Search Results (21)

The growing density of space debris in Low Earth Orbit poses significant risks to Distributed Space Systems (DSSs), where multiple satellites operate in close proximity. Conventional single-satellite collision avoidance maneuvers do not account for internal formation safety and may induce secondary conjunction risks. This work presents a formation-level trajectory optimization framework for short-term conjunction mitigation that jointly addresses external debris avoidance and inter-satellite collision prevention. The proposed Space-Debris Evasion with Internal-Collision-Avoidance (SDEICA) method formulates the problem as a sequential convex programming scheme. A probabilistic debris keep-out region is modeled as an elliptical collision tube derived from the relative position covariance at the Time of Closest Approach (TCA) and convexified via tangent-plane approximation. Internal safety constraints are incorporated through successive linearization of inter-satellite separation conditions. The framework is evaluated on 1197 conjunction scenarios derived from ESA Collision-Avoidance Challenge data for a three-satellite formation. Results demonstrate a systematic reduction in the probability of collision below the operational threshold of ${10}^{-5}$ in all cases, within numerical tolerance, eliminating intersatellite distance violations, maintaining bounded formation deviation, and requiring only moderate control effort. The median computational time is 17.12 s per scenario. These findings indicate that sequential convex optimization provides a practical approach for coordinated, fuel-efficient collision avoidance in satellite formations operating in increasingly congested orbital environments. Full article

This paper deals with a boundary control problem for the Darcy–Brinkman–Jeffreys system describing 3D (or 2D) steady-state flows of an incompressible viscoelastic fluid through a porous medium. Applying the elliptic regularization method and arguments from the topological degree theory, we prove a theorem about the weak solvability of the corresponding boundary value problem under an inhomogeneous Dirichlet boundary condition. Using this theorem, we obtain sufficient conditions for the existence of optimal weak solutions minimizing a given cost function. Moreover, it is shown that the set of all optimal weak solutions is bounded and sequentially weakly closed in an appropriate function space. Full article

This paper presents a topology optimization approach that enables the creation of void structures that reduce part weight while meeting stress constraints for additive manufacturing, using an optimization packing problem. The problem is aimed at maximizing a total area of elliptical voids within an irregular polygonal domain, subject to minimum-distance constraints. Geometric feasibility conditions are expressed analytically using the phi-function technique, ensuring exact enforcement of 3D printing standards. A corresponding nonlinear programming mathematical model is constructed. A stress condition is incorporated in the model using an equivalent mechanical stress computed from the resulting geometry. A solution strategy is proposed that integrates geometric design and solid mechanics within a unified optimization approach. To solve the constrained optimization problem, a local optimization algorithm is developed, based on feasible directions method. Gradients of geometric constraints and the objective function are computed analytically, while the stress gradient is estimated numerically using a finite difference approximation. This permits the simultaneous consideration of geometric and mechanical constraints without requiring an explicit stress function. Numerical experiments demonstrate that the approach produces optimized designing parts with controlled peak stress and achieves competitive performance compared with known topology optimization techniques. Full article

Currently, fastener installation within the narrow, confined space of a wing box must be performed manually, as existing robotic systems are unable to adequately meet the internal assembly requirements. To address this problem, a new robot with one prismatic and five revolute joints (1P5R) has been developed for entering and operating inside the wing box. Firstly, the mechanical structure and control system of the robot were designed and implemented. Then, an improved Probabilistic Roadmap (PRM) method was developed to enable rapid and smooth path planning, mainly depending on optimization of sampling strategy based on Halton sequence, an elliptical-region-based redundant point optimization strategy using control points, improving roadmap construction, and path smoothing based on B-spline curves. Finally, obstacle–avoidance path planning based on the improved PRM was simulated using the MoveIt platform, corresponding robotic motion experiments were conducted, and the improved PRM was validated. Full article

In many scientific and engineering fields, such as hydrogeology, petroleum engineering, geotechnical research, and developing renewable energy solutions, fluid flow modeling in porous media is essential. In these areas, optimizing extraction techniques, forecasting environmental effects, and guaranteeing structural safety all depend on an understanding of the behavior of single-phase flows—fluids passing through connected pore spaces in rocks or soils. Darcy’s law, which results in an elliptic partial differential equation controlling the pressure field, is usually the mathematical basis for such modeling. Analytical solutions to these partial differential equations are seldom accessible due to the complexity and variability in natural porous formations, which makes the employment of numerical techniques necessary. To approximate subsurface flow solutions, traditional methods like the finite difference method, two-point flux approximation, and multi-point flux approximation have been employed extensively. Accuracy, stability, and computing economy are trade-offs for each, though. Deep learning techniques, in particular convolutional neural networks, physics-informed neural networks, and neural operators such as the Fourier neural operator, have become strong substitutes or enhancers of conventional solvers in recent years. These models have the potential to generalize across various permeability configurations and greatly speed up simulations. The purpose of this study is to examine and contrast the mentioned deep learning and numerical approaches to the problem of pressure distribution in single-phase Darcy flow, considering a 2D domain with mixed boundary conditions, localized sources, and sinks, and both homogeneous and heterogeneous permeability fields. The result of this study shows that the two-point flux approximation method is one of the best regarding computational speed and accuracy and the Fourier neural operator has potential to speed up more accurate methods like multi-point flux approximation. Different permeability field types only impacted each methods’ accuracy while computational time remained unchanged. This work aims to illustrate the advantages and disadvantages of each method and support the continuous development of effective solutions for porous medium flow problems by assessing solution accuracy and computing performance over a range of permeability situations. Full article

This study investigates a stochastic production planning problem with regime-switching parameters, inspired by economic cycles impacting production and inventory costs. The model considers types of goods and employs a Markov chain to capture probabilistic regime transitions, coupled with a multidimensional Brownian motion representing stochastic demand dynamics. The production and inventory cost optimization problem is formulated as a quadratic cost functional, with the solution characterized by a regime-dependent system of elliptic partial differential equations (PDEs). Numerical solutions to the PDE system are computed using a monotone iteration algorithm, enabling quantitative analysis. Sensitivity analysis and model risk evaluation illustrate the effects of regime-dependent volatility, holding costs, and discount factors, revealing the conservative bias of regime-switching models when compared to static alternatives. Practical implications include optimizing production strategies under fluctuating economic conditions and exploring future extensions such as correlated Brownian dynamics, non-quadratic cost functions, and geometric inventory frameworks. In contrast to earlier studies that imposed static or overly simplified regime-switching assumptions, our work presents a fully integrated framework—combining optimal control theory, a regime-dependent system of elliptic PDEs, and comprehensive numerical and sensitivity analyses—to more accurately capture the complex stochastic dynamics of production planning and thereby deliver enhanced, actionable insights for modern manufacturing environments. Full article

The aim of this study is to solve the problem of it being difficult to obtain quantitative step signals when testing the time constant of thermocouples using the laser excitation method, thereby restricting the accuracy and repeatability of the test of the time constant of thermocouples. This paper designs a thermocouple time constant testing system in which laser power can be adjusted in real time. The thermocouple to be tested and a colorimetric thermometer with a faster response speed are placed on a pair of conjugate focal points of an elliptic mirror. By taking advantage of the aberration-free imaging characteristic of the conjugate focus, the temperature measured by the colorimetric thermometer is taken as the true value on the surface of the thermocouple so as to adjust the output power of the laser in real time, make the output curve of the thermocouple reach a steady state, and calculate the time constant of the thermocouple. This paper simulates and analyzes the effects of adjusting PID parameters using quantum neural networks. By comparing this with the method of optimizing PID parameters with BP neural networks, the superiority of the designed QNN-PID controller is proven. The designed controller was applied to the test system, and the dynamic response curves of the thermocouple reaching equilibrium at the expected temperatures of 800 °C, 900 °C, 1000 °C, 1050 °C, and 1100 °C were obtained. Through calculation, it was obtained that the time constants of the tested thermocouples were all within 150 ms, proving that this system can be used for the time constant test of rapid thermocouples. This also provides a basis for the selection of thermocouples in other subsequent temperature tests. Meanwhile, repeated experiments were conducted on the thermocouple test system at 1000 °C, once again verifying the feasibility of the test system and the repeatability of the experiment. Full article

In this paper, we study the data completion problem for the Cauchy–Stokes equation in a cylindrical domain, $\mathsf{\Omega }$. Neumann and Dirichlet boundary conditions are prescribed on part of the overdetermined boundary, ${\mathsf{\Gamma }}_0$, and the goal is to complete the data on the other part of the boundary, ${\mathsf{\Gamma }}_a$. Here, ${\mathsf{\Gamma }}_0$ and ${\mathsf{\Gamma }}_a$ represent the side faces of the cylinder $\mathsf{\Omega }$. This problem is known to be ill-posed and is formulated as an optimal control problem with a regularized cost function. To directly approximate the missing data on ${\mathsf{\Gamma }}_a$, we employ the method of factorization of elliptic boundary value problems. This technique allows the factorization of a boundary value problem into a product of parabolic problems. It is successfully applied to the optimality system in this work, yielding new and significant results. Full article

Optimization problems with PDE constraints are widely used in engineering and technical fields. In some practical applications, it is necessary to smooth the control variables and suppress their large fluctuations, especially at the boundary. Therefore, we propose an elliptic PDE-constrained optimization model with a control gradient penalty term. However, introducing this penalty term increases the complexity and difficulty of the problems. To solve the problems numerically, we adopt the strategy of “First discretize, then optimize”. First, the finite element method is employed to discretize the optimization problems. Then, a heterogeneous strategy is introduced to formulate the augmented Lagrangian function for the subproblems. Subsequently, we propose a three-block inexact heterogeneous alternating direction method of multipliers (three-block ihADMM). Theoretically, we provide a global convergence analysis of the three-block ihADMM algorithm and discuss the iteration complexity results. Numerical results are provided to demonstrate the efficiency of the proposed algorithm. Full article

In this work, we analyze a bilinear optimal control problem related to a 2D parabolic–elliptic chemo-repulsion system with a nonlinear chemical signal production term. We prove the existence of global optimal solutions with bilinear control, and applying a generic result on the existence of Lagrange multipliers in Banach spaces, we obtain first-order necessary optimality conditions and derive an optimality system for a local optimal solution. Full article

We study a time-optimal problem in the roto-translation group with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and turn with a given minimum turning radius. Our work generalizes the sub-Riemannian problem by adding a restriction on the velocity vector to lie in a circular sector. The sub-Riemannian problem is given by a special case when the sector is the full disc. The trajectories of the system are applicable in image processing to detect salient lines. We study the local and global controllability of the system and the existence of a solution for given arbitrary boundary conditions. In a general case of the sector opening angle, the system is globally but not small-time locally controllable. We show that when the angle is obtuse, a solution exists for any boundary conditions, and when the angle is reflex, a solution does not exist for some boundary conditions. We apply the Pontryagin maximum principle and derive a Hamiltonian system for extremals. Analyzing a phase portrait of the Hamiltonian system, we introduce the rectified coordinates and obtain an explicit expression for the extremals in Jacobi elliptic functions. We show that abnormal extremals are of circular type, and they correspond to motions of a car along circular arcs of minimal possible radius. The normal extremals in a general case are given by concatenation of segments of sub-Riemannian geodesics in ${SE}_2$ and arcs of circular extremals. We show that, in a general case, the vertical (momentum) part of the extremals is periodic. We partially study the optimality of the extremals and provide estimates for the cut time in terms of the period of the vertical part. Full article

In this paper, a new preconditioning method is proposed for the linear system arising from the elliptic optimal control problem. It is based on row permutations of the linear system and approximations of the corresponding Schur complement inspired by the matching strategy. The eigenvalue bounds of the preconditioned matrices are shown to be independent of mesh size and regularization parameter. Numerical results illustrate the efficiency of the proposed preconditioning methods. Full article

In this paper, elliptic optimal control problems with $L^1$-control cost and box constraints on the control are considered. To numerically solve the optimal control problems, we use the First optimize, then discretize approach. We focus on the inexact alternating direction method of multipliers (iADMM) and employ the standard piecewise linear finite element approach to discretize the subproblems in each iteration. However, in general, solving the subproblems is expensive, especially when the discretization is at a fine level. Motivated by the efficiency of the multigrid method for solving large-scale problems, we combine the multigrid strategy with the iADMM algorithm. Instead of fixing the mesh size before the computation process, we propose the strategy of gradually refining the grid. Moreover, to overcome the difficulty whereby the $L^1$-norm does not have a decoupled form, we apply nodal quadrature formulas to approximately discretize the $L^1$-norm and $L^2$-norm. Based on these strategies, an efficient multilevel heterogeneous ADMM (mhADMM) algorithm is proposed. The total error of the mhADMM consists of two parts: the discretization error resulting from the finite-element discretization and the iteration error resulting from solving the discretized subproblems. Both errors can be regarded as the error of inexactly solving infinite-dimensional subproblems. Thus, the mhADMM can be regarded as the iADMM in function space. Furthermore, theoretical results on the global convergence, as well as the iteration complexity results $o(1/k)$ for the mhADMM, are given. Numerical results show the efficiency of the mhADMM algorithm. Full article

In this research, we investigate an optimal control problem governed by elliptic PDEs with Dirichlet boundary conditions on complex connected domains, which can be utilized to model the cooling process of concrete dam pouring. A new convergence result for two-dimensional Dirichlet boundary control is proven with the Fourier finite volume element method. The Lagrange multiplier approach is employed to find the optimality systems of the Dirichlet boundary optimal control problem. The discrete optimal control problem is then obtained by applying the Fourier finite volume element method based on Galerkin variational formulation for optimality systems, that is, using Fourier expansion in the azimuthal direction and the finite volume element method in the radial direction, respectively. In this way, the original two-dimensional problem is reduced to a sequence of one-dimensional problems, with the Dirichlet boundary acting as an interval endpoint at which a quadratic interpolation scheme can be implemented. The convergence order of state, adjoint state, and Dirichlet boundary control are therefore proved. The effectiveness of the method is demonstrated numerically, and numerical data is provided to support the theoretical analysis. Full article

The transfer between two coplanar Keplerian orbits of a spacecraft with a continuous-thrust propulsion system is a classical problem of astrodynamics, in which a numerical procedure is usually employed to find the transfer trajectory that optimizes (i.e., maximizes or minimizes) a given performance index such as, for example, the delivered payload mass, the propellant mass, the total flight time, or a suitable combination of them. In the last decade, this class of problem has been thoroughly analyzed in the context of heliocentric mission scenarios of a spacecraft equipped with an Electric Solar Wind Sail as primary propulsion system. The aim of this paper is to further extend the existing related literature by analyzing the optimal transfer of an Electric Solar Wind Sail-based spacecraft with a Sun-facing attitude, a particular configuration in which the sail nominal plane is perpendicular to the Sun-spacecraft (i.e., radial) direction, so that the propulsion system is able to produce its maximum propulsive acceleration magnitude. The problem consists in transferring the spacecraft, which initially traces a heliocentric circular orbit, into an elliptic coplanar orbit of given eccentricity with a minimum-time trajectory. Using a classical indirect approach for trajectory optimization, the paper shows that a simplified version of the optimal control problem can be obtained by enforcing the typical transfer constraints. The numerical simulations show that the proposed approach is able to quantify the transfer performance in a parametric and general form, with a simple and efficient algorithm. Full article