Form Of Lagrange Multipliers Research Articles

Construction of a solution for optimal control problem with phase and integral constraints

Abstract. A method for solving the Lagrange problem with phase restrictions for processes described by ordinary differential equations without involvement of the Lagrange principle is supposed. Necessary and sufficient conditions for existence of a solution of the variation problem are obtained, feasible control is found and optimal solution is constructed by narrowing the field of feasible controls. The basis of the proposed method for solving the variation problem is an immersion principle. The essence of the immersion principle is that the original variation problem with the boundary conditions with phase and integral constraints is replaced by equivalent optimal control problem with a free right end of the trajectory. This approach is made possible by finding the general solution of a class of Fredholm integral equations of the first order. The scientific novelty of the results is that: there is no need to introduce additional variables in the form of Lagrange multipliers; proof of the existence of a saddle point of the Lagrange functional; the existence and construction of a solution to the Lagrange problem are solved together.

Research on Optimization and Integration of Railway Logistics Resources in China

The optimal allocation of railway logistics resources is the current key issue confronted by China s railway industry. Under this background, this paper researches this issue along the route of motivation mechanism model evaluation. Based on the research of internal development needs of railway logistics resources and motivation of external social needs, this paper analyzes the effect of dynamic factors generated by different resources on the logistics system, provides a direction for the optimization and integration of resources, establishes a multi-objective nonlinear programming model based on customer utility, and uses Lagrange multiplier form to explain the coupling relationship between corresponding services for customer needs and types of logistics resources. Combined with the relevant data of railway logistics in China Traffic Yearbook in 2000 to 2009 processed by DEA model, the coordination validity coefficient-Cs, development validity coefficient-Ds and comprehensiveness validity coefficient-Ss are used to represent the optimization effect of railway logistics resources in timing sequence. The research results show that China’s railway logistics resource system is at the stage of increasing returns to scale. There is a need to take measures to strengthen resource integration and increase investment scale, so as to ensure the sustainable development of railway logistics system.

An index 0 Differential-Algebraic equation formulation for multibody dynamics: Holonomic constraints

ABSTRACTThe Lagrange multiplier form of index 3 differential-algebraic equations of motion for holonomically constrained multibody systems is transformed using tangent space generalized coordinates to an index 0 form that is equivalent to an ordinary differential equation. The index 0 formulation includes embedded tolerances that assure satisfaction of position, velocity, and acceleration constraints and is solved using established explicit and implicit numerical integration methods. Numerical experiments with two spatial applications show that the formulation accurately satisfies constraints, preserves invariants due to conservation laws, and behaves as if applied to an ordinary differential equation.

General rule for boundary conditions from the action principle

We construct models where initial and boundary conditions can be found from the fundamental rules of physics, without the need to assume them, they will be derived from the action principle. Those constraints are established from physical viewpoint, and it is not in the form of Lagrange multipliers. We show some examples from the past and some new examples that can be useful, where constraint can be obtained from the action principle. Those actions represent physical models. We show that it is possible to use our rule to get those constraints directly.

The domain interface method: a general-purpose non-intrusive technique for non-conforming domain decomposition problems.

A domain decomposition technique is proposed which is capable of properly connecting arbitrary non-conforming interfaces. The strategy essentially consists in considering a fictitious zero-width interface between the non-matching meshes which is discretized using a Delaunay triangulation. Continuity is satisfied across domains through normal and tangential stresses provided by the discretized interface and inserted in the formulation in the form of Lagrange multipliers. The final structure of the global system of equations resembles the dual assembly of substructures where the Lagrange multipliers are employed to nullify the gap between domains. A new approach to handle floating subdomains is outlined which can be implemented without significantly altering the structure of standard industrial finite element codes. The effectiveness of the developed algorithm is demonstrated through a patch test example and a number of tests that highlight the accuracy of the methodology and independence of the results with respect to the framework parameters. Considering its high degree of flexibility and non-intrusive character, the proposed domain decomposition framework is regarded as an attractive alternative to other established techniques such as the mortar approach.

Analytical solution of a linear quadratic optimal control problem with control value constraints

Optimal control problem for time varying linear systems with fixed endpoints of the trajectories and quadratic functional is considered. A method for constructing feedback controls with regard to constraints imposed on them is proposed. The problem is solved using a special form of Lagrange multipliers.

Multibody dynamics modelling and system identification of a quarter-car test rig with McPherson strut suspension

In this paper, we develop a multibody dynamics model of a quarter-car test-rig equipped with a McPherson strut suspension and we apply a system identification technique on it. Constrained equations of motion in the Lagrange multiplier form are derived and employed to characterise the dynamic behaviour of the test rig modelled once as a linear system and once as a non-linear system. The system of differential algebraic equations is integrated using a Hilber–Hughes–Taylor integrator. The responses of both models (linear and non-linear) to a given displacement input are obtained and compared with the experimental response recorded using the physical quarter-car test rig equipped with a McPherson strut suspension. The system identification is performed for control purposes. The results, as well as the performance and area of applicability of the test rig models derived, are discussed.

Penalty/Barrier multiplier algorthm for semidefinit programming*

We present a generalization of the Penalty/Barrier Multiplier algorithm for semidefinite programming, based on a matrix form of Lagrange multipliers. Our approach allows to use among others logarithmic, shifted logarithmic, exponential and a very effective quadratic-logarithmic penalty/Barrier functions. We present a dual analysis of the method, based on its correspondence to a proximal point algorithm with a nonquadratic distance-like function. We give computationally tractable dual bounds, which are produced by the Legendre transformation of the penalty function. Numerical results for large-scale problems from robust control, robust truss topology design and free material design demonstrate high efficiency of the algorithm

On the Nonuniqueness of Solutions Obtained With Simplified Variational Principles

The attempt to satisfy subsidiary conditions in boundary value problems without additional independent unknowns in the form of Lagrange multipliers has led to the development of so-called “simplified variational principles.” They are based on using the Euler-Lagrange equations for the Lagrange multipliers to express the multipliers in terms of the original variables. It is shown that the conversion of a variational principle with subsidiary conditions to such a simplified variational principle may lead to the loss of uniqueness of the solution of a boundary value problem. A particularly simple form of the geometrically nonlinear theory of bending of beams is used as the vehicle for this proof. The development given in this paper is entirely analytical. Hence, the deficiencies of the investigated simplified variational principle are fundamental.

Dynamics of Flexible Multibody Systems Using Bond Graphs and Lagrange Multipliers

A procedure is presented to study the dynamics of interconnected flexible systems using bond graphs. The concept of Lagrange multipliers, which are commonly used in analysis of constrained systems, is introduced in the procedure. The overall motions of each of the component bodies are described in terms of large rigid body motions and small elastic vibrations. Bond graphs are used to represent both rigid body and flexible dynamics of each body in a unified manner. Bond graphs of such sub-systems are coupled to one another satisfying the kinematic constraints at the interfaces to get the overall system model. Constraint reactions are introduced in the form of Lagrange multipliers at the interfaces without disturbing the integral causality in the subsystem models, which leads to easy derivation of system equations. The equations of motion and higher derivatives of the constraint relations are integrated to obtain the constraint reactions and the system response. The procedure is illustrated by an example system and results are in good agreement with those presented earlier.

Substructuring Methods for Computing the Nullspace of Equilibrium Matrices

Equations of equilibrium arise in numerous areas of engineering. Applications to electrical networks, structures, and fluid flow are elegantly described in Introduction to Applied Mathematics, Wellesley Cambridge Press, Wellesley, MA, 1986 by Strang. The context in which equilibrium equations arise may be stated in two forms: Constrained Minimization Form: $\min(x^T Ax - 2x^T r)\,{\text{subject to}}\,Ex = s$, Lagrange Multiplier Form: $EA^{ - 1} E^T \lambda = s - EA^{ - 1} r\,{\text{and}}\,Ax = r - E^T \lambda $. The Lagrange multiplier form given above results from block Gaussian elimination on the $2 \times 2$ block matrix system for the constrained minimization form. Here A is generally some symmetric positive-definite matrix associated with the minimization problem. For example, A can be the element flexibility matrix in the structures application. An important approach (called the force method in structural optimization) to the solution to such problems involves dimension reduction nullspace schemes based upon computation of a basis for the nullspace for E. In our approach to solving such problems we emphasize the parallel computation of a basis for the nullspace of E and examine the applications to structural optimization and fluid flow. Several new block decomposition and node ordering schemes are suggested and reanalyses computations are investigated. Comparisons of these schemes are made with those of Storaasli et al. for structures and Hall et al. for fluids.

A New Approach to Lagrange Multipliers

We consider a mathematical programming problem on a Banach space, and we derive necessary conditions for optimality in Lagrange multiplier form. We prove further that “most mathematical programming problems are normal.” The novelty of our approach lies on the one hand in the absence of both differentiability and convexity hypotheses on the functions delimiting the problem, and on the other hand in the method of proof, which is new. The approach unifies the well-known smooth and convex cases besides treating a new general class of problems. The main analytical tool in this paper is an extension to infinite-dimensional spaces of the “generalized gradient” previously introduced by the author. The calculus of the generalized gradient is explored as a preliminary step.