Stepwise Solutions for Optimal Control Problems

Abstract

A new stepwise method for solving optimal control problems is introduced. The main motivation for developing this new approach is the limitation of the continuous-time Pontryagin Maximum Principle (PMP) where all control functions must be continuous. However, in many real-world applications such as drug injection or resource allocation problems, it is not practical to continuously change the control. In dealing with these problems it is strictly preferred to change the control only at certain moments of time and keep it constant otherwise. Clearly, in this case the resulting stepwise solution cannot be calculated optimally using PMP since it is not continuous anymore. The other advantage of stepwise solutions is that they can be obtained much easier compared to the PMP approach when the system has complex dynamics or the cost function is more complicated. Some numerical examples are solved by using both the classical PMP and the proposed stepwise method and the results are compared, which prove the high performance of the proposed method.

Keywords

• Optimal control theory,Heuristic and meta-heuristic optimization,Nonlinear dynamics

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