We analyze the sensitivity of the extremal equations that arise from the first order optimality conditions for time dependent optimization problems. More specifically, we consider parabolic PDEs with distributed or boundary control and a linear quadratic performance criterion. We prove the solution's boundedness with respect to the right-hand side of the first order optimality condition which includes initial data. If the system fulfills a particular stabilizability and detectability assumption, the bound is independent of the time horizon. As a consequence, the influence of a perturbation of the right-hand side decreases exponentially backward in time. We use this property for the construction of efficient numerical discretizations in a model predictive control scheme. Moreover, a quantitative turnpike theorem in the $W([0,T])$-norm is derived.


  1. sensitivity analysis
  2. turnpike property
  3. model predictive control

MSC codes

  1. 49K20
  2. 49K40
  3. 93D20

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Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 2753 - 2774
ISSN (online): 1095-7138


Submitted: 26 October 2018
Accepted: 21 May 2019
Published online: 6 August 2019


  1. sensitivity analysis
  2. turnpike property
  3. model predictive control

MSC codes

  1. 49K20
  2. 49K40
  3. 93D20



Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : GR 1569/17-1, SCHI 1379/5-1

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