Variations Lagrange form

We conclude that the relationship between dynamic programming and the optimum control theorem has been established on the basis that — dCjSNii the costate variable N. Dynamic programming is seen to be the Hamilton-Jacobi form of the calculus of variations, whereas the optimum control theorem was the Euler-Lagrange form. 

The use of this theory in studies of nonlinear oscillations was suggested in 1929 (by Andronov). At a later date (1937) Krylov and Bogoliubov (K.B.) simplified somewhat the method of attack by a device resembling Lagrange s method of the variation of parameters, and in this form the method became useful for solving practical problems. Most of these early applications were to autonomous systems (mainly the self-excited oscillations), but later the method was extended to... 

The formalism used to calculate the pulse shape that maximizes J is optimal control theory. This formalism can be considered to be an extension of the calculus of variations to the case where the constraints include differential equations. In general, the constraints expressed in the form of differential equations express the restriction that the amplitude must always satisfy the Schrodinger equation. In addition, there can be a variety of other constraints, such as a restriction on the total energy in the pulse or on the shape of the pulse. These constraints are accounted for by the method of Lagrange multipliers, which modify the objective functional (4.6) and thereby permit the calculation of the unconstrained maximum of the modified objective functional. When the only constraints are satisfaction of the Schrodinger equation and limitation of the pulse energy, the modified objective functional can be written in the form... 

Generalized momenta are defined by pk = and generalized forces are defined by Qk = f - When applied using t as the independent variable, Euler s variational equation for the action integral / takes the form of Lagrange s equations of motion... 

The study of minimal surfaces arose naturally in the development of the calculus of variations. The problem of finding the surface forming the smallest area for a given perimeter was first posed by Lagrange in 1762, in the appendix of a famous paper that established the variational calculus [8]. He showed that a necessary condition for the existence of such a surface is the equation... 

A form of the principle of stationary action that will prove of great use in later applications is obtained by expressing the principle of stationary action (eqn (8.79)) for an infinitesimal time interval, that is, as a variation of the Lagrange function operator. Equation (8.79) is re-expressed as... 

Equations (8.126) and (8.127) are identical in form to the corresponding results obtained for the variation of the Lagrange function operator in eqns (8.97) and (8.98). They are variational statements of the Heisenberg equation of motion for the observable F in the Schrodinger representation. When T describes a stationary state... 

The direct method of obtaining the change in the quotient eqn ( 3.6) has not even been considered as it is far too involved to be manageable. The traditional method is to use Lagrange s method of undetermined multipliers to form a linear combination of the two expressions which are required to vanish, and require this linear combination to vanish for each degree of variational freedom. In Our case this is to combine eqns ( 3.20) and ( 3.22) using a linear combination... 

By using the calculus of variations, the integral form of Lagrange s method can be obtained. This is known as Hamilton s principle (Goldstein et al. 2002). It can be stated as follows ... 

The application of direct variational method for solution of Euler-Lagrange equation is similar to that in Sect. 3.2.2.1. It ailows to obtain the free energy in the form of polarization power series with the coefficients dependent on average particles radius and the parameters of Euler-Lagrange equation (3.45). In particular, surface polarization Pd in the boundary conditions leads to appearance of built-in field Ecyi(R), which can be written as... 

Application of the direct variational method [8, 78] for the Euler-Lagrange Eq. (4.20) leads to the conventional form of the free energy with renormalized coefficients... 

In cases where equations of motion are desired for deformable bodies, methods such as the extended Hamilton s principle may be employed. The energy is written for the system and, in addition to the terms used in Lagrange s equation, strain energy would be included. Application of Hamilton s principle will yield a set of equations of motion in the form of partial differential equations as well as the corresponding boundary conditions. Derivations and examples can be found in other sources (Baruh, 1999 Benaroya, 1998). Hamilton s principle employs the calculus of variations, and there are many texts that will be of benefit (Lanczos, 1970). 

Classical methods of calculus of variations are attractive from the point of view of the opportunity to obtain solutions in analytical form. But this is feasible in simple cases, which often are far from the demands of the state-of-art practice. In complicated cases, at a large number of optimization parameters, numerical approaches are used to solve the appropriate Euler-Lagrange equations. The main obstacle arising here is related to the fact that the numerical solution of the system of differential equations may turn out to be more complicated than the solution fi-om the very beginning of the optimization problem by numerical methods of mathematical programming. 

In this section, we develop the basic optimum theorem in the Lagrangian form, to include the cases where the allowable variations of the control variables are limited. In a later section, we shall introduce Pontryagin s treatment of cases where the state variables are similarly limited to make the distinetion clear, we shall refer to constraints on the control variables but restraints on the state variables. In the next section, the Lagrange formulation will be converted to the Hamilton form. 

To apply the variational calculus methods, one should know when a gyroscopic system is Lagrangian, i.e is described by the Lagrange equations with the Lagrange function L on TAf. For this, it is necessary and sufficient that the form of the gyroscopic forces be exact, that is, F = dA where A is the differential 1-form on Af. 

Non-linear programming technique (NLP) is used to solve the problems resulting from syntheses optimisation. This NLP approach involves transforming the general optimal control problem, which is of infinite dimension (the control variables are time-dependant), into a finite dimensional NLP problem by the means of control vector parameterisation. According to this parameterisation technique, the control variables are restricted to a predefined form of temporal variation which is often referred to as a basis function Lagrange polynoms (piecewise constant, piecewise linear) or exponential based function. A successive quadratic programming method is then applied to solve the resultant NLP. 

Additional restrictions can be incorporated into the variational problem by making use of the Lagrange multiplier technique [27-31]. For example, if we want to fix the population in the atoms 1, 2,..., r, then for each atom A = l,...,r the constraint takes the form ... 

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