Titlebook: Optimal Control Theory for Applications; David G. Hull Textbook 2003 Springer Science+Business Media New York 2003 aerospace engineering.c

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Unconstrained Minimizationon begins by considering the minimization of a function of unconstrained (independent) variables. First, the conditions for minimizing a function of one variable are derived. It is shown that necessary conditions for a minimum are that the first differential of the performance index must vanish and

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Constrained Minimization: Inequality Constraintsty constraints can be imposed. To introduce the subject, conditions for locating a boundary minimal point of a function of one independent variable (see Section 2.2) are determined using the direct approach. Next, a device for converting an inequality constraint into an equality constraint, called a

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Controllabilityn admissible comparison path satisfies the differential constraints, the prescribed initial conditions, and the prescribed final conditions, and it lies in the neighborhood of the minimum. The admissible comparison path is composed of a first-order part, a second-order part, and so on. The notion of

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Fixed Final Time: First Differentialons and the natural boundary conditions. These equations are to be solved for the optimal controls and states. For problems not explicitly containing the time, a first integral is shown to exist. Several examples are presented to illustrate the application of the theory.

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Fixed Final Time: Second Differentialhe second differential is obtained by taking the differential of the first differential. Second, the Legendre-Clebsch condition is derived from the second differential. Third, for the class of nonsingular problems (.. > 0) conditions are developed for the existence of a neighboring optimal path. Fin