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

  • Jürgen Franke


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.


Cost Function Vector Space Mathematical Biology Target Condition Optimal Path 
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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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