Minimization of the energy of a vibrational system . . . . . - PowerPoint PPT Presentation

Motivation and introduction Optimality criteria Results Related results . . Minimization of the energy of a vibrational system . . . . . Ivica Nakić Department of Mathematics Faculty of Natural Sciences and Mathematics University of Zagreb Second Najman Conference on Spectral Problems for Operators and Matrices, 2009 Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . Outline . . . Motivation and introduction 1 . . . Optimality criteria 2 . . . Results 3 . . . Related results 4 Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . A class of vibrational systems We consider a damped vibrational system described by the differential equation M ¨ x + C ˙ x + Kx = 0 , x (0) = x 0 , ˙ x (0) = ˙ x 0 , where M , C , and K (called mass, damping and stiffness matrix, respectively) are real, symmetric matrices of order n with M , K positive definite, and C positive semi–definite matrices. Our aim is to optimize this vibrational system in the sense of finding an optimal damping matrix C such that the energy of the system is minimal. Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . Linearization The choice of linearization is such that the underlying phase space has energy norm. y = Ay ˙ y (0) = y 0 with the solution y ( t ) = e At y 0 . 2 , y = ( y 1 Here y 1 = L ∗ 1 x , y 2 = L ∗ x , K = L 1 L ∗ 1 , M = L 2 L ∗ y 2 ) , 2 ˙ ( L ∗ ) 1 x 0 , and y 0 = L ∗ 2 ˙ x 0 ( ) L 1 L −∗ 0 2 A = . − L − 1 − L 2 − 1 CL −∗ 2 L 1 2 Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . Linearization ( ) L 1 L −∗ 0 2 A = . − L − 1 − L 2 − 1 CL −∗ 2 L 1 2 The underlying norm is energy-norm: ∥ y ( t ) ∥ 2 = ˙ x ( t ) ∗ M ˙ x ( t ) + x ( t ) ∗ Kx ( t ) = 2 E ( t ) . This linearization is closest to normality among all linearizations of the form y 1 = W 1 x , y 2 = W 2 ˙ x , W 1 , W 2 nonsingular (in the sense F − ∑ | λ i ( A ) | 2 is minimal). that ∥ A ∥ 2 Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . Linearization ( ) L 1 L −∗ 0 2 A = . − L − 1 − L 2 − 1 CL −∗ 2 L 1 2 The underlying norm is energy-norm: ∥ y ( t ) ∥ 2 = ˙ x ( t ) ∗ M ˙ x ( t ) + x ( t ) ∗ Kx ( t ) = 2 E ( t ) . This linearization is closest to normality among all linearizations of the form y 1 = W 1 x , y 2 = W 2 ˙ x , W 1 , W 2 nonsingular (in the sense F − ∑ | λ i ( A ) | 2 is minimal). that ∥ A ∥ 2 Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . Convenient form Let L − 1 1 be SVD–decomposition of the matrix L − 1 2 L 1 = U 2 Ω U ∗ 2 L 1 , with Ω = diag ( ω 1 , . . . , ω n ) > 0 . We can assume ( ) U 1 0 ω 1 ≤ ω 2 ≤ · · · ≤ ω n . Set U = . Then 0 U 2 [ 0 ] Ω A = U ∗ AU = � , − ˆ − Ω C where ˆ 2 L − 1 2 CL −∗ C = U ∗ 2 U 2 is positive semi–definite. If we denote F = L −∗ 2 U 2 , then F ∗ MF = I , F ∗ KF = Ω . Thus we have obtained a particularly convenient, the so–called modal representation of the problem. In the following we will work in the basis in which matrix A has this form. Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . The choice of optimality criterion A number of optimality criteria in use: spectral abscissa criterion ˆ C �→ s ( A ) := max ℜ λ k → min k minimization of the number of oscillations ℜ λ k ˆ C �→ max | λ k | → min k minimization of the total energy of the system ∫ ∞ ˆ E ( t ) dt → min C �→ 0 We will concentrate on the last criterion. Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . The choice of optimality criterion A number of optimality criteria in use: spectral abscissa criterion ˆ C �→ s ( A ) := max ℜ λ k → min k minimization of the number of oscillations ℜ λ k ˆ C �→ max | λ k | → min k minimization of the total energy of the system ∫ ∞ ˆ E ( t ) dt → min C �→ 0 We will concentrate on the last criterion. Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . The choice of optimality criterion A number of optimality criteria in use: spectral abscissa criterion ˆ C �→ s ( A ) := max ℜ λ k → min k minimization of the number of oscillations ℜ λ k ˆ C �→ max | λ k | → min k minimization of the total energy of the system ∫ ∞ ˆ E ( t ) dt → min C �→ 0 We will concentrate on the last criterion. Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . The choice of optimality criterion A number of optimality criteria in use: spectral abscissa criterion ˆ C �→ s ( A ) := max ℜ λ k → min k minimization of the number of oscillations ℜ λ k ˆ C �→ max | λ k | → min k minimization of the total energy of the system ∫ ∞ ˆ E ( t ) dt → min C �→ 0 We will concentrate on the last criterion. Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . The choice of optimality criterion A number of optimality criteria in use: spectral abscissa criterion ˆ C �→ s ( A ) := max ℜ λ k → min k minimization of the number of oscillations ℜ λ k ˆ C �→ max | λ k | → min k minimization of the total energy of the system ∫ ∞ ˆ E ( t ) dt → min C �→ 0 We will concentrate on the last criterion. Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . Minimal energy criterion We want to get rid of the dependence on the initial state. (At least) three viable approaches: take maximum over all initial states ∫ ∞ max E ( t ; y 0 ) dt ∥ y 0 ∥ =1 0 take average over all initial states ∫ ∫ ∞ E ( t ; y 0 ) dt dσ ∥ y 0 ∥ =1 0 take ”smoothed” maximum over all initial states ∫ ∫ ∞ max E ( t ; z 0 ) dt dσ ∥ y 0 ∥ =1 { z 0 : ∥ z 0 ∥ =1 , ∥ z 0 − y 0 ∥≤ δ } 0 Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . Minimal energy criterion We want to get rid of the dependence on the initial state. (At least) three viable approaches: take maximum over all initial states ∫ ∞ max E ( t ; y 0 ) dt ∥ y 0 ∥ =1 0 take average over all initial states ∫ ∫ ∞ E ( t ; y 0 ) dt dσ ∥ y 0 ∥ =1 0 take ”smoothed” maximum over all initial states ∫ ∫ ∞ max E ( t ; z 0 ) dt dσ ∥ y 0 ∥ =1 { z 0 : ∥ z 0 ∥ =1 , ∥ z 0 − y 0 ∥≤ δ } 0 Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . Minimal energy criterion We want to get rid of the dependence on the initial state. (At least) three viable approaches: take maximum over all initial states ∫ ∞ max E ( t ; y 0 ) dt ∥ y 0 ∥ =1 0 take average over all initial states ∫ ∫ ∞ E ( t ; y 0 ) dt dσ ∥ y 0 ∥ =1 0 take ”smoothed” maximum over all initial states ∫ ∫ ∞ max E ( t ; z 0 ) dt dσ ∥ y 0 ∥ =1 { z 0 : ∥ z 0 ∥ =1 , ∥ z 0 − y 0 ∥≤ δ } 0 Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . Lyapunov equation It’s easy to see that ∫ ∞ E ( t ; y 0 ) dt = 1 2 y ∗ 0 Xy 0 , 0 where X is the solution of the Lyapunov equation A ∗ X + XA = − I. Hence maximum over all initial states ֌ 1 2 ∥ X ∥ average over all initial states ֌ trace ( XZ ) ”smoothed” maximum over all initial states ֌ α ∥ X ∥ + β trace ( XZ ) Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . Lyapunov equation It’s easy to see that ∫ ∞ E ( t ; y 0 ) dt = 1 2 y ∗ 0 Xy 0 , 0 where X is the solution of the Lyapunov equation A ∗ X + XA = − I. Hence maximum over all initial states ֌ 1 2 ∥ X ∥ average over all initial states ֌ trace ( XZ ) ”smoothed” maximum over all initial states ֌ α ∥ X ∥ + β trace ( XZ ) Ivica Nakić Minimization of a vibrational system

Motivation and introduction Optimality criteria Results Related results . Lyapunov equation It’s easy to see that ∫ ∞ E ( t ; y 0 ) dt = 1 2 y ∗ 0 Xy 0 , 0 where X is the solution of the Lyapunov equation A ∗ X + XA = − I. Hence maximum over all initial states ֌ 1 2 ∥ X ∥ average over all initial states ֌ trace ( XZ ) ”smoothed” maximum over all initial states ֌ α ∥ X ∥ + β trace ( XZ ) Ivica Nakić Minimization of a vibrational system