Mathematical Problems of Control Theory: An Introduction by Gennady A. Leonov PDF

By Gennady A. Leonov

ISBN-10: 9810246943

ISBN-13: 9789810246945

Indicates essentially how the examine of concrete keep watch over structures has encouraged the improvement of the mathematical instruments wanted for fixing such difficulties. The Aizerman and Brockett difficulties are mentioned and an advent to the idea of discrete regulate structures is given.

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Indicates basically how the learn of concrete regulate structures has influenced the improvement of the mathematical instruments wanted for fixing such difficulties. The Aizerman and Brockett difficulties are mentioned and an advent to the speculation of discrete keep an eye on platforms is given.

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Contents:

1 restricted minimization
1. 1 Preliminaries. .. ..
1. 2 restricted minimization
1. three twin technique . . . . . . .
1. four Minimizers with the least strength .
1. five software of twin technique . ,.
1. 6 a number of options of nonhomogeneous equation.
1. 7 units of constraints . . . . . . . .
1. eight restricted minimization for Ff .
1. nine Subcritical challenge . .. .. .
1. 10 software to the p-Laplacian .
1. eleven serious challenge . . .
1. 12 Bibliographical notes. . . . .

2 functions of Lusternik-Schnirelman idea
2. 1 Palais-Smale , case p '# q
2. 2 Duality mapping . . . . . . . . . .
2. three Palais-Smale situation, case p = q
2. four The Lustemik-Schnirelman thought .
2. five Case p > q
2. 6 Case. p < q . .. .. .. .. .. . 2. 7 Case p = q . .. .. .. .. .. . 2. eight The p-Laplacian in bounded area 2. nine Iterative development of eigenvectors 2. 10 serious issues of upper order 2. eleven Bibliographical notes. . . . . . . . . 3 Nonhomogeneous potentials 3. 1 Preliminaries and assumptions 3. 2 limited minimization . . 3. three program - compact case. 3. four Perturbation theorems - noncompact case 3. five Perturbation of the practical a - noncompact case. 3. 6 lifestyles of infinitely many ideas . . . . . . . . 3. 7 normal minimization - case p > q .
3. eight Set of constraints V . .. .. .. .
3. nine program to a severe case p = n
3. 10 Technical lemmas . . . . . . . . .
3. eleven lifestyles outcome for challenge (3. 34)
3. 12 Bibliographical notes. . . . . . .

4 Potentials with covariance
4. 1 Preliminaries and restricted minimization
4. 2 twin process . . . . . . . . . . . . .
4. three Minimization topic to constraint V . . . .
4. four Sobolev inequality . . . . . . . . . . . . .
4. five Mountain cross theorem and limited minimization
4. 6 Minimization challenge for a method of equations .
4. 7 Bibliographical notes. . . . . . . . . . . . . . .

5 Eigenvalues and point units
5. 1 point units . .. .. .. .. .. ..
5. 2 Continuity and monotonicity of a .
5. three The differentiability houses of a
5. four Schechter's model of the mountain go theorem
5. five basic for solvability of (5. eleven)
5. 6 homes of the functionality K(t) .
5. 7 Hilbert house case . . . . . . .
5. eight program to elliptic equations
5. nine Bibliographical notes. . . . . .

6 Generalizations of the mountain cross theorem
6. 1 model of a deformation lemma . . . . . .
6. 2 Mountain cross substitute . . . . . . . . .
6. three results of mountain move substitute
6. four Hampwile substitute. . . . . . . . . . . .
6. five Applicability of the mountain cross theorem
6. 6 Mountain go and Hampwile substitute
6. 7 Bibliographical notes. . . . . . . . . . .

7 Nondifferentiable functionals 167
7. 1 idea of a generalized gradient . . . . . . . . . . . . 167
7. 2 Generalized gradients in functionality areas. . . . . . . . . 172
7. three Mountain move theorem for in the community Lipschitz functionals . 174
7. four outcomes of Theorem 7. three. 1 . . . . . . . . . . . . . 181
7. five software to boundary price challenge with discontinuous nonlinearity 183
7. 6 decrease semicontinuous perturbation . . . . . . . . . . . . . . 185
7. 7 Deformation lemma for functionals gratifying (L) . . . . . . 188
7. eight program to variational inequalities
7. nine Bibliographical notes. . . . . . . . .

8 focus compactness precept - subcritical case 198
8. 1 Concentration-compactness precept at infinity - subcritical case 198
8. 2 restricted minimization - subcritical case . . . . . . . . two hundred
8. three limited minimization with b ¥= const, subcritical case . 205
8. four Behaviour of the Palais-Smale sequences . 211
8. five the outside Dirichlet challenge . . . . . . 215
8. 6 The Palais-Smale situation . . . . . . . 218
8. 7 Concentration-compactness precept I . 221
8. eight Bibliographical notes. . . . . . . . . . . 223

9 focus compactness precept - serious case 224
9. 1 severe Sobolev exponent . . . . . . . . 224
9. 2 Concentration-compactness precept II . . 228
9. three lack of mass at infinity. . . . . . . . . . . 229
9. four limited minimization - severe case . 233
9. five Palais-Smale sequences in severe case . . 237
9. 6 Symmetric suggestions . . . . . . . . . . . . . . . . . . 244
9. 7 feedback on compact embeddings into L 2* (Q) and L ok (}Rn) . 250
9. eight Bibliographical notes. . . . . . . . . . . . . . . . . . . . . . 252

Appendix
A. l Sobolev areas . . . . . . . . . . . . . . . . . . . . . .
A. 2 Embedding theorems . . . . . . . . . . . . . . . . . . .
A. three Compact embeddings of areas wI,p(}Rn) and DI,p(}Rn)
A. four stipulations of focus and uniform decay at infinity
A. five Compact embedding for H,1 (}Rn) .
A. 6 Schwarz symmetrization
A. 7 Pointwise convergence.
A. eight Gateaux derivatives

Bibliography

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Index

Additional resources for Mathematical Problems of Control Theory: An Introduction

Sample text

54) are similar in form. 54) includes a factor \ / 3 . 53) is an additional condition on the spring constant. 54) are sufficient conditions only. However in the engineering practice it is often impossible to pinpoint all parameters and the mathematical model is always a certain idealization. 54), turns out to be quite sufficient. • y n /^ ~^x pa>o ffl I o m Fig. 11 The schematic representation of transient process is given in Fig. 11. Chapter 2 Linear electric circuits. 1 D e s c r i p t i o n of l i n e a r b l o c k s In the previous chapter a nonlinear mathematical model was considered.

Transfer functions and frequency responses of linear blocks The following notation of inputs and outputs:

14) where A is a constant n x n-matrix, b and c are constant matrices of dimensions nxm and n x / respectively. The asterisk * denotes the transposition in the real case and Hermitian conjugation in the complex case. 3 8 Linear electric circuits. Transfer functions and frequency responses of linear blocks For the sequel we are needed in the following notation ' xi j] = xi, 77 = 1 2 , ... ,?? 13) may be written by ii — x2, &n — 1 — ^n? AToxi + Z{t), m-^m + 1 + Mm-lXm + . . + Mr,Xx. Hence we have / A = 0 1 0 ...

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