Explicit Solutions for LTV Systems: A New Perspective on Stability Criteria

Stability is a cornerstone in the analysis and design of complex dynamical systems, typically evaluated through the linearization of non-stationary system dynamics around a reference or attractive trajectory. While this process yields a linear time-varying (LTV) system—a representation central to modern frameworks such as contraction theory and adaptive control—general stability analysis for LTV systems remains a significant theoretical and practical challenge. Existing methodologies are often restricted to narrow cases, such as periodic or slowly varying systems, or rely on finding solutions to the time-dependent Lyapunov equation, which are frequently computationally demanding or analytically intractable for high-dimensional systems. This work introduces an alternative dynamical eigenstructure approach designed to derive explicit analytical solutions and define robust stability criteria for LTV systems. By generalizing classical time-invariant eigenvalue analysis, we define a dynamical eigenstructure consisting of time-varying eigenvalues and eigenfunctions that satisfy a fundamental differential dynamical equation. This unified perspective allows for the direct assessment of stability properties from the system’s time-evolving spectral characteristics without requiring the construction of a Lyapunov function. We provide a rigorous derivation of the stability conditions based on these dynamical trajectories. Finally, the effectiveness and versatility of this framework are demonstrated through several illustrative examples, showcasing its ability to provide deeper analytical insights into the transient behavior, and facilitate the stability analysis, of non-autonomous systems.