Optimal Problem with Discontinuity

概要

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Pontryagin's maximum principle is only applicable when right hand side functions in a differential equation of a controlled system and an integrand in criterion function as well as their partial derivatives with respect to x are all continuous for both x and u. The optimal control problem in which any above condition is not satisfied is called a problem with discontinuity. In this case the state space is divided by boundary into some regions in which the maximum principle is applicable. In this paper one approach to find out the optimal control law for the problem is discussed. That is,d optimal backward time trajectories are sought in a whole state space by solving one after another the newly established optimal problem with the maximum principle for each region. Usually, costate vector experiences a jump at the discontinuity point but in our method it arises as the result of the solution and any account for jump condition is not required. The mechanism of jump, however, is also examined in this paper.
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