WIAS Preprint No. 2626, (2019)

Optimal Neumann boundary control of a vibrating string with uncertain initial data and probabilistic terminal constraints


  • Farshbaf Shaker, Mohammad Hassan
    ORCID: 0000-0003-0543-5938
  • Gugat, Martin
  • Heitsch, Holger
    ORCID: 0000-0002-2692-4602
  • Henrion, René
    ORCID: 0000-0001-5572-7213

2010 Mathematics Subject Classification

  • 90C15 49J29 9J55


  • PDE constrained optimization, probabilistic constraints, uncertain initial data




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 controls that steer 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. We show the existence of such optimal controls. Numerical examples with optimal Neumann control of the wave equation are presented.

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