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Optimal navigation with random terminal time in the presence of phase constraints

  • Jürgen Franke
Article
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Summary

We consider the problem of controlling a system whose state at time t is given by p(t)∈Rn, where we assume that we can choose the velocity r(t) of p(t) and the terminal time of control ζ in an arbitrary manner, restricted only by the target condition Z(p(ζ))≦0, the phase constraints Gj(p(t))≦0, j=1,..., J for all t≦ζ, and the requirement that the norm of r is either essentially bounded or a.s. constant. For given cost function S the loss functional to be minimized is given by \(\mathfrak{E}S(p(T\zeta ),T \wedge \zeta )\), where T is a nonnegative random variable with known distribution P. So we control the state effectively only up to the random terminal time T ∧ζ.

By means of the technique of Dubovitskij and Milyutin for the treatment of extremum problems in locally convex topological vector spaces, which turns out to be a powerful tool in the stochastic setting too, we derive necessary conditions on optimal controls under rather general assumptions on P,S,Z and Gj, j=1,..., J. In an important special case where we consider simple phase constraints and monotone cost function S the general theorems allow a rather complete description of locally optimal paths in simple form.

Keywords

Cost Function Vector Space Mathematical Biology Target Condition Optimal Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Jürgen Franke
    • 1
  1. 1.Fachbereich MathematikUniversität FrankfurtFrankfurt a.M.Federal Republic of Germany

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