Applied Mathematics Seminar | Abdol-Reza Mansour, Topological obstructions to distributed feedback stabilization

We present topological obstructions to the existence of continuously differentiable stabilizing feedback laws for control systems of the form ${\dot x} = f(x,u)$ (with $f$ continuously differentiable), where the individual state dynamics and controls are constrained to depend only on selected subsets of the state variables. Such constraints arise naturally in the setting of distributed and multi-agent control systems. This is in contrast to the classical setting, where individual state dynamics and controls are assumed a priori to depend on all the state variables. We show that the distributed nature of the control system and stabilizing feedback law leads to additional topological obstructions compared to the non-distributed case. These obstructions are expressed in terms of the generators of the homology groups of certain topological spaces derived from the control problem.