Supervisory controller for a moving control source in a 2-D diffusion process

The objective of this note is to provide a guidance scheme of an actuator capable of moving across a 2-D spatial domain. The process under consideration is described by a 2-D diffusion equation and it is assumed that a single heat source (actuator) mounted on a robotic arm is capable of moving at a priori selected points in the spatial domain and injecting the heat input to the system at these points. Utilizing Lyapunov methods aimed at minimizing the total energy of the system, the guidance scheme provides the position of the heat source at discrete time intervals. Using LMIs, a control signal (heat intensity) is found that is common to all of the a priori selected points that the actuator can reside at during a given time interval. Numerical studies of the 2-D parabolic system are included to provide insight into the applicability of a moving actuator in thermal processes.