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|Title:||On the Smoothness of the Bellman Function|
|Author(s):||Tanner, Franz Xaver|
|Doctoral Committee Chair(s):||Albrecht, Felix|
|Department / Program:||Mathematics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|Abstract:||In Optimal Control Theory, necessary and sufficient conditions for the optimality of a given control, under the assumption that the Bellman function is continously differentiable, can be established by the method of dynamic programming. However, as simple examples show, the Bellman function is in general not differentiable everywhere on the attainable set. Boltyanskii introduced the concept of a regular synthesis for a control system which subsequently leads to the proof that the Bellman function is continously differentiable on an open and dense subset of the attainable set. The existence of regular syntheses for various control systems has since been established by Brunovsky and Sussmann. The works of these authors rely heavily on results from the theory of subanalytic sets due to Hardt and Hironaka.
In this thesis the regularity of the Bellman function is investigated directly for linear control systems with scalar controls, omitting the construction of a regular synthesis. This approach circumvents the use of the theory of subanalytic sets, yielding at the same time a more constructive existence proof for the synthesis.
The main result establishes the existence of a partition of the attainable set into a finite number of submanifolds with the property that the restriction of the Bellman function to each of these submanifolds is analytic. The elements of this partition are defined constructively and provide the essential tool for obtaining an optimal feedback.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988.
|Date Available in IDEALS:||2014-12-16|