Classical Calculus Of Variations Research Articles

On some optimization procedures for shock localization

AbstractWe consider a problem on shock wave localization in the numerical solution of one‐dimensional unsteady problems of gas dynamics in Eulerian variables obtained on the basis of finite difference shock‐capturing schemes. An optimization method for strong discontinuity localization proposed previously by Miranker and Pironneau is investigated by means of methods of classical variational calculus. This method may be difficult to implement when the entropy condition is included in the formulation of Miranker and Pironneau's optimization problem as an active constraint. In this connection we suggest an alternative optimization problem using artificial viscosity in the variational principle. It is shown theoretically that the application of such a variational principle yields a trajectory which coincides with the true discontinuity trajectory in the case of a shock wave moving at a constant speed. On the basis of this modification one more algorithm is proposed which reduces the shock localization problem to a problem of minimization of a univariate function. Numerical tests corroborate completely the theoretical conclusions. In particular, a higher shock localization accuracy is obtained on the basis of the proposed algorithms as compared to the original Miranker‐Pironneau method.

Stochastic calculus of variations

We consider an analogue of classical calculus of variations, in which deterministic comparison curves are replaced by stochastic processes xt, of the form with ut a control applied at time t and wt a brownian motion. Connections with stochastic mechanics and non-linear filtering are discussed, as well as the deterministic limit as λ → 0.

ON THE SYSTEM OF JACOBI EQUATIONS IN A TIME-OPTIMAL PROBLEM

A perturbation method is used to obtain second-order conditions in a nonautonomous time-optimal problem with vectorgram of full dimension. These conditions are formulated in the language of focal points in the classical variational calculus. Bibliography: 16 titles.

Sufficient Conditions for the Generalized Problem of Bolza

This paper presents sufficient conditions for strong local optimality in the generalized problem of Bolza. These conditions represent a unification, in the sense that they can be applied to both the calculus of variations and to optimal control problems, as well as problems with nonsmooth data. Also, this paper brings to light a new point of view concerning the Jacobi condition in the classical calculus of variations, showing that it can be considered as a condition which guarantees the existence of a canonical transformation which transforms the original Hamiltonian to a locally concave-convex Hamiltonian.

Sufficient conditions for the generalized problem of Bolza

This paper presents sufficient conditions for strong local optimality in the generalized problem of Bolza. These conditions represent a unification, in the sense that they can be applied to both the calculus of variations and to optimal control problems, as well as problems with nonsmooth data. Also, this paper brings to light a new point of view concerning the Jacobi condition in the classical calculus of variations, showing that it can be considered as a condition which guarantees the existence of a canonical transformation which transforms the original Hamiltonian to a locally concave-convex Hamiltonian.

Semi-classical quantum mechanics and stochastic calculus of variations

Within the framework of the general theory of stochastic calculus of variations, we examine mainly the notion of second variation in the stochastic mechanics of E. Nelson, a representative of quantum mechanics in which the concept of path for particles keep a sense. We show that the two approaches used in classical calculus of variation to know if a path is not only an extremum but also the minimum of the action, namely, the local one (weak minimum) and the global one (strong minimum), can be generalized to include the quantum-mechanical paths. Thus, we can prove that locally, a solution of the classical equation of motion is really the minimum, even in a large class of quantum paths containing the semi-classical trajectories. By introducing a stochastic version of the excess function of Weierstrass, we show the analogous global property. There, of course, one can speak of the principle of least action in a strict sense. Several explicit examples are discussed.

A CONDITION OF JACOBI TYPE FOR A QUADRATIC FORM TO BE NONNEGATIVE ON A FINITE-FACED CONE

The author considers the problem of nonnegativity and positive definiteness of a Legendre quadratic form on a family of cones in Hilbert space that are the intersection of a fixed finite-faced cone with a monotonically increasing family of subspaces. This problem arises in the investigation to second order of an extremal in the Lagrange-Mayer-Bolza problem of classical variational calculus in the presence of finitely many additional constraints in the form of end and integral inequalities when there is only one set of Lagrange multipliers for the given extremal. It is shown that a complete answer can be given in terms of solutions of the corresponding Euler-Jacobi equation. Bibliography: 5 titles.

THE INDEX OF THE SECOND VARIATION OF A CONTROL SYSTEM

The constancy of sign of the second variation of a nonlinear control system is investigated. Sufficient conditions for controllability of second order and also sufficient conditions for optimality with respect to fast action of the extremals of this system analogous to the Legendre-Jacobi conditions of the classical calculus of variation are obtained.Bibliography: 6 titles.

The why and how of finite elements

The ‘Why’ of the title reflects the desire in safety assessments for independent means of making realistic calculations for systems of complex shape. In principle the geometrical flexibility attainable by the deterministic finite element method being the equal of the stochastic Monte Carlo method. The development of the finite element method is traced to show how ideas from structural and fluid mechanics, the calculus of variations, functional analysis and the calculus of finite differences have been forged to provide a tool which minimizes the mismatch between the behaviour of a continuous system and that of a discrete model of the system assembled from finite elements. Geometrical flexibility of the model is achieved by the use of polygonal and curved elements. The behaviour of any point of an element is described in terms of its behaviour at discrete points or nodes of the element. In treating neutron transport the finite element method can be applied to phase-space, or as in this paper the spatial dependence can be treated by the use of finite elements in conjunction with expansions in orthogonal functions for the directional dependence. The mathematical formulation is based on a mixed parity form of the Boltzmann equation for one-group transport. The minimization of the mismatch between the system and its finite element model leads to a completely boundary-free maximum principle. This variational principle is also recast into a generalized least-squares principle. When the essential boundary conditions of the classical calculus of variations are imposed the well-known minimum and maximum principles for the even- and odd-parity second-order Boltzmann equations are obtained as special cases. The maximum principle for the second-order even-parity equation is used to demonstrate the precision and flexibility of the finite element method by solving the problems of a dog-legged duct in a shield and a cylindrical fuel element in a square lattice cell. The geometrical interpretation of the boundary-free maximum principle with the aid of a suitable Hilbert space then leads to completely boundary-free weighted residual or Galerkin schemes for both the first- and second-order forms of the Boltzmann equation. Imposing essential boundary conditions leads to classical schemes. The paper concludes with a sketch of finite element treatments of the multigroup Boltzmann equation.

Optimal Control and the True Hamiltonian

The natural Hamiltonian function in optimal control is generally not differentiable. However, it is possible to use the theory of generalized gradients (which we discuss as a preliminary) to obtain necessary conditions in the form of a “Hamiltonian inclusion”. The advantages of using the true Hamiltonian include the possibility of considering control problems that do not lie within the usual formulation, or problems with nonsmooth data, and the possibility of bringing to bear upon control problems certain Hamiltonian methods from the classical calculus of variations.

Decomposition in classical variational calculus

Decomposition in classical variational calculus

Optimal control of age-dependent populations

An optimal control problem is formulated to control the growth of an age-dependent population in a deterministic environment. Birth rate is used as a control variable. Classical variational calculus is applied to derive a set of equations representing necessary conditions, and these equations are uncoupled so as to obtain a feedback control law.

INTERTEMPORAL OPTIMIZATION IN AN AGRARIAN MODEL WITH Z‐ACTIVITIES*

AbstractThis paper examines the optimal characteristics of an agrarian growth model in which two production activities are distinguished, viz., one producing the usual agricultural (consumption) goods F, and the other consisting of labor‐using Z‐activities directed toward substitute consumption for industrial goods and augmentation of agricultural capital.Certain restrictive assumptions on the form of the utility function are made which reflect in part the nature of the three types of consumption goods being distinguished. Optimum conditions are derived using classical calculus of variations, yielding additional variables in the form of shadow prices. Non‐negativity of gross investment is shown to be the only relevant boundary constraint and is accordingly taken into account in the analysis. The phase diagram involving the capital‐labor ratio and the shadow price of Z‐goods indicates that the unique stationary point is a saddle‐point. Some segment of the two stable branches then represents the interior solution to the infinite horizon problem; if the initial capital‐labor ratio is ≪ sufficiently high ≫, the optimal path will entail zero gross investment in the initial phase. Where the period of optimization is finite, different patterns of optimum capital accumulation and valuation will emerge, depending on the initial and terminal values of the capital‐labor ratio. Two cases are illustrated, the optimal path in each being shown to exhibit the turnpike property.It follows that optimal growth may be achieved by directly controlling the rate of capital accumulation or by having the planning authority set the shadow price of Z‐goods. Other means of optimization are in fact available, e.g., controlling imports of industrial consumption goods. That one policy instrument is sufficient to attain the single objective is consistent with the well‐known proposition in the theory of economic policy.

A Monte Carlo Approach to Optimal Drilling

Abstract The variable weight-speed optimal drilling problem has been soloed rigorously using a Monte problem has been soloed rigorously using a Monte Carlo scheme to vary the drilling schedule toward the least cost per foot. Constraints are readily accommodated and the global nature of the optimum can he checked by starting with different initial paths. paths. Examples are given of optimal drilling schedules that lie partially interior to the feasible region as well as on upper weight or speed constraints. Comparison with the variational method of Galle and Woods showed good agreement using drilling equations presented here. However, in all cases studied, it was found that variable weight-speed optimization offered very little advantage over the simpler constant weight-speed approach. In view of ibis, a last computer program was developed for the constant weight-speed case using the powerful conjugate-gradient technique. This method should be very effective in connection with field applications. Introduction Although numerous papers discuss drilling optimization or provide optimization techniques based on field experience, it appears that the first analytical approach to the problem was published by Moore. He used drilling equations that were so simple it was possible to calculate directly the optimum weight corresponding to a given rotary speed. In 1959, Graham and Muench used what is sometimes called the "graphical" approach, together with more realistic drilling equations, to calculate optimum combinations of weight and speed to bearing failure. In their paper, cost per foot is computed vs weight for various depths at fixed speed. This is repeated for various speeds until the optimum is found for each depth. The most significant contribution to the field appeared in 1960. This paper by Galle and Woods culminated years of drilling mechanics research, reducing it all to a concise set of drilling equations. Necessary conditions for the optimal variable weight-speed path are found using the classical calculus of variations with integrated drilling equations acting as constraints. The final result is obtained using a numerical procedure. One conclusion was that, in most cases, the variable weight-speed method would effect a saving of at least 10 percent over the best constant weight-speed schedule. Although we do not want to detract from the importance of this work in any way, it is subject to question on two counts. First, the method of solution requires weight and speed to vary continuously as a function of time. Second, it is recommended that bearing-wear-limited schedules be computed using tooth-wear-limited theory. In 1962, Billington and Blenkarn used the equations and techniques developed by Galle and Woods to optimize the variable weight schedule when speed is fixed by rig limitations. They concluded there was little cost advantage in using a variable weight-speed schedule in preference to a constant speed program. In 1963 the second contribution of Galle and Woods to optimal drilling appeared. In this paper ordinary constrained calculus is used to solve the two-dimensional best constant weight-speed problem. They also provide the best speed for a given weight and the best weight for a given speed. Charts are furnished so that hand calculations can be rapidly made. In the introduction, Galle and Woods note that, in certain cases, the procedures in which both weight and rotary speed are varied or where weight alone is varied result in only slightly lower cost than if the bit were properly operated at constant weight and rotary speed. We have therefore seen a transition from the belief that in most cases variable weight-speed is at least 10 percent better than constant weight-speed to the opinion that there is very little difference between the two methods. At this point, one is not certain what to conclude. In this paper a new, rigorous method of optimization is developed. It provides an independent check on the approximations involved in the variational approach of Galle and Woods and compares optimal variable weight-speed schedules to the corresponding optimal constant weight-speed programs.

A Chebyshev minimax technique oriented to aerospace trajectory optimization problems.

This paper describes a method based upon the classical calculus of variations for solving directly Chebyshev minimax problems which arise in trajectory optimization. The close relationship between minimax problems and problems with state variable inequality constraints is used to gain insight into the minimax problem, and to define an order for minimax functions. The method is applicable to 1) afl problems in which the first time derivative of the minimax function does not contain control variables explicitly, and 2) all problems with a flat maximum (including problems in which the first time derivative of the minimax function contains control variables). The theory is applied to both a simple example and a formulation of the optimal re-entry heating problem whose performance index consists of a minimax term for maximum heating rate, a terminal function for maximum crossrange, and a path integral for minimum total heating. Several shooting and gradient-type numerical algorithms are suggested by the approach.

Optimum Boundaries for Elliptic Systems

In this paper a modification of the Ritz method is presented for finding optimum boundaries in which the Laplace or the Poisson equation is known to hold. The method developed utilized the classical calculus of variations to determine candidates for optimum boundaries. The method is applied to obtain the optimum tooth shape for maximum torque of a magnetic exciter used on a resonant stepping mechanism.

A Chebyshev minimax technique oriented to aerospace trajectory optimization problems

This paper describes a method based upon the classical calculus of variations for solving directly Chebyshev minimax problems which arise in trajectory optimization. The close relationship between minimax problems and problems with state variable inequality constraints is used to gain insight into the minimax problem, and to define an order for minimax functions. The method is applicable to 1) afl problems in which the first time derivative of the minimax function does not contain control variables explicitly, and 2) all problems with a flat maximum (including problems in which the first time derivative of the minimax function contains control variables). The theory is applied to both a simple example and a formulation of the optimal re-entry heating problem whose performance index consists of a minimax term for maximum heating rate, a terminal function for maximum crossrange, and a path integral for minimum total heating. Several shooting and gradient-type numerical algorithms are suggested by the approach.

The second variation and singular space trajectories†

During the last Few years the second variation of the classical calculus of variations has been used to analyse candidate arcs for optimal trajectories. It has proved useful in both analytical and computational work. This paper states the second variation for general space trajectories in terms of the pseudo-Hamiltonian of Pontryagin. A direct application of the theory is shown to be sufficient to demonstrate optimality of a singular arc which may arise in the escape manoeuvre of a space vehicle.

On the optimality of the boundary control for the non-linear control system with controls appearing linearly†

Abstract In this paper, the optimization of a non-linear control system is studied, where both the system and the performance index are linear in the control variables. This study is concerned mainly with the optimality of the totally singular control of the maximum principle. The problem is reduced to the study of & non-regular case of the classical calculus of variations, for which anew criterion is found concerning the property of its extremals By means of this new criterion, it is proved that the totally singular control cannot be optimal for the higher dimensional case. In other words, the optimal control should be the boundary control

Zur optimalitÄt der Uzawa-Pfade

On the Optimal Growth of Uzawa Paths. — In a number of recent works optimal growth paths are stated by the classical calculus of variations. In this paper we want to discuss the problems which occur in finding optimal growth paths by the help of the classical calculus of variations. You will find two critical aspects for the well-known growth model of Uzawa. First, by the calculus of variations not all possible economic restrictions are fully covered, so that the possibilities of economic interpretation of the results are not guaranteed uniquely. The results may be, for instance, negative. Secondly, the result found by Uzawa is — respecting the given aims — in no case optimal. The same may be stated to the model of Konig. Therefore the calculus of variations is characterized as a problematical method of finding optimal economic growth paths. On the contrary the maximum principle promises by its mere formulation a higher efficiency.