An example of optimal control of a system with discontinuous state
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An example of a system with discontinuities in the state vector is described. Such systems arise in manufacturing, animal and human locomotion, certain queueing problems and many other applications. Optimal control problems for such systems cannot be solved directly via dynamic programming or the maximum principle because both of these analytical tools require continuous state vectors. As a first step in the study of such problems, two alternate formulations for the dynamics of the example problem are given. Both of these formulations allow the solution of the resulting optimal control problem by elementary methods. One of the formulations produces the solution as the limit, as a parameter goes to zero, of the solutions to problems to which dynamic programming and the maximum principle apply.
The solution to the optimal control problem is given, in feedback form, throughout the state space. The optimal control includes a singular are that is not on the state boundary. The paper concludes with a brief discussion of more realistic and practical problems.
KeywordsDynamic Programming Maximum Principle Optimal Control Problem Optimal Trajectory Inelastic Collision
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