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Symmetries and Conservation Laws
in Optimal Control Systems

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A.M. Gugushvili, O.G. Khutsishvili, V.K. Sesadze, G.N. Dalakishvili,
N.A. Mchedlishvili, T.G. Khutsishvili, V.M. Kekenadze, D.F.M. Torres,
Symmetries and Conservation Laws in Optimal Control Systems
(book in Georgian), Georgian Technical University, Tbilisi, 2003.
(248 pp, ISBN 99940-14-53-6)

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Abstract

The application of symmetry and conservation laws
in optimal control systems is regarded in the present book.
Particularly, variational methods, optimal control,
and the Pontryagin maximum principle.

The problem of synchronization of systems is considered
as a problem of optimal control. Practical examples
of using the obtained theoretical results are given.

The book is intended for a wide reading public.

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Contents


Introduction

Chapter I. Group properties of differential equations
and conservation laws

1.1. Introduction
1.2. Structure, symmetry and conservation laws
1.3. Main definition of continuous groups
1.4. Noether theorem and conservation laws

Chapter II. Variation principles of optimal control
in Lagrange systems

2.1 Introduction
2.2. Lagrange systems
2.3. Symmetry in dynamic systems
2.4. Integration and decrease of order by means
     of one-parameter group
2.5. Principle of symmetry in variation methods of optimal control

Chapter III. Variation principle of optimal control
in Hamiltonian systems

3.1. Introduction
3.2. Hamiltonian systems
3.3. Invariants of the system and methods of their construction
3.4. Noether theorem and Hamiltonian systems
3.5. Control of Hamiltonian systems and Noether theorem
3.6. Stability of Hamiltonian systems
3.7. Averaging theory in Hamiltonian systems

Chapter IV. E. Noether Theorem and Pontryagin maximum principle

4.1. Introduction
4.2. Maximum principle
4.3. Symmetry and maximum principle
4.4. Pontryagin maximum principle and Lax method
4.5. On the Noether Theorem for Optimal Control
4.5.1 Noether Theorem with no transformation of time-variable
4.5.2 Noether Theorem with transformation of the time-variable

Chapter V. Principle of symmetry and stochastic maximum principle

5.1. Introduction
5.2. Stochastic maximum principle for linear convex optimal control
5.3. Stochastic models (equations) for control systems
5.4. Symmetry and principle of stochastic maximum

Chapter VI. Problems of optimal control of real objects
based on symmetry and conservation laws

6.1. Introduction
6.2. Optimal control and power systems
6.3. Optimal control of cosmic airship in the environ
     of collinear vibration point
6.4. Optimal synthesis of optoelectronic seismograph structure

Chapter VII. Principle of symmetry and synchronization
             phenomenon in power systems

7.1. Introduction
7.2. Synchronization and theory of optimal control
7.3. Canonical transformations of Hamiltonian systems
     (system-action-angle)
7.4. Synchronization and adiabatic invariants
7.5. Synchronization and perturbation theory
7.5.1. Averaging principle
7.5.2. Synchronization and averaging principle
7.6. Synchronization and theory of catastrophes
7.7. Synchronization in power systems
7.8. Theory of catastrophes in power resources

References

Appendixes