Nonlinear and optimal control theory : lectures given at the C.I.M.E. summer school held in Cetraro, Italy, June 19-29, 2004

Mathematical Control Theory is a branch of Mathematics having as one of its main aims the establishment of a sound mathematical foundation for the c- trol techniques employed in several di?erent ?elds of applications, including engineering, economy, biologyandsoforth. Thesystemsarisingfromthese- plied Sciences are modeled using di?erent types of mathematical formalism, primarily involving Ordinary Di?erential Equations, or Partial Di?erential Equations or Functional Di?erential Equations. These equations depend on oneormoreparameters thatcanbevaried, andthusconstitute thecontrol - pect of the problem. The parameters are to be chosen soas to obtain a desired behavior for the system. From the many di?erent problems arising in Control Theory, the C. I. M. E. school focused on some aspects of the control and op- mization ofnonlinear, notnecessarilysmooth, dynamical systems. Two points of view were presented: Geometric Control Theory and Nonlinear Control Theory. The C. I. M. E. session was arranged in ?ve six-hours courses delivered by Professors A. A. Agrachev (SISSA-ISAS, Trieste and Steklov Mathematical Institute, Moscow), A. S. Morse (Yale University, USA), E. D. Sontag (Rutgers University, NJ, USA), H. J. Sussmann (Rutgers University, NJ, USA) and V. I. Utkin (Ohio State University Columbus, OH, USA). We now brie?y describe the presentations. Agrachev's contribution began with the investigation of second order - formation in smooth optimal control problems as a means of explaining the variational and dynamical nature of powerful concepts and results such as Jacobi ?elds, Morse's index formula, Levi-Civita connection, Riemannian c- vature.

A.A. AgrachevA.S. MorseE.D. SontagH.J. SussmannV.I. Utkin

Geometry of Optimal Control Problems and Hamiltonian Systems	p. 1
1 Lagrange Multipliers' Geometry	p. 1
1.1 Smooth Optimal Control Problems	p. 1
1.2 Lagrange Multipliers	p. 4
1.3 Extremals	p. 6
1.4 Hamiltonian System	p. 7
1.5 Second Order Information	p. 10
1.6 Maslov Index	p. 14
1.7 Regular Extremals	p. 22
2 Geometry of Jacobi Curves	p. 25
2.1 Jacobi Curves	p. 25
2.2 The Cross-Ratio	p. 26
2.3 Coordinate Setting	p. 28
2.4 Curves in the Grassmannian	p. 29
2.5 The Curvature	p. 30
2.6 Structural Equations	p. 33
2.7 Canonical Connection	p. 35
2.8 Coordinate Presentation	p. 38
2.9 Affine Foliations	p. 39
2.10 Symplectic Setting	p. 41
2.11 Monotonicity	p. 44
2.12 Comparison Theorem	p. 49
2.13 Reduction	p. 51
2.14 Hyperbolicity	p. 53
References	p. 58
Lecture Notes on Logically Switched Dynamical Systems	p. 61
1 The Quintessential Switched Dynamical System Problem	p. 62
1.1 Dwell-Time Switching	p. 62
1.2 Switching Between Stabilizing Controllers	p. 65
1.3 Switching Between Graphs	p. 66
2 Switching Controls with Memoryless Logics	p. 67
2.1 Introduction	p. 67
2.2 The Problem	p. 67
2.3 The Solution	p. 67
2.4 Analysis	p. 68
3 Collaborations	p. 68
4 The Curse of the Continuum	p. 69
4.1 Process Model Class	p. 69
4.2 Controller Covering Problem	p. 73
4.3 A Natural Approach	p. 74
4.4 A Different Approach	p. 75
4.5 Which Metric?	p. 75
4.6 Construction of a Control Cover	p. 76
5 Supervisory Control	p. 76
5.1 The System	p. 77
5.2 Slow Switching	p. 86
5.3 Analysis	p. 87
5.4 Analysis of the Dwell Time Switching Logic	p. 102
6 Flocking	p. 110
6.1 Leaderless Coordination	p. 111
6.2 Symmetric Neighbor Relations	p. 142
6.3 Measurement Delays	p. 148
6.4 Asynchronous Flocking	p. 155
6.5 Leader Following	p. 158
References	p. 159
Input to State Stability: Basic Concepts and Results	p. 163
1 Introduction	p. 163
2 ISS as a Notion of Stability of Nonlinear I/O Systems	p. 163
2.1 Desirable Properties	p. 164
2.2 Merging Two Different Views of Stability	p. 165
2.3 Technical Assumptions	p. 166
2.4 Comparison Function Formalism	p. 166
2.5 Global Asymptotic Stability	p. 167
2.6 0-GAS Does Not Guarantee Good Behavior with Respect to Inputs	p. 168
2.7 Gains for Linear Systems	p. 168
2.8 Nonlinear Coordinate Changes	p. 169
2.9 Input-to-State Stability	p. 171
2.10 Linear Case, for Comparison	p. 172
2.11 Feedback Redesign	p. 173
2.12 A Feedback Redesign Theorem for Actuator Disturbances	p. 174
3 Equivalences for ISS	p. 176
3.1 Nonlinear Superposition Principle	p. 176
3.2 Robust Stability	p. 177
3.3 Dissipation	p. 178
3.4 Using "Energy" Estimates Instead of Amplitudes	p. 180
4 Cascade Interconnections	p. 180
4.1 An Example of Stabilization Using the ISS Cascade Approach	p. 182
5 Integral Input-to-State Stability	p. 183
5.1 Other Mixed Notions	p. 183
5.2 Dissipation Characterization of iISS	p. 184
5.3 Superposition Principles for iISS	p. 185
5.4 Cascades Involving iISS Systems	p. 186
5.5 An iISS Example	p. 188
6 Input to State Stability with Respect to Input Derivatives	p. 190
6.1 Cascades Involving the D[superscript k]ISS Property	p. 190
6.2 Dissipation Characterization of D[superscript k]ISS	p. 191
6.3 Superposition Principle for D[superscript k]ISS	p. 191
6.4 A Counter-Example Showing that D[superscript 1]ISS [not equal] ISS	p. 192
7 Input-to-Output Stability	p. 192
8 Detectability and Observability Notions	p. 194
8.1 Detectability	p. 195
8.2 Dualizing ISS to OSS and IOSS	p. 196
8.3 Lyapunov-Like Characterization of IOSS	p. 196
8.4 Superposition Principles for IOSS	p. 197
8.5 Norm-Estimators	p. 197
8.6 A Remark on Observers and Incremental IOSS	p. 198
8.7 Variations of IOSS	p. 199
8.8 Norm-Observability	p. 200
9 The Fundamental Relationship Among ISS, IOS, and IOSS	p. 201
10 Systems with Separate Error and Measurement Outputs	p. 202
10.1 Input-Measurement-to-Error Stability	p. 202
10.2 Review: Viscosity Subdifferentials	p. 203
10.3 RES-Lyapunov Functions	p. 204
11 Output to Input Stability and Minimum-Phase	p. 205
12 Response to Constant and Periodic Inputs	p. 205
13 A Remark Concerning ISS and H[subscript infinity] Gains	p. 206
14 Two Sample Applications	p. 207
15 Additional Discussion and References	p. 209
References	p. 213
Generalized Differentials, Variational Generators, and the Maximum Principle with State Constraints	p. 221
1 Introduction	p. 221
2 Preliminaries and Background	p. 222
2.1 Review of Some Notational Conventions and Definitions	p. 222
2.2 Generalized Jacobians, Derivate Containers, and Michel-Penot Subdifferentials	p. 228
2.3 Finitely Additive Measures	p. 229
3 Cellina Continuously Approximable Maps	p. 230
3.1 Definition and Elementary Properties	p. 231
3.2 Fixed Point Theorems for CCA Maps	p. 234
4 GDQs and AGDQs	p. 243
4.1 The Basic Definitions	p. 244
4.2 Properties of GDQs and AGDQs	p. 246
4.3 The Directional Open Mapping and Transversality Properties	p. 255
5 Variational Generators	p. 267
5.1 Linearization Error and Weak GDQs	p. 267
5.2 GDQ Variational Generators	p. 269
5.3 Examples of Variational Generators	p. 270
6 Discontinuous Vector Fields	p. 277
6.1 Co-Integrably Bounded Integrally Continuous Maps	p. 277
6.2 Points of Approximate Continuity	p. 280
7 The Maximum Principle	p. 281
References	p. 285
Sliding Mode Control: Mathematical Tools, Design and Applications	p. 289
1 Introduction	p. 289
2 Examples of Dynamic Systems with Sliding Modes	p. 289
3 VSS in Canonical Space	p. 296
3.1 Control of Free Motion	p. 298
3.2 Disturbance Rejection	p. 300
3.3 Comments for VSS in Canonical Space	p. 301
3.4 Preliminary Mathematical Remark	p. 302
4 Sliding Modes in Arbitrary State Spaces: Problem Statements	p. 303
5 Sliding Mode Equations: Equivalent Control Method	p. 305
5.1 Problem Statement	p. 305
5.2 Regularization	p. 306
5.3 Boundary Layer Regularization	p. 311
6 Sliding Mode Existence Conditions	p. 313
7 Design Principles	p. 316
7.1 Decoupling and Invariance	p. 316
7.2 Regular Form	p. 318
7.3 Block Control Principle	p. 320
7.4 Enforcing Sliding Modes	p. 322
7.5 Unit Control	p. 325
8 The Chattering Problem	p. 327
9 Discrete-Time Systems	p. 330
9.1 Discrete-Time Sliding Mode Concept	p. 331
9.2 Linear Discrete-Time Systems with Known Parameters	p. 333
9.3 Linear Discrete-Time Systems with Unknown Parameters	p. 335
10 Infinite-Dimensional Systems	p. 336
10.1 Distributed Control of Heat Process	p. 337
10.2 Flexible Mechanical System	p. 338
11 Control of Induction Motor	p. 340
References	p. 344
List of Participants	p. 349

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