# Sensitivity Analysis for Hybrid Systems and Systems with Memory

## TL;DR

This paper introduces an adjoint sensitivity analysis method for hybrid systems with memory, enabling efficient computation of sensitivities in complex systems with mode switching and history dependence.

## Contribution

It extends existing sensitivity methods to hybrid systems with memory and provides solvability conditions for mode transitions and state initialization.

## Key findings

- Developed an adjoint sensitivity method for hybrid systems with memory.
- Provided solvability conditions for mode switching and initialization.
- Demonstrated the method with numerical examples including a mechanics problem.

## Abstract

We present an adjoint sensitivity method for hybrid discrete -- continuous systems, extending previously published forward sensitivity methods. We treat ordinary differential equations and differential-algebraic equations of index up to two (Hessenberg) and provide sufficient solvability conditions for consistent initialization and state transfer at mode switching points, for both the sensitivity and adjoint systems. Furthermore, we extend the analysis to so-called hybrid systems with memory where the dynamics of any given mode depend explicitly on the states at the last mode transition point. We present and discuss several numerical examples, including a computational mechanics problem based on the so-called exponential model constitutive material law for steel reinforcement under cyclic loading.