Phase Portraits of Control Dynamical Systems

'Et moi, ... , si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point alIe.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Klappentext

'Et moi, ... , si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point alIe.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non­ The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non­ linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com­ puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Inhalt

1. Introduction.- 2. Controllable Dynamical System (CDS.- 3. Differential Inclusion. Equivalence Classes of CDS.- 4. Transformation of CDS to Phase Velocity Unit Vectors.- 5. Indicatrix of CDS.- 6. Degree of Freedom in Control of CDS.- 7. The Hamiltonian of CDS as a Support function.- 8. Types of Cones of Admissible Direction of CDS.- 9. Domain of Free Trajectories of CDS.- 10. The Principle of Inclusion in the Event Space.- 11. The Boundary of an Integral Funnels of a Differential Inclusion.- 12. Relationship of the Boundary of an Integral Funnel with the Hamiltonian-Jacobi Equation.- 13. The Principle of Inclusion for an Autonomous CDS in the State Space Integration of CDS.- 14. Boundary of the Trajectory Funnel of CDS.- 15. Geometrical Construction of the Boundary of Trajectory Funnel.- 16. The Euler-Lagrange Equation for the Boundary of Trajectory Funnel.- 17. Hatched Boundaries of Trajectory Funnels Manifold of Reverse Hatching (MRH.- 18. Singular Manifolds of CDS in State Space.- 19. Invariant Manifold of CDS.- 20. Singular and Invariant Manifolds of Linear CDS.- 21. Domain of Free Trajectories on Invariant Manifolds.- 22. Trajectories Funnel on Invariant Manifolds.- 23. Separating Hypersurfaces in State Spaces.- 24. Admissible Manifolds of CDS in State Space.- 25. Phase Portrait of CDS.- 26. Transient and Relative Motion of CDS.- 27. CDS with Ellipsoidal Indicatrix.- 28. CDS and Continuous Non Linear Media. The Principle of Maximum of Flows of Substance. Laplace Operator of CDS.- 29. CDS and Finsler Metric.- 30. Optical Analogue of CDS.- 31. Correspondence between CDSs and Mechanical Systems.- 32. CDS Subject to Phase Constraints.- 33. Singular Sets of Two-Dimensional CDSs.- 34. Phase Portrait of CDS on Two-Dimensional Manifolds.- 35. Examples on Construction ofPhase Portrait of Two-Dimensional CDS.- 36. Phase Portrait of Two-Level Quantum-Mechanical CDS.- 37. Example of Decomposable Bilinear CDS in Three-Dimensional Space.- 38. Controllability of General Bilinear CDS on Plane.- 39. Trajectory Funnel in Backward Time.- 40. Optimal Control.