For certain source terms linear systems cannot be stabilized from the boundary!

What is it about?

It is important to know the limits of what can possibly be achieved with the control tools that are available. In this paper we show that a source term in a networked linear system can make the boundary feedback stabilization impossible. We consider a star-shaped network and show that for certain source terms the stabilizability becomes impossible if one of the edges is too long, but also if the sum of all the lengths is too long.

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Photo by Joseph Barrientos on Unsplash

Photo by Joseph Barrientos on Unsplash

Why is it important?

In engineering models, different types of source terms appear, in particular to model friction. In linear models, the properties of the corresponding matrix have an essential influence on the stabilizatbility properties of the systems. If the eigenvalues of the symmetrized matrix do not have the same and the right sign, it can happen that boundary feedback stabilization is only possible if the space interval is sufficiently small. This has been pointed out by Georges Bastin and Jean-Michel Coron in their book on stabilizability. They have studied for example the boundary feedback stabilization of water channels that are governed by the Saint Venant equations. In this paper, we study the corresponding phenomenon for networked systems.

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