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

这个笑话是这样的：“核聚变的力量只有20年的时间，而且永远都会是。”为什么是20年？因为这就是建造托卡马克聚变反应堆所需的时间。为什么总是这样？由于托卡马克要产生能量，它必须利用磁场控制10M度的等离子体，这是一个由两个空间变量的六个耦合的非线性偏微分方程组(PDE)控制的动态过程，在这类系统的反馈控制方面一直没有取得真正的进展。因此，该项目的目标是在控制耦合偏微分方程方面取得实质性进展。这种进步是通过数学和软件基础设施方面的基础性进步来实现的，这将使更广泛的PDE控制社区得以发展。具体地说，通过开发一个通用的、易于理解的和实用的PDE控制计算框架，这项研究将提供所需的工具，使具有有限或没有PDE经验的控制工程师为这些系统设计可靠而有效的控制器。由于核聚变反应堆没有长期的放射性废物，而且这些反应堆的燃料可以从海水中提取，这一领域的进展有可能创造无限的清洁能源供应。此外，这项工作有可能推进核聚变以外的几个应用，包括高超声速飞行器、交通管理、软机器人、空化、流动控制和振动抑制。该项目的目标是用偏积分方程组(PIE)取代偏微分方程组(PDE)。在历史上，PDE由三组有时相互矛盾的方程和约束来定义：PDE本身，它移动状态；边界条件(BCS)，隐含地约束状态的运动；以及连续性条件，将BCS耦合到对状态运动的约束。相比之下，PIE将PDE、BCS和连续性条件组合到由有界部分积分(PI)算子定义的单个方程中，其任何解都满足原始PDE，并且不需要边界条件或连续性约束。PI运算符是有界的，形成一个代数，并且可以使用矩阵进行参数化。此外，PI算子的正性可以使用线性矩阵不等式(LMI)来加强。这意味着为使用LMI控制ODE而开发的算法可以被推广到相对较少的努力来控制Pie。这个项目将把几个这样的LMI推广到PDE。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.