1. Imagine that your system is an airplane and you need to land it. Formulate and optimization task that will allow you to safely touch the ground at the airport nearby. What is the objective function? What are the control parameters, and the restrictions?
The optimization task is to bring the plane to height = 0 at the location of the runway, with the plane's vector parallel with the runway and the change in altitude of the plane as close to zero as possible.
The objective function is the change in altitude of the plane at the time of contact. The control parameters are the speed and course of the plane leading up to landing. The constraints are the location and direction of travel of the plane at the time of landing.
2. Think of an optimization of your own. Define the system, the objective function, the controls and restrictions.
Let's say we want to optimize the hydrology below a surface mine. The system includes precipitation, upland runoff, the topography, geology, and vegetation of the mine, and the topography, geology, and ecology--including, potentially, humans--below the mine.
Let's make the objective function the difference between post-mining flow and pre-mining flow. The controls include placement of overburden, compaction of replaced topsoil, vegetation planted, and potential settling ponds and/or dykes downstream of the mine. The constraints include the surrounding topography, precipitation and other climatic variables, and upstream hydrology.