A variational derivation of LQR for piecewise time-varying systems

Alex Ansari, Todd David Murphey

Research output: Chapter in Book/Report/Conference proceedingConference contribution


This paper provides a complete derivation for LQR optimal controllers and the optimal value function using basic principles from variational calculus. As opposed to alternatives, the derivation does not rely on the Hamilton-Jacobi-Bellman (HJB) equations, Pontryagin's Maximum Principle (PMP), or the Euler Lagrange (EL) equations. Because it requires significantly less background, the approach is educationally instructive. It provides a different perspective of how and why key quantities such as the adjoint variable and Riccati equation show up in optimal control computations and their connection to the optimal value function. Additionally, the derivation presented requires fewer regularity assumptions than necessary in applying the HJB or EL equations. As with PMP, the methods in this paper apply to systems and controls that are piecewise continuous in time.

Original languageEnglish (US)
Title of host publicationACC 2015 - 2015 American Control Conference
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages6
ISBN (Electronic)9781479986842
StatePublished - Jan 1 2015
Event2015 American Control Conference, ACC 2015 - Chicago, United States
Duration: Jul 1 2015Jul 3 2015


Other2015 American Control Conference, ACC 2015
Country/TerritoryUnited States

ASJC Scopus subject areas

  • Electrical and Electronic Engineering


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