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.