# Online Estimation and Adaptive Control for a Class of History Dependent Functional Differential Equations

## TL;DR

This paper develops conditions for convergence and stability of online estimation and adaptive control methods for history dependent functional differential equations, relevant to robotic systems with complex nonlinearities.

## Contribution

It introduces new theoretical results on convergence and stability for distributed parameter systems with history dependence, including finite-dimensional approximation guarantees.

## Key findings

- Convergence of online estimation methods is established.
- Stability of adaptive control strategies is proven using Barbalat's lemma.
- Finite-dimensional approximations converge to infinite-dimensional solutions.

## Abstract

This paper presents sufficient conditions for the convergence of online estimation methods and the stability of adaptive control strategies for a class of history dependent, functional differential equations. The study is motivated by the increasing interest in estimation and control techniques for robotic systems whose governing equations include history dependent nonlinearities. The functional differential equations in this paper are constructed using integral operators that depend on distributed parameters. As a consequence the resulting estimation and control equations are examples of distributed parameter systems whose states and distributed parameters evolve in finite and infinite dimensional spaces, respectively. suWell-posedness, existence, and uniqueness are discussed for the class of fully actuated robotic systems with history dependent forces in their governing equation of motion. By deriving rates of approximation for the class of history dependent operators in this paper, sufficient conditions are derived that guarantee that finite dimensional approximations of the online estimation equations converge to the solution of the infinite dimensional, distributed parameter system. The convergence and stability of a sliding mode adaptive control strategy for the history dependent, functional differential equations is established using Barbalat's lemma.