An example in linear quadratic optimal control

George Weiss, Hans Zwart

Research output: Contribution to journalArticlepeer-review


We construct a simple example of a quadratic optimal control problem for an infinite-dimensional linear system based on a shift semigroup. This system has an unbounded control operator. The cost is quadratic in the input and the state, and the weighting operators are bounded. Despite its extreme simplicity, this example has all the unexpected features discovered recently by O. Staffans (and also by M. Weiss and G. Weiss). More precisely, in the formula linking the optimal feedback operator to the optimal cost operator, as well as in the Riccati equation, the weighting operator of the input has to be replaced by another operator, which can be derived from the spectral factorization of the Popov function.

Original languageEnglish
Pages (from-to)339-349
Number of pages11
JournalSystems and Control Letters
Issue number5
StatePublished - 16 Apr 1998
Externally publishedYes


  • Infinite-dimensional systems
  • Linear quadratic optimal control
  • Riccati equations
  • Shift semigroups
  • Spectral factorization


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