Applied Mathematics and Optimization最新文献

We advance a combined filtered/phase-field approach to topology optimization in the setting of linearized elasticity. Existence of minimizers is proved and rigorous parameter asymptotics are discussed by means of variational convergence techniques. Moreover, we investigate an abstract space discretization in the spirit of conformal finite elements. Eventually, stationarity is equivalently reformulated in terms of a Lagrangian.

The paper is devoted to the study of a new class of optimal control problems governed by discontinuous constrained differential inclusions of the sweeping type involving the duration of the dynamic process into optimization. We develop a novel version of the method of discrete approximations of its own qualitative and numerical values with establishing its well-posedness and strong convergence to optimal solutions of the controlled sweeping process. Using advanced tools of first-order and second-order variational analysis and generalized differentiation allows us to derive new necessary conditions for optimal solutions of the discrete-time problems and then, by passing to the limit in the discretization procedure, for designated local minimizers in the original problem of sweeping optimal control. The obtained results are illustrated by a numerical example.

We consider Prandtl’s 1933 model for calculating circulation distribution function (Gamma ) of a finite wing which minimizes induced drag, under the constraints of prescribed total lift and moment of inertia. We prove existence of a global minimizer of the problem without the restriction of nonnegativity (Gamma ge 0) in an appropriate function space. We also consider an improved model, where the prescribed moment of inertia takes into account the bending moment due to the weight of the wing itself, which leads to a more efficient solution than Prandtl’s 1933 result.

This paper aims to compare and evaluate various obstacle approximation techniques employed in the context of the steady incompressible Navier–Stokes equations. Specifically, we investigate the effectiveness of a standard volume penalization approximation and an approximation method utilizing high viscosity inside the obstacle region, as well as their composition. Analytical results concerning the convergence rate of these approaches are provided, and extensive numerical experiments are conducted to validate their performance.

The primary objective of this paper is to directly establish the observability inequality for stochastic parabolic equations from measurable sets. In an immediate practical application, our focus centers on the investigation of optimal actuator placement to achieve minimum norm controls in the context of approximative controllability for stochastic parabolic equations. We introduce a comprehensive formulation of the optimization problem, encompassing both the determination of the actuator location and the corresponding minimum norm control. More precisely, we reformulate the problem into a two-player zero-sum game scenario, resulting in the derivation of four equivalent formulations. Ultimately, we substantiate the pivotal outcome that the solution to the relaxed optimization problem serves as the optimal actuator placement for the classical problem.

The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush–Kuhn–Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived.

We consider an investor who is dynamically informed about the future evolution of one of the independent Brownian motions driving a stock’s price fluctuations. With linear temporary price impact the resulting optimal investment problem with exponential utility turns out to be not only well posed, but it even allows for a closed-form solution. We describe this solution and the resulting problem value for this stochastic control problem with partial observation by solving its convex-analytic dual problem.

The class of convex–concave bilinear saddle point problems encompasses many important convex optimization models arising in a wide array of applications. The most of existing primal–dual dynamical systems for saddle point problems are based on first order ordinary differential equations (ODEs), which only own the ({mathcal {O}}(1/t)) convergence rate in the convex case, and fast convergence rate analysis always requires some additional assumption such as strong convexity. In this paper, based on second order ODEs, we consider a general inertial primal–dual dynamical system, with damping, scaling and extrapolation coefficients, for a convex–concave bilinear saddle point problem. By the Lyapunov analysis approach, under appropriate assumptions, we investigate the convergence rates of the primal–dual gap and velocities, and the boundedness of the trajectories for the proposed dynamical system. With special parameters, our results can recover the Polyak’s heavy ball acceleration scheme and Nesterov’s acceleration scheme. We also provide numerical examples to support our theoretical claims.

This work is devoted to Markovian-switching systems. In particular, backward stochastic differential equations (BSDEs), forward-backward stochastic differential equations (FBSDEs), such equations with mean-field interactions, and related nonzero-sum stochastic mean-field games. First, BSDEs with Markovian switching, FBSDEs with Markovian-switching, and FBSDEs with both mean-field interactions and regime-switching are examined. Unique solvability of the underlying equations is obtained under monotonicity conditions without assuming non-degeneracy condition for the forward equation. Then the existence of open-loop Nash equilibrium points for nonzero-sum linear-quadratic stochastic differential games with random coefficients is investigated.

This paper is concerned with an evolutionary quasi-variational hemivariational inequality in which both the convex and nonconvex energy functionals depend on the unknown solution. The inequality serves as a direct problem of the inverse problem of parameters identification. Employing a fixed point argument and tools from nonlinear analysis, we establish the solvability and weak compactness of the solution set to the direct problem. Then, general existence and weak compactness results for the regularized optimization inverse problem have been proved. Moreover, we illustrate the applicability of the results by an identification problem for an initial-boundary value problem of parabolic type with mixed multivalued and nonmonotone boundary conditions and a state constraint.