Abstract
Consider the random motion in the plane of a pointM, whose velocityv=(v 1,v 2) is perturbed by an ℝ2-valued Gaussian white noise. Only noisy nonlinear observations taken on the point location (state) are available toM. The velocityv is of the formv(y)=∭ u (u 1,u 2)μ y (du), wherey denotes the value of the observed signal,U is the range of the velocity, and, for eachy, μ y is a probability measure on ℬ(U). Using the available observations, the pointM wishes to steer itself into a given target set by choosing a randomized strategy μ={μ y :y ∈ ∝2}. Sufficient conditions on weak optimal randomized strategies are derived. An algorithm for computing weak suboptimal randomized strategies is suggested, and the strategies are computed for a variety of cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Stroock, D. W., andVaradhan, S. R. S.,Diffusion Processes with Continuous Coefficients, Communications on Pure and Applied Mathematics, Vol. 22, pp. 345–400 and pp. 479–530, 1969.
Vermes, D.,Extremality Properties of the Optimal Strategy in Markovian Control Problems, Analysis and Optimization of Stochastic Systems, Edited by O. L. R. Jacobs, M. H. A. Davis, M. A. H. Dempster, C. J. Harris, and P. C. Parks, Academic Press, New York, New York, pp. 35–47, 1980.
Fleming, W. H.,Optimal Control of Partially Observable Diffusions, SIAM Journal on Control, Vol. 6, pp. 194–214, 1968.
Davis, M. H. A., andVaraiya, P.,Dynamic Programming Conditions for Partially Observable Stochastic Systems, SIAM Journal on Control, Vol. 11, pp. 226–261, 1973.
Ahmed, N. U., andTeo, K. L.,Necessary Conditions for Optimality of Cauchy Problems for Parabolic Partial Differential Systems, SIAM Journal on Control, Vol. 13, pp. 981–993, 1975.
Ahmed, N. U.,Optimal Control of Stochastic Systems, Probabilistic Analysis and Related Topics, Edited by A. T. Bharucha-Reid, Academic Press, New York, New York, Vol. 2, pp. 1–68, 1979.
Christopeit, N.,Existence of Optimal Stochastic Controls under Partial Observation, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete Vol. 51, pp. 201–213, 1980.
Christopeit, N.,Optimal Stochastic Control with Special Information Patterns, SIAM Journal on Control and Optimization, Vol. 18, pp. 559–575, 1980.
Elliott, R. J., andVaraiya, P. P.,A Sufficient Condition for the Optimal Control of a Partially Observed Stochastic System, Analysis and Optimization of Stochastic Systems, Edited by O. L. R. Jacobs, M. H. A. Davis, M. A. H. Dempster, C. J. Harris, and P. C. Parks, Academic Press, New York, New York, pp. 11–20, 1980.
Fleming, W. H.,Stochastic Control under Partial Observations, Analysis and Optimization of Systems, Springer-Verlag, Berlin, Germany, pp. 308–317, 1980.
Yavin, Y., andReuter, G. W.,Computation of Nash Equilibrium Pairs of a Stochastic Differential Game, Optimal Control Applications and Methods, Vol. 2, pp. 225–238, 1981.
Huisman, W. C., andYavin, Y.,Numerical Studies of the Performance of an Optimally Controlled Nonlinear Stochastic Oscillator, Computer Methods in Applied Mechanics and Engineering, Vol. 21, pp. 171–191, 1980.
Author information
Authors and Affiliations
Additional information
Communicated by G. Leitmann
This work was partially supported by a grant from Control Data.
Rights and permissions
About this article
Cite this article
Yavin, Y. Computation of suboptimal randomized strategies for steering the random motion of a point under partial observation. J Optim Theory Appl 44, 159–179 (1984). https://doi.org/10.1007/BF00934899
Issue Date:
DOI: https://doi.org/10.1007/BF00934899