Title page for ETD etd-04212010-100919


Type of Document Dissertation
Author Rautenberg, Carlos Nicolas
URN etd-04212010-100919
Title A Distributed Parameter Approach to Optimal Filtering and Estimation with Mobile Sensor Networks
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Burns, John A. Committee Chair
Borggaard, Jeffrey T. Committee Member
Cliff, Eugene M. Committee Member
Herdman, Terry L. Committee Member
Zietsman, Lizette Committee Member
Keywords
  • Riccati Equation
  • Kalman Filter
  • Mobile Sensor Networks
  • Optimal Filtering
Date of Defense 2010-03-31
Availability unrestricted
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 $\I_p$-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 $\I_p$. 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 $\I_p$-valued functions improves

convergence properties of strongly continuous approximating mappings

and specifically approximating $C_0$-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 $\I_p$-norm. Numerical examples are

given to illustrate the theory.

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