The common experience in solving control problems shows that optimal control as a function of time proves to be piecewise analytic, having a finite number of jumps (called switches) on any finite-time interval. Meanwhile there exists an old example proposed by A.T. Fuller [1961) in which optimal control has an infinite number of switches on a finite-time interval. This phenomenon is called chattering. It has become increasingly clear that chattering is widespread. This book is devoted to its exploration. Chattering obstructs the direct use of Pontryagin's maximum principle because of the lack of a nonzero-length interval with a continuous control function. That is why the common experience appears misleading. It is the hidden symmetry of Fuller's problem that allows the explicit solution. Namely, there exists a one-parameter group which respects the optimal trajectories of the problem. When published in 1961, Fuller's example incited curiosity, but it was considered only "interesting" and soon was forgotten. The second wave of attention to chattering was raised about 12 years later when several other examples with optimal chattering trajectories were 1 found. All these examples were two-dimensional with the one-parameter group of symmetries.

Inhalt

1.Introduction.- 1.1 The Subject of the Book.- 1.2 Hamiltonian Systems and Singular Extremals.- 1.3 The Semi-Canonical Form of Hamiltonian Systems.- 1.4 Integral Varieties with Chattering Arcs.- 1.5 An Example of Designing a Lagrangian Manifold.- 2.Fuller's Problem.- 2.1 Statement of Puller's Problem.- 2.2 Chattering Arcs.- 2.3 Untwisted Chattering Arcs.- 2.4 The Geometry of Trajectories of Hamiltonian Systems.- 3.Second Order Singular Extremals and Chattering.- 3.1 Preliminaries.- 3.2 Manifolds with Second Order Singular Trajectories.- 3.3 The Connection with Fuller's Problem.- 3.4 Resolution of the Singularity of the Poincaré Mapping.- 3.5 The Connection with the Problem of C. Marchal.- 3.6 Fixed Points of the Quotient Mapping.- 3.7 The Hyperbolic Structure of the Quotient Mapping.- 3.8 Non-Degeneracy of the Fixed Point.- 3.9 Bundles with Chattering Arcs.- 3.10 Lagrangian Manifolds.- 3.11 Synthesis with Locally Optimal Chattering Arcs.- 3.12 Regular Projection of Chattering Varieties.- 4.The Ubiquity of Fuller's Phenomenon.- 4.1 Kupka's Results.- 4.2 Codimension of the Set of Fuller Points.- 4.3 Structural Stability of the Optimal Synthesis in the Two-Dimensional Fuller Problem.- 5.Higher Order Singular Extremals.- 5.1 Conjectures Concerning Higher Order Singular Modes.- 5.2 Problems with Linear Constraints.- 5.3 Problems with Symmetries.- 5.4 Bi-Constant Ratio Solutions of Fuller's Problems.- 5.5 Optimality of b.c.r. Solutions.- 5.6 Numerical Verification of the Conjecture on the Number of Cycles in the Orbit Space.- 5.7 Three-Dimensional Puller Problems.- 6.Applications.- 6.1 Fibrations in Three-Dimensional Space.- 6.2 Stabilization of a Rigid Body.- 6.3 The Resource Allocation Problem.- 6.4 Control of Two Interdependent Oscillators.- 6.5 Lowden's Problem.- 6.6 Robot Control.- 7.Multidimensional Control and Chattering Modes.- 7.1 Multidimensional Problems with a Polyhedral Indicatrix.- 7.2 Multidimensional Problems with a Smooth Indicatrix.- Epilogue.- List of Figures.