# Controllability, Stabilization, and the Regulator Problem for Random Differential Systems

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*Russell Johnson; Mahesh Nerurkar*

This volume develops a systematic study of time-dependent control processes. The basic problem of null controllability of linear systems is first considered. Using methods of ergodic theory and topological dynamics, general local null controllability criteria are given. Then the subtle question of global null controllability is studied. Next, the random linear feedback and stabilization problem is posed and solved. Using concepts of exponential dichotomy and rotation number for linear Hamiltonian systems, a solution of the Riccati equation is obtained which has extremely good robustness properties and which also preserves all the smoothness and recurrence properties of the coefficients. Finally, a general version of the local nonlinear feedback stabilization problem is solved.

#### Table of Contents

# Table of Contents

## Controllability, Stabilization, and the Regulator Problem for Random Differential Systems

- Contents vii8 free
- §0. Introduction 110 free
- §1. Basic dynamical notions 211 free
- §2. Random linear control processes 413
- §3. Some facts about random linear systems 615
- §4. Sufficiency conditions for uniform controllability 1221
- §5. Dependence of controllability on the dynamics of the flow 1726
- §6. Global null controllability 1928
- §7. The feedback stabilization problem for random linear systems 2534
- §8. The rotation number 2938
- §9. The solution of the linear regulator and the stabilization problem 3140
- §10. Linearization of the regulator and the stabilization problem 3746