Optimization of measurement points positioning in a border control synthesis problem for the process of heating a rod

With the example of boundary control for the rod heating process, we study the synthesis problem for controlling boundary conditions in systems with distributed parameters. Feedback is provided by measuring the state of the process at internal points whose values are used to construct the control. We propose a formulation of the synthesis problem where both measurement points and the time of measurements are optimized. To solve the problem, we derive formulas necessary to apply efficient first order numerical optimization methods. We show the results of numerical experiments.