in the context of mathematical finance and actuarial science that takes into account model uncertainty in the distribution of a possibly infinite-dimensional risk factor. We study convex risk functionals that incorporate a safety margin with respect to nonparametric uncertainty by penalizing perturbations from a given baseline model using Wasserstein distance. We investigate to which extent this form of probabilistic imprecision can be approximated by restricting to a parametric family of models. The particular form of the parametrization allows to develop numerical methods based on neural networks, which give both the value of the risk functional and the worst-case perturbation of the reference measure. Moreover, we consider additional constraints on the perturbations, namely, mean and martingale constraints. We show that, in both cases, under suitable conditions on the loss function, it is still possible to estimate the risk functional by passing to a parametric family of perturbed models, which again allows for numerical approximations via neural networks.

In this paper, we study the stability in partial data inverse problems of determining the time-dependent viscosity and potential terms appearing in the Moore–Gibson–Thompson (MGT) equation in dimension (nge 2). The MGT equation, which is third order in time and of hyperbolic type, arises as a linearization of a model for nonlinear ultrasound wave propagation in viscous thermally relaxing fluids. By directly establishing some key Carleman estimates for the MGT equation and its dual, some suitable geometric optics solutions of exponential type are constructed. Then, the stability results in recovering the coefficients from partial observations on the boundary are obtained by means of the suitable geometric optics solutions together with the light ray and Fourier transforms.