{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71272"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71272","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On the Smoothness of the Bellman Function","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Tanner, Franz Xaver"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Albrecht, Felix"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:28Z","date_published":"2014-12-16T06:18:28Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8908864"],"render_values":[{"text":"(UMI)AAI8908864","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71272","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Albrecht, Felix"]},{"key":"dc:creator","label":"Author","values":["Tanner, Franz Xaver"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:28Z","10000-01-01","1988"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71272","(UMI)AAI8908864"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Made available in DSpace on 2014-12-16T06:18:28Z (GMT). No. of bitstreams: 1 8908864.pdf: 2057709 bytes, checksum: 19aa9abde2c06d31d56a3080ca626fae (MD5) Previous issue date: 1988","Embargo set by: Seth Robbins for item 71438 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","69 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988."]},{"key":"dc:title","label":"Title","values":["On the Smoothness of the Bellman Function"]}]}],"canonical_facts":{"dc:contributor":["Albrecht, Felix"],"dc:creator":["Tanner, Franz Xaver"],"dc:date":["2014-12-16T06:18:28Z","10000-01-01","1988"],"dc:description":["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.","Made available in DSpace on 2014-12-16T06:18:28Z (GMT). No. of bitstreams: 1 8908864.pdf: 2057709 bytes, checksum: 19aa9abde2c06d31d56a3080ca626fae (MD5) Previous issue date: 1988","Embargo set by: Seth Robbins for item 71438 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","69 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988."],"dc:identifier":["http://hdl.handle.net/2142/71272","(UMI)AAI8908864"],"dc:subject":["Mathematics"],"dc:title":["On the Smoothness of the Bellman Function"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}