Stochastic Lagrangian Adaptation

Contents 1 Introduction 1.1 Methodology and Principal Results 2 Problem Statement 3 Asymptotic Maximum Likelihood Identification 3.1 Asymptotic Maximum Likelihood Estimates 3.2 The Characterization of the Limit Set of AML Estimates 3.3 An Alternative Definition of the Limit Set I 3.4 A Class of AML Estimates 4 Geometric Results 4.1 A Conflict Between Identification and Control 4.2 The Unique Stationary Point of J over I 4.3 The Scalar Example 5 Lagrangian Adaptation 5.1 Motivation and Sketch of the Conceptual Adaptive Scheme 5.2 An Estimate of the B-Derivative of the Closed-Loop Matrix 5.3 Derivation of the Estimate and the Adjoint Processes Flows 5.4 The Validity of the Lagrangian Adaptive Control Scheme 5.5 From Convergence to Consistency to Optimal Performance 5.6 A Discussion of the Adaptive Scheme's Initialization Time t0 5.7 A Summary of the Lagrangian Adaptation Scheme 5.8 Computation of the Regret Rate 6 Proof of Theorem 5.2 6.1 Proof Methodology 6.2 Preliminary Results 6.3 A Zero-Order Approximation of 6.4 A First-Order Approximation of 6.5 Generalized Stochastic Picard Iterations 6.6 Convergence of {tn, ζtn t≥0} to Solutions of (5.10, 5.11) The Consistency of { t , t≥0} References Index