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1

Optimal generalized Hohmann transfer with plane change using lagrange multipliers

Kamel O. M., Soliman A. S., Amin M. R.

Mechanics and Mechanical Engineering

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2017

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Vol. 21, nr 1

11--16

EN

The optimized orbit transfer of a space vehicle, revolving initially around the primary, in a similar orbit to that of the Earth around the Sun, in an elliptic trajectory, to another similar elliptic orbit of an adequate outer planet is studied in this paper. We assume the elements of the initial orbit to be that of the Earth, and the elements of the final orbit to be that of an outer adequate planet, Mars for instance. We consider the case of two impulse generalized Hohmann non coplanar orbits. We need noncoplanar (plane change) maneuvers mainly because: 1) a launch-site location restricts the initial orbit inclination for the vehicle; 2) the direction of the launch can influence the amount of velocity the booster must supply, so certain orientations may be more desirable; and 3) timing constraints may dictate a launch window that isn’t the best, from which we must make changes[3]. We used the Lagrange multipliers method to get the optimum of the total minimum energy required ΔVT , by optimizing the two plane change angles 1 and 2, where 1 is the plane change at the first instantaneous impulse at peri-apse, and 2 the plane change at the second instantaneous thrust at apo-apse. We adopt the case of Earth - Mars, as a numerical example.

2

Inventory stocks management under the limited capital conditions – nonlinear analysis

Ślaski P.

Gospodarka Materiałowa i Logistyka

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2015

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nr 5

27--31

EN

The article presents the use of Lagrange multipliers functions to inventory stocks management under the limited capital conditions. The example verified the method based on the process analysis, the nonlinear programming and the Solver application.

PL

W artykule przedstawiono możliwość wykorzystania mnożników Lagrangea do zarządzania zapasami w warunkach ograniczonego kapitału. Zaproponowano weryfikację ww. metody w oparciu o programowanie nieliniowe i aplikację Solver.

3

Inventory stocks management under the limited capital conditions – nonlinear analysis

Ślaski P.

Gospodarka Materiałowa i Logistyka

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2015

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nr 5

678--686

EN

The article presents the use of Lagrange multipliers functions to inventory stocks management under the limited capital conditions. The example verified the method based on the process analysis, the nonlinear programming and the SOLVER application.

PL

W artykule przedstawiono możliwość wykorzystania mnożników Lagrangea do zarządzania zapasami w warunkach ograniczonego kapitału. Zaproponowano weryfikację ww. metody w oparciu o programowanie nieliniowe i aplikację SOLVER.

4

Numerical model of elastic laminated glass beams under finite strain

Zemanova A., Zeman J., Sejnoha M.

Archives of Civil and Mechanical Engineering

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2014

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Vol. 14, no. 4

734--744

EN

Laminated glass structures are formed by stiff layers of glass connected with a compliant plastic interlayer. Due to their slenderness and heterogeneity, they exhibit a complex mechanical response that is difficult to capture by single-layer models even in the elastic range. The purpose of this paper is to introduce an efficient and reliable finite element approach to the simulation of the immediate response of laminated glass beams. It proceeds from a refined plate theory due to Mau (1973), as we treat each layer independently and enforce the compatibility by the Lagrange multipliers. At the layer level, we adopt the finite-strain shear deformable formulation of Reissner (1972) and the numerical framework by Ibrahimbegović and Frey (1993). The resulting system is solved by the Newton method with consistent linearization. By comparing the model predictions against available experimental data, analytical methods and two-dimensional finite element simulations, we demonstrate that the proposed formulation is reliable and provides accuracy comparable to the detailed two-dimensional finite element analyzes. As such, it offers a convenient basis to incorporate more refined constitutive description of the interlayer.

5

Single and Double Lagrange multipliers approaches applied to Scalar potential formulation in magnetostatic FEM

Henneron T., Aubertin M., Boiteau O., Piriou F., Guerin P., Mipo J.-C.

Przegląd Elektrotechniczny

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2010

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Nr 5

119-122

EN

In this paper, we propose to introduce the single and double Lagrange multipliers approaches in the case of the finite element method (FEM). These approaches allow non-conforming meshes to be linked together. The methods introduced are developed in the case of a magnetostatic problem solved by the scalar potential formulation. An application is studied and the results obtained by both approaches are compared.

PL

W artykule przedstawiono zastosowanie pojedynczych i podwójnych mnożników Lagrangea stosowanych w metodzie elementów skończonych. To podejście pozwala na połączenie niezgodnych siatek. Metodę rozwinięto dla problemu magnetostatycznego rozwiązywanego z użyciem potencjału skalarnego. Porównano wyniki otrzymane z zastosowaniem proponowanej metody i metod klasycznych.

6

Incremental value of information for discrete-time partially observed stochastic systems

Banek T.

Control and Cybernetics

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2010

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Vol. 39, no 3

769-781

EN

A discrete-time stochastic control problem for general (nonlinear in state, control, observation and noise) models is considered. The same noise can enter into the state and into the observation equations, and the state/observation does not need to be affine with respect to the noise. Under mild assumptions the joint distribution function of the state/observation processes is obtained and used for computing the Gateaux and Frechet derivatives of the cost function. Under partial observation the control actions are restricted by the measurability requirement and we compute the Lagrange multiplier associated with this "information constraint". The multiplier is called a "dual", or "shadow" price, and in the literature of the subject is interpreted as an incremental value of information . The present and the future are two factors appearing in the multiplier and we study how they are balanced as time goes on. An algorithm for computing extremal controls in the spirit of R. Rishel (1985) is also obtained.

7

Sequential target tracking using constrained nonlinear estimators

Stubberud A.R.

Systems Science

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2003

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Vol. 29, nr 2

17-29

EN

An estimator is presented which generates sequential estimates for nonlinear, time-variable discrete-time dynamic systems in which the system state estimates are subject to an instantaneous constraint. That is, at each sample time the state estimate is constrained to lie in a given region of the state space. This nonlinear sequential estimator is an extended version of an optimal sequential estimator for linear, time-variable discrete-time systems with state estimates constrained to a given region of the state space. The linear estimator was developed from a non-probabilistic weighted linear least squares basis with the constraints added through the mechanism of Lagrange multipliers; therefore, the estimator produces "hard" constraints on the state estimate. The solution of the constrained estimation problem, at each instant of time, requires only the unconstrained state estimate at that time instant and the instantaneous constraints which define the constraint region. If the unconstrained sequential estimate satisfies the constraints, then that solution is also the constrained solution. On the other hand, if the unconstrained estimate does not satisfy the constraints, then the constrained solution is generated from the solution of a set of static equations. The constrained estimation problem is thus reduced to a sequence of nonlinear programming problems. The estimator for the state of a nonlinear system was developed by quasi-linearization of the optimal constrained linear estimator. The estimates resulting from this estimator are "optimal in the small" for nonlinear systems and are optimal for linear systems.

8

Modeling and motion planning for a class of weight handling equipment

Kiss B., Levine J., Mullhaupt P.

Systems Science

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2000

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Vol. 26, nr 4

79-92

EN

A unified framework for the modeling of a class of weight handling equipment (WHE) is presented. The dynamic equations are obtained using Lagrange multipiers associated to geometric constraints between generalized coordinates. This approach provides a simple way to show differential flatness for all WHEs of the class. The flatness property can then be exploited for motion planning.

9

Lagrange multipliers determination for optimum control problem solution

Grevtsov N., Efimov O., Melts I.

Research Bulletin / Warsaw University of Technology, Institute of Aeronautics and Applied Mechanics

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1998

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nr 8

209-214

EN

This paper considers the problem of Lagrange multipliers initial proximity determination for iterative procedure of second order method of optimum control problem solution. The control problem corresponds to the optimum control of dynamic system described by finite - difference equations with account of terminal conditions, control and phase variables or mixed inequality constraints. The time of control process is free or fixed. The problem of Lagrange multipliers determination is formulated as the problem of finding the minimum of the norm square of the relations describing optimality conditions at initial proximity of control. This problem is reduced to the special two point boundary value problem (TPBVP). The method of this TPBVP solution for initial proximity of Lagrange multipliers determination is proposed.