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UBC Theses and Dissertations

Sufficient conditions for optimal control and the generalized problem of Bolza Zeidan, Vera Michel


We develop in this thesis four sufficiency criteria for the generalized problem of Bolza. These results represent a unification, in the sense that they can be applied to both the calculus of variations and to optimal control problems, as well as to problems with nonsmooth data. The first criterion, "point convexity", extends the convexity approach of Rockafellar. However, we derive a "Hamiltonian-Jacobi" approach which can be applied when the point convexity assumption fails to be satisfied. The method employed for this criterion brings to light a new point of view concerning the Jacobi condition in the classical calculus of variations. The latter can be considered as a condition which guarantees the existence of a canonical transformation transforming the original Hamiltonian to a locally concave-convex Hamiltonian. The third sufficiency criterion is an extension of the Hamilton-Jacobi approach from optimal control to the generalized problem of Bolza. This result gives rise to another sufficiency criterion in terms of a certain inequality. Our theorems on sufficient conditions are closely related. We prove that under certain assumptions the last three approaches can be unified. By this we mean that their hypotheses are equivalent. However, the point convexity, and hence the convexity, criterion turns out to have the most restrictive hypotheses of the four. The generality of the theorems proven stems to a great extent from the fact that not only non-differentiable but even infinite- valued functions are allowed in the treatment. The utility of using such functions appears when we apply these theorems to optimal control problems.

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