Control theory optimum

Optimal control theory A method for determining the optimum laser field used to maximize a desired product of a chemical reaction. The optimum field is derived by maximizing the objective function, which is the sum of the expectation value of the target operator at a given time and the cost penalty function for the laser field, under the constraint that quantum states of the reactants satisfy the Schrodinger equation. 

Indicative of this tendency in scientific forestry is the substantial literature on optimum control theory, which is imported from management science. For an application and bibliography, see D. M. Donnelly and D. R. Betters, Optimum Control for Scheduling Final Harvest in Even-Aged Forest Stands, Forest Ecology and Management 46 (1991) 135-49. 

Optimum Nuclear Reactor Control Theory Jeffery Lewins and Albert L. Babb... 

The mathematical model of the system to which this optimum control theory is applicable is a model based on a finite number of ordinary differential equations, as in, for example, the point kinetics equations of nuclear reactor control. (The final section of this review extends the general theory to distributed systems the applicability of this more general theory will inevitably be limited by the difficulties of solving the distributed model equations.)... 

Optimum control theory was developed in the main by Pontryagin and... 

On one rather technical point, the optimum theory does differ from classical calculus of variations. This is in regard to the Weierstrass- or end condition, which is not satisfied by the rigorous optimum control theory. The reason for this nicety in the sufficient conditions for an optimum (and the mathematics will not be pursued further) is that optimum control theory is based on the utilitarian search for a greatest or a least value of a functional rather than a maximum or a minimum. These latter terms have precise mathematical meaning in relation to their turning values, whereas all we may require in application is the least value of a cost functional. The example of a finite straight line may make this point clearer, for such a line has neither maximum nor minimum and yet will have, at its ends, greatest and least values. 

Obviously, there is the danger that, in discussing only one physical situation of interesting but admittedly limited practical significance, the wide applicability of the theory may be overlooked. To try and compensate for any such deficiency, we shall discuss the application of optimum control theory in the remainder of this introductory section. 

This problem was considered by Kerr and Lennox (36) and by Fredsall and Babb (37), although the application of optimum control theory is due to Rosztoczy (48), whose arguments we follow, including the treatment of... 

To demonstrate a connection between dynamic programming and optimum control theory, we suppose that — dCjdN is the adjoint function N. Then dynamic programming takes the form... 

To apply dynamic programming, it is convenient to work backwards from the possible final conditions where the cost function is known, just as in the optimum control theory. We therefore introduce the complimentary cost function, C, which is the optimum remaining cost of going to an end state. The basic relation becomes... 

We can discuss these advantages further after a comparison of the application of dynamic progranuning and optimum control theory to the xenon shutdown problem (41-45). 

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