Systemspotential

Systemspotential, often referred to as system potential or potential function in the context of dynamical systems, is a scalar function defined on the state space that encodes a quantity such as potential energy or an energy-like measure. It is used to understand system behavior, stability, and certain classes of control or optimization problems.

Mathematically, let x be the state and V: R^n → R be a differentiable function. If the system dynamics can be written as dx/dt = f(x) and f(x) = -∇V(x), the system is a gradient system and V decreases along trajectories: dV/dt = ∇V·dx/dt = -||∇V||^2 ≤ 0. Global existence of V as a potential function implies conservative or dissipative structure; in non-dissipative cases V may remain constant along motion, while in dissipative cases it typically decreases toward equilibria. Not all systems admit a global potential; some exhibit non-conservative forces or path-dependent behavior.

In physics and engineering, V often represents potential energy, as in gravitational, electrostatic, or spring systems. In control theory and energy-based modeling, potential functions are used as Lyapunov functions to assess stability, or as part of port-Hamiltonian formulations that separate energy storage, dissipation, and interconnection. In machine learning, energy-based models use similar potential concepts to assign low energy to desired configurations.

Practical use involves selecting or constructing V to capture the desired equilibria and stability properties. Limitations arise when a global potential cannot be defined due to non-conservative forces or complex interactions, necessitating alternative methods for analysis or control. See also potential energy, Lyapunov function, and gradient systems.