Geometric Control and Nonsmooth Analysis: In Honor of the 73rd Birthday of H Hermes and of the 71st Birthday of R T Rockafellar (Series on Advances in Mathematics for Applied Sciences)

The aim of this volume is to provide a synthetic account of past research, to give an up-to-date guide to current intertwined developments of control theory and nonsmooth analysis, and also to point to future research directions. Contents: Multiscale Singular Perturbations and Homogenization of Optimal Control Problems (M Bardi et al.); Patchy Feedbacks for Stabilization and Optimal Control: General Theory and Robustness Properties (F Ancona & A Bressan); Sensitivity of Control Systems with Respect to Measure-Valued Coefficients (Z Artstein); Systems with Continuous Time and Discrete Time Components (A Bacciotti); A Review on Stability of Switched Systems for Arbitrary Switchings (U Boscain); Regularity Properties of Attainable Sets under State Constraints (P Cannarsa et al.); A Generalized Hopf Lax Formulas: Analytical and Approximations Aspects (I Capuzzo Dolcetta); Regularity of Solutions to One-Dimensional and Multi-Dimensional Problems in the Calculus of Variations (F H Clarke); Stability Analysis of Sliding Mode Controllers (F H Clarke & R B Vinter); Generalized Differentiation of Parameterized Families of Trajectories (M Garavello et al.); Sampled-Data Redesign for Nonlinear Multi-Input Systems (L Gr??ne & K Worthmann); On the Definition of Trajectories Corresponding to Generalized Controls on the Heisenberg Group (P Mason); Characterization of the State Constrained Bilateral Minimal Time Function (C Nour); Existence and a Decoupling Technique for the Neutral Problem of Bolza with Multiple Varying Delays (N L Ortiz); Stabilization Problem for Nonholonomic Control Systems (L Rifford); Proximal Characterization of the Reachable Sets for a Discontinuous Differential Inclusion (V R Rios & P R Wolenski); Linear-Convex Control and Duality (R T Rockafellar & R Goebel); Strong Optimality of Singular Trajectories (G Stefani); High-Order Point Variations and Generalized Differentials (H Sussmann). CONTENTS......Page 10 Preface......Page 6 Conference Committees......Page 9 Multiscale Singular Perturbations and Homogenization of Optimal Control Problems 0. Alvarez, M. Bardi and C. Marchi......Page 12 1. Introduction......Page 13 2. Standing assumptions......Page 16 3. Ergodicity, stabilization and the effective problem......Page 17 3.1. Ergodicity and the effective Hamiltonian......Page 18 3.2. Stabilization and the eflective initial data......Page 19 4. Regular perturbation of singular perturbation problems......Page 20 5.1. The three scale case......Page 24 5.2. The general case......Page 29 6. Iterated homogenization for coercive equations......Page 31 7.1. Singular perturbation of a differential game......Page 32 7.2. Homogenization of a deterministic optimal control problem......Page 34 7.3. Multiscale singular perturbation under a nonresonance condition......Page 35 References......Page 37 1. Introduction......Page 39 2. Patchy vector fields and patchy feedbacks......Page 42 3. Stabilizing feedback controls......Page 48 4. Nearly optimal patchy feedbacks......Page 52 5. Robustness......Page 60 6. Stochastic perturbations......Page 67 References......Page 74 1. Introduction......Page 76 3. The chattering parameters model......Page 78 4. The Prohorov metric......Page 81 5 . Sensitivity for relaxed controls......Page 83 6. A matching result......Page 86 7. Sensitivity for chattering parameters......Page 87 8. Remarks and examples......Page 89 References......Page 91 1. Introduction......Page 93 2. Description of the model......Page 94 3. Oscillatory systems: an example......Page 97 4. Stability notions......Page 102 5. A sufficient condition for stability......Page 103 6. Sufficient conditions for asymptotic stability......Page 107 References......Page 110 1. Introduction......Page 111 2. General properties of multilinear systems......Page 112 3. Common Lyapunov functions......Page 115 4. Two-dimensional bilinear systems......Page 118 4.1. The diagonalisable case......Page 121 4.1.1. Normal forms in the diagonalizable case......Page 122 4.1.2. Stability conditions in the diagonalizable case......Page 123 4.2. The nondiagonalizable case......Page 125 4.2.1. Normal forms in the nondiagonalizable case......Page 126 4.2.2. Stability conditions in the nondiagonalizable case......Page 127 5. An open problem......Page 128 References......Page 129 1. Introduction......Page 131 2. Maximum principle under state constraints......Page 133 3. Perimeter estimates for the attainable set......Page 142 References......Page 146 1. Introduction......Page 147 2. A generalized eikonal equation......Page 149 3. The generalized Hopf-Lax formula......Page 151 4. The Hopf-Lax formula for the Heisenberg Hamiltonian......Page 153 4.1. A singular perturbation problem on the Heisenberg group......Page 154 4.2. Convergence rate of finite diflerences approximation......Page 157 References......Page 159 1. Introduction......Page 162 2. The theorem of De Giorgi......Page 164 3. Hilbert-Haar theory......Page 166 4.1. Interior regularity......Page 167 4.2. Continuity at the boundary......Page 168 5. The one-dimensional case......Page 169 References......Page 173 1. Introduction......Page 175 3. Lyapunov Functions for Sliding Mode Control......Page 179 4. Sufficient Conditions for Stability......Page 180 5 . An Example......Page 183 References......Page 187 1. Introduction......Page 188 2. Basic definitions......Page 191 3. Approach (a)......Page 194 3.1. Technical proofs......Page 198 4. Approach (b)......Page 203 5. Applications of the main results......Page 206 References......Page 215 1. Introduction......Page 217 2. Problem formulation......Page 219 3. Fliess series expansion......Page 221 4. Necessary and sufficient conditions......Page 225 5. Examples......Page 233 References......Page 238 1. Introduction......Page 239 1.1. Previous approaches and results......Page 241 2. Functional spaces and topologies......Page 242 3. The Heisenberg example......Page 244 References......Page 252 1. Introduction......Page 254 2. Main result......Page 256 References......Page 257 1. Introduction......Page 259 2. Main Assumptions and Preliminaries......Page 260 3. Existence of Solutions......Page 261 4. The Decoupling Technique......Page 264 References......Page 270 1.1. Stabilization of nonholonomic control systems......Page 271 1.2. Stabilization problem for nonholonomic distributions......Page 272 1.3. Two obstructions......Page 273 2.1. The Nonholonomic integrator......Page 274 2.2. The Riemannian case......Page 275 3.1. SRSz,s vector fields......Page 276 3.3. Existence results of SRSz sections......Page 277 References......Page 279 1. Introduction......Page 281 2. DL dynamics and invariance......Page 283 3. Main result......Page 287 References......Page 289 1. Introduction......Page 291 2.1. The (finite horizon) Linear- Convex Regulator......Page 292 2.2. Convex conjugate functions......Page 293 2.3. General duality framework......Page 294 2.4. The primal and the dual value functions......Page 295 2.5. The Hamiltonian and open-loop optimality conditions......Page 298 2.6. Hamilton-Jacobi results......Page 299 2.7. Feedback optimality conditions......Page 300 3.1. Convex-valued and single-valued optimal feedback......Page 301 3.2. Convex functions with locally Lipschitz gradients......Page 303 3.3. Locally Lipschitz continuous optimal feedback......Page 305 4. Infinite horizon problems......Page 306 References......Page 310 1. Introduction......Page 311 2.1. Notations......Page 314 2.2. The Weak Maximum Principle......Page 315 2.3. Geometry near singular extremals of the first kind......Page 316 3.1. The Hamiltonian x......Page 318 4. Sufficient conditions from Hamiltonian viewpoint......Page 320 4.1. Remarks on Theorem 4.1......Page 323 5. The extended second variation......Page 324 5.1. Reduction to a non-singular problem......Page 328 6. Coercivity of J&......Page 330 6.1. Proof of the Main Theorem......Page 333 6.2. A particular case......Page 334 7. Final remarks......Page 335 References......Page 336 1. Introduction......Page 338 1.1. Preliminary remarks on notation......Page 339 1.2.1. Properties of AGDQs......Page 341 1.2.2. Uniform AGDQs......Page 344 1.3. l’ransversality of cones and multicones.......Page 345 1.4. The nonseparation theorem.......Page 346 2.1. State space bundles and their sections......Page 348 2.2.1. Comparison of maps and flows......Page 349 2.3.1. Compatible selections......Page 350 3.1. Variations of set-valued maps......Page 351 3.3. Summability......Page 352 4. The AGDQ maximum principle......Page 353 5. Generalized Bianchini-Stefani IIVs and the summability theorem......Page 355 5.2. GBS IIVs......Page 356 6. Proof of Theorem 5.1......Page 358 References......Page 368 List of Partcipants......Page 370 Author Index......Page 372