**Contemporary Mathematics**

Volume: 619;
2014;
247 pp;
Softcover

MSC: Primary 34; 35; 49; 60; 68; 78; 90; 91; 92; 93;

Print ISBN: 978-1-4704-1077-3

Product Code: CONM/619

List Price: $91.00

AMS Member Price: $72.80

MAA member Price: $81.90

**Electronic ISBN: 978-1-4704-1713-0
Product Code: CONM/619.E**

List Price: $91.00

AMS Member Price: $72.80

MAA member Price: $81.90

# Variational and Optimal Control Problems on Unbounded Domains

Share this page *Edited by *
*Gershon Wolansky; Alexander J. Zaslavski*

A co-publication of the AMS and Bar-Ilan University

This volume contains the proceedings of the
workshop on Variational and Optimal Control Problems on Unbounded
Domains, held in memory of Arie Leizarowitz, from January 9–12, 2012,
in Haifa, Israel.

The workshop brought together a select group of worldwide experts
in optimal control theory and the calculus of variations, working on
problems on unbounded domains.

The papers in this volume cover many different areas of optimal
control and its applications. Topics include needle variations in
infinite-horizon optimal control, Lyapunov stability with some
extensions, small noise large time asymptotics for the normalized
Feynman-Kac semigroup, linear-quadratic optimal control problems with
state delays, time-optimal control of wafer stage positioning, second
order optimality conditions in optimal control, state and time
transformations of infinite horizon problems, turnpike properties of
dynamic zero-sum games, and an infinite-horizon variational problem on
an infinite strip.

#### Readership

Graduate students and research mathematicians interested in variational and optimal control problems.

# Table of Contents

## Variational and Optimal Control Problems on Unbounded Domains

- Preface v6 free
- Biography and Bibliography of Arie Leizarowitz vii8 free
- Workshop Program xi12 free
- Needle Variations in Infinite-Horizon Optimal Control 118 free
- Comments on Lyapunov 𝛼-stability with Some Extensions 1936
- 1. Introduction 1936
- 2. Notation and preliminary results 2037
- 3. Some facts about convex cones 2138
- 4. Proof of Theorem 2.2 2239
- 5. Common Lyapunov 𝛼-stability 2441
- 6. Common 𝛼-stability over cones 2542
- 7. The constrained Laypunov problem as a common Lyapunov factor problem 2643
- 8. Conclusion 2744
- Acknowledgement 2744
- On Arieh Leizarowitz 2744
- References 2744

- Small Noise Large Time Asymptotics for the Normalized Feynman-Kac Semigroup 3148
- Linear Constraints for Convex Approximation of the Stability Domain of a Polynomial in Coefficients Space 4966
- Singular Solution of an Infinite Horizon Linear-Quadratic Optimal Control Problem with State Delays 5976
- 1. Introduction 5976
- 2. Problem formulation and main assumptions 6178
- 3. Regularization of the OOCP 6380
- 4. Asymptotic solution of (3.8)-(3.11) 6582
- 5. \boldmath{𝜖}-Free conditions for the existence and uniqueness of solution to the CCP 7996
- 6. Asymptotic expansion of the CCP cost functional optimal value 8097
- 7. Proof of Theorem 4.4 8097
- 8. Proof of Lemma 4.6 87104
- 9. Suboptimal state-feedback control of the CCP 90107
- 10. Solution of the OOCP 94111
- References 96113

- Time-Optimal Control of Wafer Stage Positioning Using Simplified Models 99116
- Robust Stability and Monitoring Threshold Functions 109126
- 1. Introduction –the tale of two problems 109126
- 2. Robust stability radius–computation 111128
- 3. Monitoring threshold functions through the convex hull 113130
- 4. Monitoring threshold functions through 𝑙_{𝑝} balls 115132
- 5. Text Mining application 122139
- 6. Numerical results 124141
- 7. Additional enhancements 126143
- 8. Future research directions 128145
- 9. Conclusion 129146
- References 130147

- One Dimensional Singular Calculus of Variations in Infinite Horizon and Applications 131148
- Second Order Optimality Conditions in Optimal Control Problems with Mixed Inequality Type Constraints on a Variable Time Interval 141158
- An Infinite-Horizon Variational Problem on an Infinite Strip 157174
- State and Time Transformations of Infinite Horizon Optimal Control Problems 189206
- 1. Introduction 189206
- 2. Weighted Lebesgue and Sobolev spaces 190207
- 3. Motivating examples 191208
- 4. Problem formulation 192209
- 5. The first logarithmic transformation: a transformation of the state 195212
- 6. The second logarithmic transformation: a transformation of the time 200217
- 7. Applications 202219
- 8. Conclusions 206223
- References 207224

- Turnpike Properties of Approximate Solutions of Discrete-Time Optimal Control Problems on Compact Metric Spaces 209226
- Turnpike Theory for Dynamic Zero-Sum Games 225242