ABSTRACT
This chapter considres 'continuous control' in which the local characteristics depend on an additional control parameter. This is then selected at each instant of time in order to steer the process in favourable directions or to influence the jump behaviour in favourable ways. The chapter provides a simple formulation of dynamic programming for Piecewise-Deterministic Processes which is adequate to solve some problems, but is too restrictive to qualify as a general theory; the development of one is the goal. It aims to dynamic programming and show that a weak form of the Bellman equation gives a necessary and sufficient condition for optimality. The chapter shows how the ideas of dynamic programming can be used to provide strong sufficient conditions for optimality. This is useful on two counts: first, practical problems can be solved by applying them, and second, we identify the appropriate form of the Bellman equation, which will reappear in a more general context.
