Hamilton density

Note that if the interaction between the two fermions is described with the following effective interaction Hamilton density... 

Likewise, the Hamiltonian of a system is denoted as H and can be obtained from the Hamilton density H via... 

At this point, we introduce the Hamiltonian description of the system and corresponding form of the optimum theorem. This is done firstly because the Hamiltonian is a concise way to express the state and costate equations. But, more than this conciseness, it turns out that the Hamilton density itself has an interesting and useful property in the optimum system. 

In analogy with the use of the terms Lagrangian and Lagrange density, we use the term Hamiltonian for the integral of a Hamilton density (recognizing that this usage is not universal). 

Define the Hamilton density as that part of the Lagrange density not containing the differential operator ... 

We can further show that, in autonomous or time invariant systems, the Hamilton density just defined is a constant of an optimum control program. By autonomous is meant the absence of any direct dependence of the properties of the system (including extreme values of constrained controls) on time. Consider the rate of change of the Hamilton density ... 

It follows that the Hamilton density has the property of being a constant in an autonomous optimum system. There is an even more striking property of the Hamilton density in a system in which the end time, t, is free and subject to perturbations of either sign. Consider such a perturbation in the original Lagrangian by differentiating with respect... 

The term in Nf dNf vanishes, for it is just the transversality condition imposed as a final boundary condition on the adjoint function such that the Lagrangian should be stationary. For an optimum system, SL is nonnegative. If, as is true with a free end time, St is arbitrary in sign, then Hf must be zero. Being zero at the final time and constant during the control period, the Hamilton density of an autonomous optimum free end time system is identically zero. 

This result of a constant Hamilton density is of preu tical use as an algorithm to seek an optimum control program. We may follow a path from an assumed end point and, continually calculating the necessary elements of the Hamilton density, select that control pattern that yields a constant H. Such a technique is in use, particularly with analog computers. It follows from the theorem itself that this value, Hq say, is a least value, which may itself provide a further test for optimality. 

We can now discuss more clearly the minor dilferences in the development of the optimum theorem in this review as compared with that given originally by Pontryagin (7). Consider om definition of the Hamilton density ... 

Although only one adjoint boundary condition is available from the transversality condition, the adjoint equation and switching function are homogeneous in the costate variable, so that the overall normalization is immaterial in determining an optimum condition. We now have the freedom to impose the additional result of the free end time problem that the Hamilton density vanishes. Boundary conditions to secure this result as well as the transversality condition are... 

Since this is a free end time problem, the Hamilton density is to vanish, and we have... 

Again, this change in the cost functional may not be negative for an optimum. The augmented Hamilton density is... 

Because of the relaxation of the boundary condition previously imposed at 2, we cannot say that this Lagrangian is stationary to perturbations in the density. We must, in fact, carry both the cause du and the effect 5N in evaluating 5C. Into Eq. (48) we substitute not only the augmented adjoint equation, Eq. (42) and the definition of the augmented Hamilton density Eq. (44), but also the equation for the density perturbation... 

This is indeed the solution obtained by manipulation of the Hamilton density (55). (The negative sign arises since we are computing the time taken to decrease the iodine population by one atom.)... 

In the b form, limiting ourselves to a time optimal problem for simplicity, the control period is fixed, and the Hamilton density does not now vanish. Since the end of the trajectory is not bound to any particular target curve, we must take both adjoint functions to vanish at the end time if the Lagrangian is to be stationary for arbitrary errors in the density. On the other hand, the cost functional is now the post-shutdown xenon peak, which is determined only by the end state Nf(tf). Thus, the integrand of the cost function has a delta function form ... 

Suppose we limit ourselves to the steady state (which is meaningful in the context of a distributed system). Since the time derivative vanishes, Lagrange and Hamilton densities are the same. Included in the formalism of the optimum control theorem are a number of optimum theories of... 

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