Abstract
Consider the problem minu E L 0[z] subject to constraints of the form E L i [z]<-(=)0 and dz=F(t, z, u)dt+σ(t,z) dw. The controls u can be open-loop or feed-back; w is a Brownian motion. Using Girsanov’s approach to define solutions and Neustadt’s theory of extremals, a maximum principle is derived wherein the adjoint variables are the integrands of the stochastic integrals which represent certain martingales determined by the problem.
This work was supported by the National Research Council of Canada under grant A 8051.
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References
V. E. Benes, “Existences of optimal strategies based on specified information, for a class of stochastic decision problems”, SIAM Journal on Control 8 (1970) 179–188.
V. E. Benes, “Existence of optimal stochastic control laws”, SIAM Journal on Control 9 (1971) 446–472.
J. M. Bismut, “An example of optimal stochastic control with constraints”, SIAM Journal on Control 12 (1974) 401–418.
D. L. Burkholder, “Distribution function inequalities for martingales”, The Annals of Probability 1 (1973) 19–42.
J. M. C. Clark, “The representation of functionals of Brownian motion by stochastic integrals”, The Annals of Mathematical Statistics 41 (1970) 1282–1295.
M. H. A. Davis, “On the existence of optimal policies in stochastic control”, SIAM Journal on Control 11 (1973) 587–594.
M. H. A. Davis and P. Varaiya, “Dynamic programming conditions for partially observable stochastic systems”, SIAM Journal on Control, 11 (1973) 226–261.
I. V. Girsanov, “On transforming a certain class of stochastic processes by absolutely continuous substitution of measures”, Theory of Probability and its Applications 5 (1960) 285–301.
P. R. Halmos, Measure theory (Van Nostrand, New York 1950).
U. G. Haussmann, W. J. Anderson and A. Boyarsky, “A new stochastic time optimal control problem, SIAM Journal on Control, to appear.
H. J. Kushner, “Necessary conditions for continuous parameter stochastic optimization problems”, SIAM Journal on Control 10 (1972) 550–565.
L. W. Neustadt, “A general theory of extremals”, Journal of Computer and Systems Sciences 3 (1969) 57–92.
M. Nisio, “On stochastic optimal control laws’, Nagoya Mathematical Journal 52 (1973) 1–30.
K. Yamada, “Continuity of cost functionals in diffusion processes and its application to an existence theorem of optimal controls, Proc. IEEE Conference on Decision and Control, Dec. 1973, San Diego.
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© 1976 The Mathematical Programming Society
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Haussmann, U.G. (1976). General necessary conditions for optimal control of stochastic systems. In: Wets, R.J.B. (eds) Stochastic Systems: Modeling, Identification and Optimization, II. Mathematical Programming Studies, vol 6. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120743
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DOI: https://doi.org/10.1007/BFb0120743
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