An energy control problem is analyzed in the PDE (partial differential equation) setting. As opposed to the existing literature, the present control goal addresses not only decreasing the plant energy but also its increasing what is important, e.g., in vibrational technologies, in studying wave motion, etc. A benchmark linear wave equation, governing 1-D (one-dimensional) string oscillations, is chosen for exposition. A distributed control input, independently enforcing the underlying string over its entire spatial location, is assumed to be available. The speed-gradient method (Fradkov, 1996) is presently developed and justified in the above PDE setting. The applicability of the Krasovskii-LaSalle principle is established for the resulting sliding-mode closed-loop system in the infinite-dimensional setting. By applying this principle, all the closed-loop trajectories, initialized beyond the origin, are shown to approach the desired energy level set. Capabilities of the proposed speed-gradient algorithm of reaching the energy goal are supported by numerical simulations. The obtained results constitute the first step to justify properties of feedback energy control for PDE systems important for application in physics.
Scopus subject areas
- Applied Mathematics
- Electrical and Electronic Engineering