WIAS Preprint No. 1231, (2007)

Optimal control problems with delays in state and control and mixed control-state constraints


  • Göllmann, Laurenz
  • Kern, Daniela
  • Maurer, Helmut

2010 Mathematics Subject Classification

  • 49K15 49K25


  • Retarded optimal control problems, delays in state and control, mixed control-state inequality constraints, Pontryagin's minimum principle, discretization methods, optimal control of a CSTR reactor, optimal fishing


Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control-state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods for the delayed control problem are discussed which amount to solving a large-scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and two numerical examples from chemical engineering and economics illustrate the results.

Appeared in

  • Optimal Control Appl. Methods, 30 (2009) pp. 341--365.

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