Necessary conditions for singular extremals in the calculus of variations
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
The original objective of this research was to derive necessary conditions which would enable us to determine the nature of the intermediate - thrust arcs of optimal rocket trajectories, which Lawden had obtained in closed form. From the calculus of variations point of view these arcs are known as singular extremals. Such extremals can satisfy the classical Clebsch-Legendre condition marginally (equality). In such a case, it is necessary to discover further in, equality conditions by reference to which the nature of these extremals can be decided. Several authors have previously obtained such generalized Clebsch conditions, which are however limited to the variation of one control variable at the time. Here the complete generalized Clebsch condition is presented and this is stated as theorem 2 in chapter 4. In section 4.8 it is shown that the positive definiteness of this generalized Clebsch condition plays a role in the derivation of the singular extremals similar to that of the positive definiteness of the classical Clebsch condition in the derivation of regular extremals. The theory is applied mostly to problems from the theory of optimal control.