Topological obstructions in the control of partial differential equations

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Control systems governed by partial differential equations are of key theoretical interest as well as of key importance for industrial applications . Such systems abound in real-world applications, from the control of water flow for navigation in locks and channels to the control of heat flow in industrial processes. The control properties of systems governed by partial differential equations is intimately tied to the nature of the governing partial differential equations themselves, and this stands in stark contrast to the finite-dimensional case, where the only relevant line of divide is between (time-invariant) linear and nonlinear systems. Two major problems related to the control of systems are controllability and stabilization; Whereas these are quite well settled and understood for finite-dimensional systems, this is far from being the case for systems governed by nonlinear partial differential equations. In particular, whereas it is now known that there are topological obstructions to local asymptotic stabilization of nonlinear finite dimensional systems via regular feedback laws, no such obstruction is yet known for feedback stabilization of nonlinear partial differential equations. Similarly, whereas the controllability properties of finite-dimensional systems are very well understood, topological obstructions to controllability of nonlinear partial differential equations have been uncovered so far only for very restricted classes of nonlinear partial differential equations. The current proposal is aiming at bridging this gap between finite-dimensional and infinite-dimensional control systems. In particular, it has two goals: The primary goal is to uncover obstructions to stabilization of a system governed by a nonlinear partial differential equation using a suitably regular static state feedback law. Even in this setting, the obstructions are expected to be of a topological nature owing to the isotropy group of the problem. The secondary goal of this proposal is to uncover obstructions to the controllability of nonlinear partial differential equations beyond the classes that have been considered so far in the literature. The problems considered in this proposal are of fundamental theoretical importance in furthering our understanding of systems governed by partial differential equations; they are of fundamental practical significance as well, owing to their numerous real-world applications.