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Optimization and dynamical systems (Communications and control engineering series), 1st ed 1994, corr. 2nd printing 1996, 1994 Communications and Control Engineering Series

Langue : Anglais

Auteurs :

Préfacier : Brockett R.

Couverture de l’ouvrage Optimization and dynamical systems (Communications and control engineering series), 1st ed 1994, corr. 2nd printing 1996
This study addresses both classical and previously unsolved optimization tasks in linear algebra, linear systems theory, control theory and signal processing. Exploiting developments in Computation such as Parallel Computing and Neural Networks, it gives a dynamical systems approach for tackling a wide class of constrained optimization tasks.Optimization and Dynamical Systems will be of interest to engineers and mathematicians. Engineers will learn the mathematics and the technical approach necessary to solve a wide class of constrained optimization tasks. Mathematicians will see how techniques from global analysis and differential geometry can be developed to achieve useful construction procedures for optimization on manifolds.
1 Matrix Eigenvalue Methods.- 1.1 Introduction.- 1.2 Power Method for Diagonalization.- 1.3 The Rayleigh Quotient Gradient Flow.- 1.4 The QR Algorithm.- 1.5 Singular Value Decomposition (SVD).- 1.6 Standard Least Squares Gradient Flows.- 2 Double Bracket Isospectral Flows.- 2.1 Double Bracket Flows for Diagonalization.- 2.2 Toda Flows and the Riccati Equation.- 2.3 Recursive Lie-Bracket Based Diagonalization.- 3 Singular Value Decomposition.- 3.1 SVD via Double Bracket Flows.- 3.2 A Gradient Flow Approach to SVD.- 4 Linear Programming.- 4.1 The Rôle of Double Bracket Flows.- 4.2 Interior Point Flows on a Polytope.- 4.3 Recursive Linear Programming/Sorting.- 5 Approximation and Control.- 5.1 Approximations by Lower Rank Matrices.- 5.2 The Polar Decomposition.- 5.3 Output Feedback Control.- 6 Balanced Matrix Factorizations.- 6.1 Introduction.- 6.2 Kempf-Ness Theorem.- 6.3 Global Analysis of Cost Functions.- 6.4 Flows for Balancing Transformations.- 6.5 Flows on the Factors X and Y.- 6.6 Recursive Balancing Matrix Factorizations.- 7 Invariant Theory and System Balancing.- 7.1 Introduction.- 7.2 Plurisubharmonic Functions.- 7.3 The Azad-Loeb Theorem.- 7.4 Application to Balancing.- 7.5 Euclidean Norm Balancing.- 8 Balancing via Gradient Flows.- 8.1 Introduction.- 8.2 Flows on Positive Definite Matrices.- 8.3 Flows for Balancing Transformations.- 8.4 Balancing via Isodynamical Flows.- 8.5 Euclidean Norm Optimal Realizations.- 9 Sensitivity Optimization.- 9.1 A Sensitivity Minimizing Gradient Flow.- 9.2 Related L2-Sensitivity Minimization Flows.- 9.3 Recursive L2-Sensitivity Balancing.- 9.4 L2-Sensitivity Model Reduction.- 9.5 Sensitivity Minimization with Constraints.- A Linear Algebra.- A.1 Matrices and Vectors.- A.2 Addition and Multiplication of Matrices.- A.3 Determinant and Rank of a Matrix.- A.4 Range Space, Kernel and Inverses.- A.5 Powers, Polynomials, Exponentials and Logarithms.- A.6 Eigenvalues, Eigenvectors and Trace.- A.7 Similar Matrices.- A.8 Positive Definite Matrices and Matrix Decompositions.- A.9 Norms of Vectors and Matrices.- A.10 Kronecker Product and Vec.- A.11 Differentiation and Integration.- A.12 Lemma of Lyapunov.- A.13 Vector Spaces and Subspaces.- A.14 Basis and Dimension.- A.15 Mappings and Linear Mappings.- A.16 Inner Products.- B Dynamical Systems.- B.1 Linear Dynamical Systems.- B.2 Linear Dynamical System Matrix Equations.- B.3 Controllability and Stabilizability.- B.4 Observability and Detectability.- B.5 Minimality.- B.6 Markov Parameters and Hankel Matrix.- B.7 Balanced Realizations.- B.8 Vector Fields and Flows.- B.9 Stability Concepts.- B.10 Lyapunov Stability.- C Global Analysis.- C.1 Point Set Topology.- C.2 Advanced Calculus.- C.3 Smooth Manifolds.- C.4 Spheres, Projective Spaces and Grassmannians.- C.5 Tangent Spaces and Tangent Maps.- C.6 Submanifolds.- C.7 Groups, Lie Groups and Lie Algebras.- C.8 Homogeneous Spaces.- C.9 Tangent Bundle.- C.10 Riemannian Metrics and Gradient Flows.- C.11 Stable Manifolds.- C.12 Convergence of Gradient Flows.- References.- Author Index.

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