Work

Dissipative energy functionals of passive linear time‐varying systems

Year: 2026

Type: article

Abstract: The concept of dissipativity plays a crucial role in the analysis of control systems. Dissipative energy functionals, also known as Hamiltonians, storage functions, or Lyapunov functions, depending on the setting, are extremely valuable to analyze and control the behavior of dynamical systems, but in general circumstances they are very difficult to compute, and not fully understood. In this paper, we consider passive linear time‐varying (LTV) systems, under mild regularity assumptions, and their associated storage functions, as a necessary step to analyze general nonlinear systems. We demonstrate that every passive LTV system must have at least one time‐varying positive semidefinite quadratic storage function, greatly reducing our search scope. Now, focusing on quadratic storage functions, we analyze in detail their necessary regularity, which is lesser than continuous. Moreover, we prove that the rank of quadratic storage functions is non‐increasing in time, allowing us to introduce a novel null space decomposition, under much weaker assumptions than the one needed for general matrix functions. Additionally, we show a necessary kernel condition for the quadratic storage function, allowing us to reduce our search scope even further.

Source: ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

DOI: 10.1002/zamm.70325

Authors: Riccardo Morandin, Dorothea Hinsen

Institutions: Otto-von-Guericke-Universität Magdeburg, Max Planck Institute for Dynamics of Complex Technical Systems

Language: English

Topic: Control and Stability of Dynamical Systems

Subfield: Control and Systems Engineering

Field: Engineering

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,460

Funders: Deutsche Forschungsgemeinschaft, Bundesministerium für Bildung und Forschung, Berlin Mathematics Research Center MATH+