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New proof and some generalizations of the minimum principle in optimal control

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An optimal control problem is reduced to the finite-dimensional problem of minimizing the terminal payoff over the intersection of the target set with the reachable set. The pointwise Pontryagin minimum principle is derived from two simple preliminary results: the first states that the intersection of two inseparable derived cones at a common point of two given sets is contained in the quasitangent cone (hence, in the contingent cone) to their intersection; the second identifies a derived cone to the reachable set. The standard variants of the minimum principle are easily generalized to problems defined by non-differentiable terminal payoffs on arbitrary target sets.

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Communicated by L. D. Berkovitz

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MiricĂ, S. New proof and some generalizations of the minimum principle in optimal control. J Optim Theory Appl 74, 487–508 (1992).

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Key Words

  • Optimal control problems
  • minimum principle
  • derived cones
  • quasitangent cones
  • contingent cones
  • superdifferentials
  • generalized gradients