A convex sum-of-squares approach to analysis, state feedback and output feedback control of parabolic PDEs

Aditya Gahlawat, Matthew Peet

Research output: Contribution to journalArticlepeer-review

25 Scopus citations


We present an optimization-based framework for analysis and control of linear parabolic partial differential equations (PDEs) with spatially varying coefficients without discretization or numerical approximation. For controller synthesis, we consider both full-state feedback and point observation (output feedback). The input occurs at the boundary (point actuation). We use positive-definite matrices to parameterize positive Lyapunov functions and polynomials to parameterize controller and observer gains. We use duality and an invertible state variable transformation to convexify the controller synthesis problem. Finally, we combine our synthesis condition with the Luenberger observer framework to express the output feedback controller synthesis problem as a set of LMI/SDP constraints. We perform an extensive set of numerical experiments to demonstrate the accuracy of the conditions and to prove the necessity of the Lyapunov structures chosen. We provide numerical and analytical comparisons with alternative approaches to control, including Sturm-Liouville theory and backstepping. Finally, we use numerical tests to show that the method retains its accuracy for alternative boundary conditions.

Original languageEnglish (US)
Article number7517344
Pages (from-to)1636-1651
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number4
StatePublished - Apr 2017


  • Control design
  • distributed parameter systems
  • partial differential equations (PDEs)
  • sum of squares

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


Dive into the research topics of 'A convex sum-of-squares approach to analysis, state feedback and output feedback control of parabolic PDEs'. Together they form a unique fingerprint.

Cite this