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 AbstractIn this thesis we develop a rigorous mathematical framework foranalyzing 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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