War on Boundary Conditions - A Control-Oriented Framework for Partial Differential Equations

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The joke goes like this: "Fusion power is just 20 years away and always will be". Why 20 years? Because that is how long it takes to build a tokamak fusion reactor. Why always? Because for a tokamak to produce energy, it must control a 10M degree plasma using a magnetic field, a dynamic process governed by six coupled nonlinear Partial-Differential Equations (PDEs) in two spatial variables and no real progress has ever been made on feedback control of such systems. The goal of the project, then, is to make substantial progress on the control of coupled PDEs. Such progress is through foundational advances in mathematics and software infrastructure which will enable the broader PDE controls community to grow. Specifically, by developing a universal, easily-understood and applied computational framework for the control of PDEs, this research will provide the tools needed to allow controls engineers with limited or no PDE experience to design reliable and effective controllers for these systems. Since nuclear fusion reactors have no long-term radioactive waste, and since fuel for these reactors can be extracted from sea water, progress in this area has the potential for creating an unlimited supply of clean energy. Furthermore, this work has the potential to advance several applications beyond nuclear fusion including hypersonic vehicles, traffic management, soft robotics, cavitation, flow control, and vibration suppression.The goal of the project is to replace Partial Differential Equations (PDEs) with Partial Integral Equations (PIEs). Historically, PDEs are defined by three sometimes contradictory sets of equations and constraints: the PDE itself, which moves the state; the Boundary Conditions (BCs), which implicitly constrain the motion of the state; and the continuity conditions, which couple the BCs to constraints on the motion of the state. By contrast, PIEs combine PDE, BCs, and continuity conditions into a single equation, defined by bounded Partial-Integral (PI) operators, any solution to which satisfies the original PDE and requires no boundary conditions or continuity constraints. PI operators are bounded, form an algebra, and can be parameterized using matrices. Further, positivity of PI operators can be enforced using Linear Matrix Inequalities (LMIs). This means that algorithms developed for control of ODEs using LMIs can be generalized to control of PIEs with relatively little effort. This project will pursue the generalization of several such LMIs to PDEs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.