Summary
We consider the problem of controlling a system whose state at time t is given by p(t)∈R n, 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 G j (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 G j, 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.
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Franke, J. Optimal navigation with random terminal time in the presence of phase constraints. Z. Wahrscheinlichkeitstheorie verw Gebiete 60, 453–484 (1982). https://doi.org/10.1007/BF00535710
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DOI: https://doi.org/10.1007/BF00535710