{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/27103"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/27103","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"A Distributed Parameter Approach to Optimal Filtering and Estimation with Mobile Sensor Networks","abstract":"In this thesis we develop a rigorous mathematical framework for analyzing and approximating optimal sensor placement problems for distributed parameter systems and apply these results to PDE problems defined by the convection-diffusion equations. The mathematical problem is formulated as a distributed parameter optimal control problem with integral Riccati equations as constraints. In order to prove existence of the optimal sensor network and to construct a framework in which to develop rigorous numerical integration of the Riccati equations, we develop a theory based on Bochner integrable solutions of the Riccati equations. In particular, we focus on ℐ<sub>p</sub>-valued continuous solutions of the Bochner integral Riccati equation. We give new results concerning the smoothing effect achieved by multiplying a general strongly continuous mapping by operators in ℐ<sub>p</sub>. These smoothing results are essential to the proofs of the existence of Bochner integrable solutions of the Riccati integral equations. We also establish that multiplication of continuous ℐ<sub>p</sub>-valued functions improves convergence properties of strongly continuous approximating mappings and specifically approximating C₀-semigroups. We develop a Galerkin type numerical scheme for approximating the solutions of the integral Riccati equation and prove convergence of the approximating solutions in the ℐ<sub>p</sub>-norm. Numerical examples are given to illustrate the theory.","abstract_html":"In this thesis we develop a rigorous mathematical framework for analyzing and approximating optimal sensor placement problems for distributed parameter systems and apply these results to PDE problems defined by the convection-diffusion equations. The mathematical problem is formulated as a distributed parameter optimal control problem with integral Riccati equations as constraints. In order to prove existence of the optimal sensor network and to construct a framework in which to develop rigorous numerical integration of the Riccati equations, we develop a theory based on Bochner integrable solutions of the Riccati equations. In particular, we focus on ℐ&lt;sub&gt;p&lt;/sub&gt;-valued continuous solutions of the Bochner integral Riccati equation. We give new results concerning the smoothing effect achieved by multiplying a general strongly continuous mapping by operators in ℐ&lt;sub&gt;p&lt;/sub&gt;. These smoothing results are essential to the proofs of the existence of Bochner integrable solutions of the Riccati integral equations. We also establish that multiplication of continuous ℐ&lt;sub&gt;p&lt;/sub&gt;-valued functions improves convergence properties of strongly continuous approximating mappings and specifically approximating C₀-semigroups. We develop a Galerkin type numerical scheme for approximating the solutions of the integral Riccati equation and prove convergence of the approximating solutions in the ℐ&lt;sub&gt;p&lt;/sub&gt;-norm. Numerical examples are given to illustrate the theory.","abstract_has_math":false,"creators":["Rautenberg, Carlos Nicolas"],"institution":"Virginia Tech","degree_name":"Ph. 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We develop a Galerkin type numerical scheme for approximating the solutions of the integral Riccati equation and prove convergence of the approximating solutions in the ℐ<sub>p</sub>-norm. Numerical examples are given to illustrate the theory."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["A Distributed Parameter Approach to Optimal Filtering and Estimation with Mobile Sensor Networks"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Burns, John A."],"dc:contributor.committeemember":["Zietsman, Lizette","Herdman, Terry L.","Cliff, Eugene M.","Borggaard, Jeffrey T."],"dc:contributor.department":["Mathematics"],"dc:creator":["Rautenberg, Carlos Nicolas"],"dc:date.accessioned":["2014-03-14T20:10:22Z"],"dc:date.available":["2014-03-14T20:10:22Z","2010-05-05"],"dc:date.issued":["2010-03-31"],"dc:description.abstract":["In this thesis we develop a rigorous mathematical framework for analyzing and approximating optimal sensor placement problems for distributed parameter systems and apply these results to PDE problems defined by the convection-diffusion equations. The mathematical problem is formulated as a distributed parameter optimal control problem with integral Riccati equations as constraints. In order to prove existence of the optimal sensor network and to construct a framework in which to develop rigorous numerical integration of the Riccati equations, we develop a theory based on Bochner integrable solutions of the Riccati equations. In particular, we focus on ℐ<sub>p</sub>-valued continuous solutions of the Bochner integral Riccati equation. We give new results concerning the smoothing effect achieved by multiplying a general strongly continuous mapping by operators in ℐ<sub>p</sub>. These smoothing results are essential to the proofs of the existence of Bochner integrable solutions of the Riccati integral equations. We also establish that multiplication of continuous ℐ<sub>p</sub>-valued functions improves convergence properties of strongly continuous approximating mappings and specifically approximating C₀-semigroups. 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