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In mathematics or engineering, a control problem consists of finding a time-varying control strategy such that a dynamical system behaves in a desired way. A real-world example would be an electric network with a large share of renewable energies (hence with a lot of temporal variation in energy production), in which a large group of batteries is charged or uncharged in order to keep the time-varying mismatch between energy production and demand approximately constant. The control strategy would then be a rule, depending on time and on the past, current and predicted power production and consumption, when to charge or discharge each of the batteries.

Optimal control then means that one tries to find a control strategy that minimizes or maximizes a certain value. In the example above, this could for instance be the aggregated deviation of the mismatch from a given reference value over a certain time interval. In this lecture I will introduce several different types of optimal control problems by means of examples and then explain the most important algorithms for solving them.