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An existence theorem for optimal control systems with state variable inC, and stochastic control problems

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Abstract

We consider an existence theorem for control systems whose state variables for everyt are inC, the set of continuous functions varying over a given setI. The dependence of the state variables upona ε I is induced by their dependence upon the initial state and the state equation governing the system. In contrast, the controlu=u(t) is taken as a measurable function oft alone. The usual space constraints and boundary conditions are also allowed to vary overaεI, and the cost functional is now taken to be a continuous functional over a suitable class of continuous functions. We also discuss an application of these results to control systems with stochastic boundary conditions.

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References

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    Kelley, J. L.,General Topology, D. Van Nostrand Company, Princeton, New Jersey, 1955.

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    Cesari, L.,Existence Theorems for Weak and Usual Optimal Solutions in Lagrange Problems with Unilateral Constraints, I, Transactions of the American Mathematical Society, Vol. 124, No. 3, 1966.

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    Baum, R.,Optimal Control Systems with Stochastic Boundary Conditions, University of Michigan, Ph.D. Thesis, 1969.

Additional Bibliography

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    Fleming, W. H.,Optimal Continuous-Parameter Stochastic Control, SIAM Review, Vol. 11, No. 4, 1969.

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    Kushner, H. J.,On the Existence of Optimal Stochastic Controls, SIAM Journal on Control, Vol. 3, No. 3, 1965.

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Additional information

This research was accomplished under Grant No. AF-AFOSR-942-65. The author is grateful to Dr. Lamberto Cesari for his suggestions and assistance in the preparation of this paper.

Communicated by L. Cesari

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Baum, R.F. An existence theorem for optimal control systems with state variable inC, and stochastic control problems. J Optim Theory Appl 5, 335–346 (1970). https://doi.org/10.1007/BF00928670

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Keywords

  • Boundary Condition
  • Control System
  • Continuous Function
  • Control Problem
  • Measurable Function