ISSN:
1573-2878
Schlagwort(e):
Optimal stochastic control
;
hitting probabilities
Quelle:
Springer Online Journal Archives 1860-2000
Thema:
Mathematik
Notizen:
Abstract A two-dimensional random motion of a point is dealt with. The point velocity (v cos ϑ,v sin ϑ) is subjected to two different kinds of perturbations, the first represented by a vector of independent standard Wiener processes and the second by a generalized type of Poisson process. The control function is ϑ, whilev is kept fixed. We assume given a configuration ofn+1 target sets,A 0,...,A n , in the plane, all of these sets being surrounded by an open and bounded setD. We denote by ℙ x (ℙ(ϑ)∈A i the probability thatX t , the location of the point, whereX 0=x∈D, will reach the setA i beforeX i reaches any other setA j ,i≠j, and before it leavesD. The problem dealt with here is to find an optimal control law ϑ*, ϑ*=ϑ*(x),x∈D, such that the function $$V\left( {x;\theta } \right) = \sum\limits_{i = 0}^n {\lambda _i \mathbb{P}_x \left( {X_\tau \left( \theta \right) \in A_i } \right),}$$ where λ1,i=0,...,n, are given nonnegative numbers, will be maximized on a given class of admissible control laws. Sufficient conditions on optimal controls, of a dynamic programming type, are derived. These conditions require the existence of a smooth solution to a nonlinear partial integrodifferential equation, which is solved here by applying a finite-difference scheme. Two examples are dealt with numerically.
Materialart:
Digitale Medien
URL:
http://dx.doi.org/10.1007/BF00935891
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