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