In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe an exact terminal state. Instead, we are looking for a control that steers the system into a given neighborhood of the desired terminal state with sufficiently high probability. This neighborhood is described in terms of an inequality for the terminal energy. The probabilistic constraint in the considered optimal control problem leads to optimal controls that are robust against the inevitable uncertainties of the initial state. Numerical examples with optimal Neumann control of the wave equation are presented.


  1. terminal constraint
  2. uncertain initial data
  3. probabilistic constraint
  4. optimal control
  5. boundary control
  6. wave equation

MSC codes

  1. 49J20
  2. 49J55
  3. 49M37

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Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 2288 - 2311
ISSN (online): 1095-7138


Submitted: 24 June 2019
Accepted: 12 June 2020
Published online: 5 August 2020


  1. terminal constraint
  2. uncertain initial data
  3. probabilistic constraint
  4. optimal control
  5. boundary control
  6. wave equation

MSC codes

  1. 49J20
  2. 49J55
  3. 49M37



M. Hassan Farshbaf-Shaker

Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : B04, C03, C05
Fondation Mathématique Jacques Hadamard https://doi.org/10.13039/501100007493

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