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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | Print version: Banks, H.T. A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering. Hoboken : CRC Press, ©2012 |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
H T Banks |

ISBN: | 9781439880845 1439880840 |

OCLC Number: | 798535753 |

Description: | 1 online resource (278 pages) |

Contents: | Front Cover; Contents; Preface; Acknowledgments; 1. Introduction to Functional Analysis in Applications; 2. Semigroups and Infinitesimal Generators; 3. Generators; 4. Adjoint Operators and Dual Spaces; 5. Gelfand Triple, Sesquilinear Forms, and Lax-Milgram; 6. Analytic Semigroups; 7. Abstract Cauchy Problems; 8. General Second-Order Systems; 9. Weak Formulations for Second-Order Systems; 10. Inverse or Parameter Estimation Problems; 11. ""Weak"" or ""Variational Form""; 12. Finite Element Approximations and the Trotter-Kato Theorems; 13. Delay Systems: Linear and Nonlinear. 14. Weak* Convergence and the Prohorov Metric in Inverse Problems15. The Prohorov Metric in Optimization and Optimal Design Problems; 16. Control Theory for Distributed Parameter Systems; 17. Families of Approximate Control Problems; References. |

### Abstract:

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Publisher Synopsis

A Modern Framework Based on Time-Tested MaterialA Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering presents functional analysis as a tool for understanding and treating distributed parameter systems. Drawing on his extensive research and teaching from the past 20 years, the author explains how functional analysis can be the basis of modern partial differential equation (PDE) and delay differential equation (DDE) techniques.Recent Examples of Functional Analysis in Biology, Electromagnetics, Materials, and MechanicsThrough numerous application examples, the book illustrates the role that functional analysis-a classical subject-continues to play in the rigorous formulation of modern applied areas. The text covers common examples, such as thermal diffusion, transport in tissue, and beam vibration, as well as less traditional ones, including HIV models, uncertainty in noncooperative games, structured population models, electromagnetics in materials, delay systems, and PDEs in control and inverse problems. For some applications, computational aspects are discussed since many problems necessitate a numerical approach. Introduction to Functional Analysis in Applications Example 1: Heat Equation Some Preliminaries: Hilbert, Banach, and Other Spaces Useful in Operator Theory Return to Example 1: Heat Equation Example 2: General Transport Equation Example 3: Delay Systems-Insect/Insecticide Models Example 4: Probability Measure Dependent Systems - Maxwell's Equations Example 5: Structured Population ModelsSemigroups and Infinitesimal Generators Basic Principles of Semigroups Infinitesimal Generators Generators Introduction to Generation TheoremsHille-Yosida Theorems Results from the Hille-Yosida ProofCorollaries to Hille-Yosida Lumer-Phillips and Dissipative Operators Examples Using Lumer-Phillips TheoremAdjoint Operators and Dual Spaces Adjoint Operators Dual Spaces and Strong, Weak, and Weak* Topologies Examples of Spaces and Their Duals Return to Dissipativeness for General Banach Spaces More on Adjoint Operators Examples of Computing Adjoints Gelfand Triple, Sesquilinear Forms, and Lax-MilgramExample 6: The Cantilever Beam The Beam Equation in the Form x derivative = Ax + F Gelfand TriplesSesquilinear Forms Lax-Milgram (Bounded Form)Lax-Milgram (Unbounded Form)Summary Remarks and MotivationAnalytic SemigroupsExample 1: The Heat Equation (again) Example 2: The Transport Equation (again) Example 6: The Beam Equation (again) Summary of Results on Analytic Semigroup Generation by Sesquilinear FormsTanabe Estimates (on "Regular Dissipative Operators") Infinitesimal Generators in a General Banach Space Abstract Cauchy ProblemsGeneral Second-Order SystemsIntroduction to Second-Order SystemsResults for Ï 2 V-elliptic Results for Ï 2 H-semiellipticStronger Assumptions for Ï 2Weak Formulations for Second-Order SystemsModel Formulation Discussion of the Model Theorems 9.1 and 9.2: ProofsInverse or Parameter Estimation ProblemsApproximation and ConvergenceSome Further Remarks"Weak" or "Variational Form"Finite Element Approximations and the Trotter-Kato TheoremsFinite Elements Trotter-Kato Approximation TheoremDelay Systems: Linear and Nonlinear Linear Delay Systems and ApproximationModeling of Viral Delays in HIV Infection DynamicsNonlinear Delay Systems State Approximation and Convergence for Nonlinear Delay Systems Fixed Delays versus Distributed DelaysWeak* Convergence and the Prohorov Metric in Inverse Problems Populations with Aggregate Data, Uncertainty, and PBMA Prohorov Metric Framework for Inverse Problems Metrics on Probability SpacesExample 5: The Growth Rate Distribution Model and Inverse Problem in Marine PopulationsThe Prohorov Metric in Optimization and Optimal Design Problems Two Player Min-Max Games with Uncertainty Optimal Design Techniques Generalized Curves and Relaxed Controls of Variational Theory Preisach Hysteresis in Smart Materials NPML and Mixing Distributions in Statistical EstimationControl Theory for Distributed Parameter Systems MotivationAbstract Formulation Infinite Dimensional LQR Control: Full State Feedback The Finite Horizon Control ProblemThe Infinite Horizon Control Problem Families of Approximate Control ProblemsThe Finite Horizon Problem Approximate Control GainsThe Infinite Horizon Problem Approximate Control GainsReferencesIndex "The book under review has the valuable advantage of being of interest to both mathematicians and engineers. â ¦ appropriate tools are carefully introduced and discussed in detail, and they are readily applied to practical situations related to the models derived from the generic examples. The main thrust of the book consists of those parts and topics of functional analysis that are fundamental to rigorous discussions of practical differential equations and delay systems as they arise in diverse applications and in particular in control and estimation."-Larbi Berrahmoune, Mathematical Reviews, May 2013 H.T. Banks is a Distinguished University Professor and Drexel Professor of Mathematics at North Carolina State University, where he is also the director of the Center for Research in Scientific Computation and co-director of the Center for Quantitative Sciences in Biomedicine. He currently serves on the editorial boards of 14 journals and has published over 425 papers in applied mathematics and engineering journals. A fellow of the IEEE, IoP, SIAM, and AAAS, Dr. Banks has received numerous honors, including the W.T. and Idalia Reid Prize in Applied Mathematics from SIAM, the Lord Robert May Prize from the Journal of Biological Dynamics, and Best Paper Awards from the ASME and ACS. Introduction to Functional Analysis in Applications Example 1: Heat Equation Some Preliminaries: Hilbert, Banach, and Other Spaces Useful in Operator Theory Return to Example 1: Heat Equation Example 2: General Transport Equation Example 3: Delay Systems-Insect/Insecticide Models Example 4: Probability Measure Dependent Systems - Maxwell's Equations Example 5: Structured Population Models Semigroups and Infinitesimal Generators Basic Principles of Semigroups Infinitesimal Generators Generators Introduction to Generation Theorems Hille-Yosida Theorems Results from the Hille-Yosida Proof Corollaries to Hille-Yosida Lumer-Phillips and Dissipative Operators Examples Using Lumer-Phillips Theorem Adjoint Operators and Dual Spaces Adjoint Operators Dual Spaces and Strong, Weak, and Weak* Topologies Examples of Spaces and Their Duals Return to Dissipativeness for General Banach Spaces More on Adjoint Operators Examples of Computing Adjoints Gelfand Triple, Sesquilinear Forms, and Lax-Milgram Example 6: The Cantilever Beam The Beam Equation in the Form x derivative = Ax + F Gelfand Triples Sesquilinear Forms Lax-Milgram (Bounded Form) Lax-Milgram (Unbounded Form) Summary Remarks and Motivation Analytic Semigroups Example 1: The Heat Equation (again) Example 2: The Transport Equation (again) Example 6: The Beam Equation (again) Summary of Results on Analytic Semigroup Generation by Sesquilinear Forms Tanabe Estimates (on "Regular Dissipative Operators") Infinitesimal Generators in a General Banach Space Abstract Cauchy Problems General Second-Order Systems Introduction to Second-Order Systems Results for sigma2 V-elliptic Results for sigma2 H-semielliptic Stronger Assumptions for sigma2 Weak Formulations for Second-Order Systems Model Formulation Discussion of the Model Theorems 9.1 and 9.2: Proofs Inverse or Parameter Estimation Problems Approximation and Convergence Some Further Remarks "Weak" or "Variational Form" Finite Element Approximations and the Trotter-Kato Theorems Finite Elements Trotter-Kato Approximation Theorem Delay Systems: Linear and Nonlinear Linear Delay Systems and Approximation Modeling of Viral Delays in HIV Infection Dynamics Nonlinear Delay Systems State Approximation and Convergence for Nonlinear Delay Systems Fixed Delays versus Distributed Delays Weak* Convergence and the Prohorov Metric in Inverse Problems Populations with Aggregate Data, Uncertainty, and PBM A Prohorov Metric Framework for Inverse Problems Metrics on Probability Spaces Example 5: The Growth Rate Distribution Model and Inverse Problem in Marine Populations The Prohorov Metric in Optimization and Optimal Design Problems Two Player Min-Max Games with Uncertainty Optimal Design Techniques Generalized Curves and Relaxed Controls of Variational Theory Preisach Hysteresis in Smart Materials NPML and Mixing Distributions in Statistical Estimation Control Theory for Distributed Parameter Systems Motivation Abstract Formulation Infinite Dimensional LQR Control: Full State Feedback The Finite Horizon Control Problem The Infinite Horizon Control Problem Families of Approximate Control Problems The Finite Horizon Problem Approximate Control Gains The Infinite Horizon Problem Approximate Control Gains References Index H.T. Banks is a Distinguished University Professor and Drexel Professor of Mathematics at North Carolina State University, where he is also the director of the Center for Research in Scientific Computation and co-director of the Center for Quantitative Sciences in Biomedicine. He currently serves on the editorial boards of 14 journals and has published over 425 papers in applied mathematics and engineering journals. A fellow of the IEEE, IoP, SIAM, and AAAS, Dr. Banks has received numerous honors, including the W.T. and Idalia Reid Prize in Applied Mathematics from SIAM, the Lord Robert May Prize from the Journal of Biological Dynamics, and Best Paper Awards from the ASME and ACS. A Modern Framework Based on Time-Tested Material A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering presents functional analysis as a tool for understanding and treating distributed parameter systems. Drawing on his extensive research and teaching from the past 20 years, the author explains how functional analysis can be the basis of modern partial differential equation (PDE) and delay differential equation (DDE) techniques. Recent Examples of Functional Analysis in Biology, Electromagnetics, Materials, and Mechanics Through numerous application examples, the book illustrates the role that functional analysis-a classical subject-continues to play in the rigorous formulation of modern applied areas. The text covers common examples, such as thermal diffusion, transport in tissue, and beam vibration, as well as less traditional ones, including HIV models, uncertainty in noncooperative games, structured population models, electromagnetics in materials, delay systems, and PDEs in control and inverse problems. For some applications, computational aspects are discussed since many problems necessitate a numerical approach. Read more...

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