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)

CONTENTS Preface Conference Committees Multiscale Singular Perturbations and Homogenization of Optimal Control Problems 0. Alvarez, M. Bardi and C. Marchi 1. Introduction 2. Standing assumptions 3. Ergodicity, stabilization and the effective problem 3.1. Ergodicity and the effective Hamiltonian 3.2. Stabilization and the eflective initial data 4. Regular perturbation of singular perturbation problems 5. Singular perturbations with multiple scales 5.1. The three scale case 5.2. The general case 6. Iterated homogenization for coercive equations 7. Examples 7.1. Singular perturbation of a differential game 7.2. Homogenization of a deterministic optimal control problem 7.3. Multiscale singular perturbation under a nonresonance condition References Patchy Feedbacks for Stabilization and Optimal Control: General Theory and Robustness Properties F. Ancona and A. Bressan 1. Introduction 2. Patchy vector fields and patchy feedbacks 3. Stabilizing feedback controls 4. Nearly optimal patchy feedbacks 5. Robustness 6. Stochastic perturbations References Sensitivity of Control Systems with Respect to Measure- Valued Coefficients Z. Artstein 1. Introduction 2. Standing hypotheses 3. The chattering parameters model 4. The Prohorov metric 5 . Sensitivity for relaxed controls 6. A matching result 7. Sensitivity for chattering parameters 8. Remarks and examples References Systems with Continuous Time and Discrete Time Components A. Bacciotti 1. Introduction 2. Description of the model 3. Oscillatory systems: an example 4. Stability notions 5. A sufficient condition for stability 6. Sufficient conditions for asymptotic stability References A Review on Stability of Switched Systems for Arbitrary Switchings U. Boscain 1. Introduction 2. General properties of multilinear systems 3. Common Lyapunov functions 4. Two-dimensional bilinear systems 4.1. The diagonalisable case 4.1.1. Normal forms in the diagonalizable case 4.1.2. Stability conditions in the diagonalizable case 4.2. The nondiagonalizable case 4.2.1. Normal forms in the nondiagonalizable case 4.2.2. Stability conditions in the nondiagonalizable case 5. An open problem Acknowledgments References Regularity Properties of Attainable Sets under State Constraints P. Cannarsa, M. Castelpietra and P. Cardaliaguet 1. Introduction 2. Maximum principle under state constraints 3. Perimeter estimates for the attainable set References A Generalized Hopf-Lax Formula: Analytical and Approxi- mations Aspects I. Capuzzo Dolcetta 1. Introduction 2. A generalized eikonal equation 3. The generalized Hopf-Lax formula 4. The Hopf-Lax formula for the Heisenberg Hamiltonian 4.1. A singular perturbation problem on the Heisenberg group 4.2. Convergence rate of finite diflerences approximation References Regularity of Solutions to One-Dimensional and Multi- Dimensional Problems in the Calculus of Variations F.H. Clarke 1. Introduction 2. The theorem of De Giorgi 3. Hilbert-Haar theory 4. New boundary hypotheses 4.1. Interior regularity 4.2. Continuity at the boundary 4.3. More general Lagrangians 5. The one-dimensional case References Stability Analysis of Sliding Mode Controllers F.H. Clarke and R.B. Vinter 1. Introduction 2. System Description 3. Lyapunov Functions for Sliding Mode Control 4. Sufficient Conditions for Stability 5 . An Example References Generalized Differentiation of Parameterized Families of Trajectories M. Garavello, E. Girejko and B. Piccoli 1. Introduction 2. Basic definitions 3. Approach (a) 3.1. Technical proofs 4. Approach (b) 5. Applications of the main results Acknowledgments References Sampled-Data Redesign for Nonlinear Multi-Input Systems L. Griine and K. Worthmann 1. Introduction 2. Problem formulation 3. Fliess series expansion 4. Necessary and sufficient conditions 5. Examples References On the Definition of Trajectories Corresponding to Generali- zed Controls on the Heisenberg Group P. Mason 1. Introduction 1.1. Previous approaches and results 2. Functional spaces and topologies 3. The Heisenberg example 4. Conclusion Acknowledgements References Characterization of the State Constrained Bilateral Minimal Time Function C. Nour 1. Introduction 2. Main result References Existence and a Decoupling Technique for the Neutral Prob- lem of Bolza with Multiple Varying Delays N.L. Ortiz 1. Introduction 2. Main Assumptions and Preliminaries 3. Existence of Solutions 4. The Decoupling Technique 5 . Conclusion References Stabilization Problem for Nonholonomic Control Systems L. Rifford 1. Introduction 1.1. Stabilization of nonholonomic control systems 1.2. Stabilization problem for nonholonomic distributions 1.3. Two obstructions 2 . Examples 2.1. The Nonholonomic integrator 2.2. The Riemannian case 3. Smooth repulsive stabilization 3.1. SRSz,s vector fields 3.2. Existence results of SRSz feedbacks 3.3. Existence results of SRSz sections 3.4. A Nonholonomic dream References Proximal Characterization of the Reachable Set for a Discon- tinuous Differential Inclusion V.R. Rios and P.R. Wolenski 1. Introduction 2. DL dynamics and invariance 3. Main result References Linear-Convex Control and Duality R.T. Rockafellar and G. Goebel 1. Introduction 2. Duality in finite horizon optimal control 2.1. The (finite horizon) Linear- Convex Regulator 2.2. Convex conjugate functions 2.3. General duality framework 2.4. The primal and the dual value functions 2.5. The Hamiltonian and open-loop optimality conditions 2.6. Hamilton-Jacobi results 2.7. Feedback optimality conditions 3. Regularity of the value function 3.1. Convex-valued and single-valued optimal feedback 3.2. Convex functions with locally Lipschitz gradients 3.3. Locally Lipschitz continuous optimal feedback 4. Infinite horizon problems Acknowledgments References Strong Optimality of Singular Trajectories G. Stefani 1. Introduction 2. Notations and preliminary results 2.1. Notations 2.2. The Weak Maximum Principle 2.3. Geometry near singular extremals of the first kind 3. Hamiltonian approach to strong local optimality 3.1. The Hamiltonian x 4. Sufficient conditions from Hamiltonian viewpoint 4.1. Remarks on Theorem 4.1 5. The extended second variation 5.1. Reduction to a non-singular problem 6. Coercivity of J& 6.1. Proof of the Main Theorem 6.2. A particular case 7. Final remarks References High-Order Point Variations and Generalized Differentials H. Sussmann 1. Introduction 1.1. Preliminary remarks on notation 1.2. Approximate Generalized Dinerential Quotients 1.2.1. Properties of AGDQs 1.2.2. Uniform AGDQs 1.3. l’ransversality of cones and multicones. 1.4. The nonseparation theorem. 2. Flows, trajectories, and generalized differentials of flows. 2.1. State space bundles and their sections 2.2. Flows and trajectories 2.2.1. Comparison of maps and flows 2.2.2. Dajec tories 2.3. AGDQs of ftows along trajectories 2.3.1. Compatible selections 2.3.2. Fields of variational vectors and adjoint covectors 3. Variations, impulse variations, summability 3.1. Variations of set-valued maps 3.2. Infinitesimal impulse variations 3.3. Summability 4. The AGDQ maximum principle 5. Generalized Bianchini-Stefani IIVs and the summability theorem 5.1. Times of right and left regularity 5.2. GBS IIVs 5.3. The summability theorem for GBS IIVs 6. Proof of Theorem 5.1 References List of Partcipants Author Index