Abstract
For finite dimensional linear systems, under very mild regularity assumptions, the reachable set for a compact control set is closed and convex, regardless of the initial state. This fact is significant in understanding the time optimal control problem and in the design of computational algorithms for producing optimal controls. For bilinear systems the reachable set is typically not convex and not even simply connected although Filippov’s theorem [1) shows that it is closed if the control set is compact and convex. Sussmann [2] has shown that it need not be closed for a compact control set, and examples abound indicating that its connectivity depends heavily on the initial state.
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References
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Brockett, R.W. (1975). On the Reachable Set for Bilinear Systems. In: Ruberti, A., Mohler, R.R. (eds) Variable Structure Systems with Application to Economics and Biology. Lecture Notes in Economics and Mathematical Systems, vol 111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-47457-6_3
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DOI: https://doi.org/10.1007/978-3-642-47457-6_3
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