This paper derives linear quadratic regulator (LQR) results for boundary-controlled parabolic partial differential equations (PDEs) via weak variations. Research on optimal control of PDEs has a rich 40-year history. This body of knowledge relies heavily on operator and semigroup theory. Our research distinguishes itself by deriving existing LQR results from a more accessible set of mathematics, namely weak-variational concepts. Ultimately, the LQR controller is computed from a Riccati PDE that must be derived for each PDE model under consideration. Nonetheless, a Riccati PDE is a significantly simpler object to solve than an operator Riccati equation, which characterizes most existing results. To this end, our research provides an elegant and accessible method for practicing engineers who study physical systems described by PDEs. Simulation examples, closed-loop stability analyses, comparisons to alternative control methods, and extensions to other models are also included.
Optimal Boundary Control of Reaction–Diffusion Partial Differential Equations via Weak Variations
Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 19, 2012; final manuscript received November 21, 2012; published online February 21, 2013. Assoc. Editor: Rama K. Yedavalli.
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Moura, S. J., and Fathy, H. K. (February 21, 2013). "Optimal Boundary Control of Reaction–Diffusion Partial Differential Equations via Weak Variations." ASME. J. Dyn. Sys., Meas., Control. May 2013; 135(3): 034501. https://doi.org/10.1115/1.4023071
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