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Second order necessary conditions in optimal control Zheng, Harry H.


This thesis presents second order necessary conditions for the standard deterministic optimal control problem without conventional regularity assumptions. It reports three different advances in the second-order theory. First, we study conjugate points in smooth optimal control. We describe a set of conditions under which the second variation along an extremal trajectory is certain to be negative: when these are satisfied, they identify a certain point in the basic interval as a "Generalized Conjugate Point (GCP)." The GCP's include all conjugate points in the classical sense, and all points where the Legendre conditionis violated. We also find GCP's in many problems left unresolved by the published literature, and deduce that the associated extremals are not optimal. Second, we use a second-order tangent set to predict the second-order variations along a trajectory to a differential inclusion. Assuming only that the problem's data are Lipschitz continuous, we obtain a generalized form of the familiar statement that the second variation associated with a minimum point is nonnegative. The result is expressed in terms of a generalized second-order derivative introduced by Aubin, and reproduces the classical necessary conditions in the smooth case. Third, we apply Rockafellar's theory of epi-differentiability to integral functionals. Using At touch's theorem, we show that certain nonconvex integral functionals of interest in optimal control are twice epi-differentiable. From this we deduce necessary conditions for optimality for problems without endpoint constraints.

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