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UBC Theses and Dissertations

A Hamilton-Jacobi approach to the differential inclusion problem Offin, Daniel C.


In the classical calculus of variations, the Hamilton - Jacobi theory leads, under general hypotheses, to sufficient conditions for a local minimum. The optimal control problem as well has its own Hamilton -Jacobi approach to sufficient conditions for optimality. In this thesis we extend this approach to the differential inclusion problem; a general, nonconvex, nondifferentiable control problem. In particular, the familiar Hamilton - Jacobi equation is generalized and a corresponding necessary condition (chapter 2) is obtained. The sufficiency condition (chapter 3) is derived and an example is presented where it is shown how this result may lead to considerable simplification. Finally, we show (chapter 4) how the classical theory of canonical transformations may be brought to bear on certain Hamiltonian inclusions associated with the differential inclusion problem. Our main tool will be the generalized gradient, a set valued derivative for Lipschitz functions which reduces to the subdifferential of convex analysis in the convex case and the familiar derivative in the C¹ case.

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