Lagrange Multiplier Research Articles

Existence and asymptotic behavior of normalized solutions for fourth‐order equations of Kirchhoff type

In this paper, we consider the following fourth‐order equation of Kirchhoff type under the normalized constraint , where are constants, and is a Lagrange multiplier. Firstly, we obtain the existence of ground state solutions for or with . Moreover, we show the nonexistence of solutions for . Finally, we investigate the asymptotic behavior of normalized solutions obtained above with respect to the parameter .

An approach for regime-switching stochastic control problems with memory and terminal conditions

In this research article, we focus on a stochastic optimal control problem with two types of terminal constraints. These specific conditions provide real-valued and stochastic Lagrange multipliers. Our model evolves according to a Markov regime-switching jump diffusion model with memory. In this context, the memory is represented by a Stochastic Differential Delay Equation. We present two theorems for each constraint within the general formulation of stochastic optimal control theory in a Lagrangian environment. We approach to this task from a theoretical perspective and provide mild technical assumptions, which make our theorems applicable for a broad class of stochastic control problems as well as for a wide range of disciplines such as engineering, biology, operations research, medicine, computer science and economics. In this work, we apply Stochastic Maximum Principle to demonstrate an optimal dividend policy corresponding to a time-delayed wealth process of a company. Moreover, we determine the real-valued Lagrange multiplier of this control problem explicitly.

Research and application of migration and plugging mechanism of temporary plugging balls in multicluster fracturing of horizontal wells

Plugging and diverting stimulation technology using temporary plugging balls (PDSTB) in horizontal wells is one of the effective measures to enhance production in heterogeneous carbonate reservoirs. However, the migration and plugging mechanism of temporary plugging balls in horizontal wells remain unclear, leading to suboptimal field application. Based on the Lagrangian method, the migration and plugging models of temporary plugging balls were established. The study investigated the migration and plugging mechanism of temporary plugging balls. It analyzed the effects of parameters such as displacement on the plugging efficiency. The results show that the flow field in the perforation section is significantly affected by the displacement. Temporary plugging balls show three states of plugging: one ball plugging one shot-hole, multiple balls plugging one shot-hole, and balls remaining in the wellbore. The plugging efficiency initially increases, and then, decreases with the increase in displacement and the ball diameter. The plugging efficiency initially increases, and then, stabilizes with the increase in the viscosity of the carrying fluid and the usage ratio of balls. Plugging efficiency decreases with increasing density of balls. The greater the pressure difference at the outlet of the perforation section, the higher the plugging efficiency in the advantageous inlet section. Based on these results, the field construction of Well M was guided. The gas production was 142.66 × 104 m3/d after stimulation, and the production increase was significant. This study provides a theoretical basis for optimizing and designing the parameters of PDSTB in carbonate long horizontal wells.

Why the action?

In this note, I present two variations of the principle of stationary action in order to provide alternative ways for students to think about the action that they may find more intuitive than the traditional approach. These methods involve Lagrange multiplier techniques and are, therefore, best suited for students who are encountering the stationary action principle for a second or third time, perhaps in an advanced undergraduate mathematical methods course or a graduate classical mechanics course.

Numerical study of the effects of unmatched pressure on the supersonic particle-laden mixing layer

The dispersion of monodisperse, inertial particles in a supersonic mixing layer consisting of two sheared flows with differing pressures (P1 for the particle-laden jet flow and P2 for the airflow) is numerically investigated using large Eddy simulation and Euler–Lagrange methods. The calculations reveal the following insights: The pressure disparity between the two flows induces a transverse gas flow effect, which swiftly deflects the mixing layer from the high-pressure side to the low-pressure side. The growth rate of mixing layer increases with the ratio of P2/P1 and while the deflected displacement correlates with the pressure difference |P2-P1|. However, the particles exhibit delayed tracking characteristics to the deflected mixing layer because of their relative relaxation to the transverse gas velocity, particularly in the upstream region of the mixing layer (also known as the Kelvin–Helmholtz instability developing zone or KH zone). Notably, when the P2 exceeds that of the P1, particles can more easily penetrate into the vortices of KH zone, significantly enhancing the downstream gas–particle mixing. This mixing enhancement is particularly pronounced for larger particles due to their increased inertia, which allows them to advance into the vortices of KH zone more effectively than smaller ones.

A Unified Collision Avoidance Trajectory Planning with Dual Variables for Collaborative Aerial Transportation Systems

As an essential application of unmanned aerial vehicle (UAV) systems, payload transportation has garnered significant attention in recent years. Collaborative payload transportation utilizing multiple UAVs can effectively increase the payload capacity of the transportation system. Nevertheless, the incorporation of multiple UAVs makes the dynamic model of the transportation system more complex due to the coupled UAV and payload states. In the immediate disaster relief response, the collaborative system is often required to promptly deliver supplies to the target site while avoiding obstacles to ensure the system’s safety. Consequently, devising fast delivery trajectories that avoid collisions for such complicated systems poses a considerable challenge. To this end, a novel trajectory planning method is presented for collaborative transportation systems. Specifically, the dynamic model of the collaborative transportation system is derived by utilizing the Euler–Lagrange method. Then, the trajectory planning problem is formulated as an optimization problem with considerations of dynamics, actuation, safety, and formation constraints. To expedite the optimization process, the collision avoidance safety constraint is constructed using dual variables. The efficacy of this trajectory planning approach is confirmed through multiple real-world flight experiments involving collaborative aerial transportation systems of two and three UAVs.

Shaped pattern synthesis for hybrid analog–digital arrays via manifold optimization-enabled block coordinate descent

Hybrid analog–digital (HAD) architecture is a promising means to realize large-scale arrays owing to the judicious trade-off between system performance and hardware complexity. This paper investigates the beampattern shaping of HAD arrays through minimizing the beampattern matching error between desired and actual patterns. Due to the nonconvex constraints on analog and digital weights and the nonconvex nonsmooth objective function, the resultant codesign is NP-hard. To address this issue, we create an efficient algorithm by leveraging block coordinate descent (BCD) and Riemannian manifold optimization. We first equivalently represent the objective function as a smooth bi-quadratic function by introducing a uni-modulus auxiliary variable. Subsequently, we alternatingly optimize the scaling factor, digital weights, analog weights and auxiliary variable under the BCD framework, where the simultaneous update of analog weights and auxiliary variable is recast as uni-modular constrained quadratic programming which is efficiently solved by Riemannian Newton method, and the digital weights have a global optimal solution via Lagrangian multiplier method. We also derive an explicit convergence condition of this algorithm. Numerical results demonstrate that the proposed algorithm has superior performance and faster convergence speed than alternative algorithms, and produces nearly the same mainlobe gain as fully digital arrays with significantly fewer radio-frequency chains.

Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations

In this paper, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we form a new system which incorporates the energy evolution process but is still equivalent to the original equations. Such nonlinear system is then discretized in time based on the backward differentiation formulas, resulting in a dynamically regularized Lagrange multiplier (DRLM) method. First- and second-order DRLM schemes are derived and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting accuracy and stability of the numerical solutions. Fully discrete energy stability is also proved with the Marker-and-Cell (MAC) discretization in space. Various numerical experiments in two and three dimensions verify the convergence and energy dissipation as well as demonstrate the accuracy and robustness of the proposed DRLM schemes.

Mathematical and numerical analysis of reduced order interface conditions and augmented finite elements for mixed dimensional problems

In this paper, we are interested in the mathematical properties of methods based on a fictitious domain approach combined with reduced-order interface coupling conditions, which have been recently introduced to simulate 3D-1D fluid-structure or structure-structure coupled problems. To give insights on the approximation properties of these methods, we investigate them in a simplified setting by considering the Poisson problem in a two-dimensional domain with non-homogeneous Dirichlet boundary conditions on small inclusions. The approximated reduced problem is obtained using a fictitious domain approach combined with a projection on a Fourier finite-dimensional space of the Lagrange multiplier associated to the Dirichlet boundary constraints, obtaining in this way a Poisson problem with defective interface conditions. After analyzing the existence of a solution of the reduced problem, we prove its convergence towards the original full problem, when the size of the holes tends towards zero, with a rate which depends on the number of modes of the finite-dimensional space. In particular, our estimates highlight the fact that to obtain a good convergence on the Lagrange multiplier, one needs to consider more modes than the first Fourier mode (constant mode). This is a key issue when one wants to deal with real coupled problems, such as fluid-structure problems for instance. Next, the numerical discretization of the reduced problem using the finite element method is analyzed in the case where the computational mesh does not fit the small inclusion interface. As is standard for these types of problem, the convergence of the solution is not optimal due to the lack of regularity of the solution. Moreover, convergence exhibits a well-known locking effect when the mesh size and the inclusion size are of the same order of magnitude. This locking effect is more apparent when increasing the number of modes and affects the Lagrange multiplier convergence rate more heavily. To resolve these issues, we propose and analyze a stabilized method and an enriched method for which additional basis functions are added without changing the approximation space of the Lagrange multiplier. Finally, the properties of numerical strategies are illustrated by numerical experiments.

Active Disturbance Rejection Control for Flux Weakening in Interior Permanent Magnet Synchronous Motor Based on Full Speed Range

To address the impact of load disturbances on the full-speed-range control of an interior permanent magnet synchronous motor (PMSM), an active disturbance rejection control (ADRC) method is proposed. The speed loop employs phased field-weakening control (FW) based on ADRC, while the current loop utilizes proportional-integral-derivative (PID) control. Starting from the motor parameters, the Lagrange multiplier method was used to derive the critical speeds for the maximum torque per ampere (MTPA) and maximum torque per voltage (MTPV) ratios, and the timing for the field-weakening control was analyzed. A full-speed-range control model of the motor was established, and an ADRC-based speed loop controller was designed to achieve smooth transitions between high speeds and anti-disturbance solid capabilities. Based on the proposed control strategy, a 21 kW PMSM was used as the research object, and a full-speed-range control simulation model was developed in MATLAB/SIMULINK to verify the strategy. Compared to the traditional PID control, the simulation results demonstrate that the proposed strategy effectively observes and compensates for load disturbances, significantly reducing initial torque oscillations under three different operating conditions. After a sudden load increase, torque oscillations were reduced by 16%, with the stator current reaching steady state 0.03 s faster, response speed improving by 0.02 s, smooth transitions between speed ranges, and enhanced anti-disturbance performance.

An Introduction to Lagrange Multipliers: Theory and Applications in Economics

The purpose of this paper is to explore the basic applications of the Lagrange multiplier method in economics and to help beginners build their understanding of this mathematical tool. Tesla Inc. is used as a case study to examine how it maximizes profits by optimizing pricing and marketing expenses under given market conditions and constraints. The context of the study includes the importance of maintaining Tesla’s leadership position as the world’s leading electric car manufacturer in a competitive market. In this paper, the Lagrange multiplier method is used to incorporate specific market share and marketing expense constraints into the profit function and to find the extreme values by constructing the Lagrange function and solving for the derivatives. However, the results of the study show that the solutions obtained are extreme minima rather than the expected maxima. This result highlights the possible limitations of the Lagrange multiplier method in economic optimization problems. Nonetheless, this paper provides valuable lessons for understanding and applying the Lagrange multiplier method and points the way for further research on the application of optimization methods in economic decision making.

Elliptic Hermite-Gaussian soliton and transformations in nonlocal media induced by linear anisotropy.

Elliptic Hermite-Gaussian (HG) soliton clusters in nonlocal media with anisotropic diffractions are studied comprehensively. The relations among solitons parameters, diffraction indices, and the degree of nonlocality are derived analytically with the Lagrangian method. Stable elliptic HG soliton clusters can be obtained when linear diffraction is anisotropic. When the solitons are launched with an initial orientation angle, we also demonstrate numerically mode transformations between HG and Laguerre-Gaussian (LG) solitons induced by linear anisotropy. Our results will enrich the soliton phenomenon with linear anisotropic diffraction and may lead to novel applications in all-optical switching, interconnection, etc.

Stochastic optimal control of pre-exposure prophylaxis for HIV infection for a jump model.

We analyze a stochastic optimal control problem for the PReP vaccine in a model for the spread of HIV. To do so, we use a stochastic model for HIV/AIDS with PReP, where we include jumps in the model. This generalizes previous works in the field. First, we prove that there exists a positive, unique, global solution to the system of stochastic differential equations which makes up the model. Further, we introduce a stochastic control problem for dynamically choosing an optimal percentage of the population to receive PReP. By using the stochastic maximum principle, we derive an explicit expression for the stochastic optimal control. Furthermore, via a generalized Lagrange multiplier method in combination with the stochastic maximum principle, we study two types of budget constraints. We illustrate the results by numerical examples, both in the fixed control case and in the stochastic control case.

A hybrid finite element method for moving-habitat models in two spatial dimensions

Moving-habitat models track the density of a population whose suitable habitat shifts as a consequence of climate change. Whereas most previous studies in this area consider 1-dimensional space, we derive and study a spatially 2-dimensional moving-habitat model via reaction-diffusion equations. The population inhabits the whole space. The suitable habitat is a bounded region where population growth is positive; the unbounded complement of its closure is unsuitable with negative growth. The interface between the two habitat types moves, depicting the movement of the suitable habitat poleward. Detailed modelling of individual movement behaviour induces a nonstandard discontinuity in the density across the interface. For the corresponding semi-discretised system we prove well-posedness for a constant shifting velocity before constructing an implicit-explicit hybrid finite element method. In this method, a Lagrange multiplier weakly imposes the jump discontinuity across the interface. For a stationary interface, we derive optimal a priori error estimates over a conformal mesh with nonconformal discretisation. We demonstrate with numerical convergence tests that these results hold for the moving interface. Finally, we demonstrate the strength of our hybrid finite element method with two biologically motivated cases, one for a domain with a curved boundary and the other for non-constant shifting velocity.

Enhanced PINNs with augmented Lagrangian method and transfer learning for hydrodynamic lubrication analysis

Purpose By seamlessly integrating physical laws, physics-informed neural networks (PINNs) have flexibly solved a wide variety of partial differential equations (PDEs). However, encoding PDEs and constraints as soft penalties in the loss function can cause gradient imbalances, leading to training and accuracy issues. This study aims to introduce the augmented Lagrangian method (ALM) and transfer learning to address these challenges and enhance the effectiveness of PINNs for hydrodynamic lubrication analysis. Design/methodology/approach The loss function was reformatted by ALM, adaptively adjusting the loss weights during training. Transfer learning was used to accelerate the convergence of PINNs under similar conditions. Additionally, the iterative process for load balancing was reframed as an inverse problem by extending film thickness as a trainable variable. Findings ALM-PINNs significantly reduced the maximum absolute boundary error by almost 80%. Transfer learning accelerated PINNs for solving the Reynolds equation, reducing training epochs by an order of magnitude. The iterative process for load balancing was effectively eliminated by extending the thickness as a trainable parameter, achieving a maximum percentage error of 2.31%. These outcomes demonstrated strong agreement with FDM results, analytical solutions and experimental data. Originality/value This study proposes a PINN-based approach for hydrodynamic lubrication analysis that significantly improves boundary accuracy and the training process. Additionally, it effectively replaces the load balancing procedure. This methodology demonstrates considerable potential for broader applications across various boundary value problems and iterative processes. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-07-2024-0277/

A Three-Block Inexact Heterogeneous Alternating Direction Method of Multipliers for Elliptic PDE-Constrained Optimization Problems with a Control Gradient Penalty Term

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.

Mathematical Model of a Nonlinear Electromagnetic Circuit Based on the Modified Hamilton–Ostrogradsky Principle

This paper presents a mathematical model of a typical lumped-parameter electromagnetic assembly, which consists of two subassemblies: one includes a magnetic circuit and the other with selected elements of electric circuits. An interdisciplinary research approach is used, which assumes the use of a modified integral method based on the variational Hamilton–Ostrogradsky principle. The modification of the method is the extension of the Lagrange function by two components. The first one reflects the dissipation of electromagnetic energy in the system, while the second one reflects the effect of external non-potential forces acting on the electromagnetic system. This approach allows for the avoidance of the inconvenience of the classical theory, which assumes the decomposition of the entire integrated system into individual electrical subsystems. The state equations of the electromagnetic subassembly are presented solely on the basis of the energy approach, which in turn allows taking into account various latent motions in the system, because the equations are derived based on non-stationary constraints between subsystems. The adopted theory allows for the formulation of the model of the system in a vector form, which gives much more possibilities for the analysis of higher-order electromagnetic circuits. Another important advantage is that the state equations of the considered electrical object are given in Cauchy normal form. In this way, the equations can be integrated both explicitly and implicitly. The results of computer simulations are presented in graphical form, analysed, and discussed.

Theoretical and Numerical Modeling of Optical Switching of Epitaxial Nanostructures Based on Iron-Garnet Films

The paper presents a theoretical analysis of magnetization switching in a gadolinium ferrite garnet film due to the demagnetizing effect of a femtosecond laser pulse. Using the Lagrange formalism for a two-sublattice ferrimagnet, the effective Lagrangian, thermodynamic potential, and Rayleigh dissipative function are obtained. The phase diagram of the ferrite film is analyzed, and the main states of the system are identified. Magnetization switching diagrams and trajectories of the order parameter dynamics of the magnet are constructed. The ranges of magnetic fields, temperatures, and demagnetization values for the most efficient magnetization switching are analyzed.

Linear Programming-Based Sparse Kernel Regression with L1-Norm Minimization for Nonlinear System Modeling

This paper integrates L1-norm structural risk minimization with L1-norm approximation error to develop a new optimization framework for solving the parameters of sparse kernel regression models, addressing the challenges posed by complex model structures, over-fitting, and limited modeling accuracy in traditional nonlinear system modeling. The first L1-norm regulates the complexity of the model structure to maintain its sparsity, while another L1-norm is essential for ensuring modeling accuracy. In the optimization of support vector regression (SVR), the L2-norm structural risk is converted to an L1-norm framework through the condition of non-negative Lagrange multipliers. Furthermore, L1-norm optimization for modeling accuracy is attained by minimizing the maximum approximation error. The integrated L1-norm of structural risk and approximation errors creates a new, simplified optimization problem that is solved using linear programming (LP) instead of the more complex quadratic programming (QP). The proposed sparse kernel regression model has the following notable features: (1) it is solved through relatively simple LP; (2) it effectively balances the trade-off between model complexity and modeling accuracy; and (3) the solution is globally optimal rather than just locally optimal. In our three experiments, the sparsity metrics of SVs% were 2.67%, 1.40%, and 0.8%, with test RMSE values of 0.0667, 0.0701, 0.0614 (sinusoidal signal), and 0.0431 (step signal), respectively. This demonstrates the balance between sparsity and modeling accuracy.

Constraint Realization‐Based Hamel Field Integrator for Geometrically Exact Planar Euler–Bernoulli Beam Dynamics

ABSTRACTIn this article, we first introduce a Hamel field integrator designed for a geometrically exact Euler–Bernoulli beam with infinite‐dimensional holonomic constraints, constructed using a Lagrange multiplier. This method addresses the complexities introduced by constraints, but the additional multiplier introduces a new degree of freedom and hence results in a system with mixed‐type partial differential equations. To address this issue, we further propose a constraint realization method based on perturbation theory for infinite‐dimensional mechanical systems within the framework of Hamel's formalism. This method circumvents the use of additional Lagrange multiplier, significantly reducing the computational complexity of modeling problems. Building on this, we construct a perturbed Hamel field integrator optimized for parallel computing and incorporate artificial viscosity to accelerate constraint convergence. While applicable to three dimensions, our method is demonstrated in a simplified context using planar Euler–Bernoulli beam examples to illustrate the effectiveness of the unified mathematical framework.