Extremization of Linear Integrals by Green's Theorem

Publisher Summary A new procedure has been developed to solve linear problems of both the simple type and the isoperimetric type. This procedure is based on Green's theorem relative to the transformation of line integrals into surface integrals. In these problems the functional form to be extremized and the possible isoperimetric constraint are linear in the derivative of the unknown function. When dealing with the extremization of linear integrals, it is important to study the behavior of the fundamental function (ω) within the admissible domain. The equation of the extremal arc is related to the sign of the function ω. The sense in which this extremal arc is to be traveled depends on the sign of the function ω and whether a maximum or a minimum value is desired for the integral. On the other hand, if the function ω changes sign within the admissible domain, the extremal arc is discontinuous and is composed of subarcs. Linear problems are frequently observed in the dynamics of flight of turbojet, turbofan, ramjet, and rocket vehicles. This chapter illustrates several linear problems associated with the flight trajectory of a short-range, air-to-air missile arid with the re-entry of a variable-geometry ballistic missile.