Continuetijd

Continuetijd, literally Dutch for "continuous time," is a concept used to describe models in which time is treated as a continuous variable rather than as a sequence of discrete moments. In Continuetijd models, the state of a system changes smoothly over time, typically described by ordinary differential equations (ODEs) or by stochastic differential equations for random dynamics.

Mathematically, continuous-time dynamical systems are expressed as dx/dt = f(x,t,u(t)) in the deterministic case, or by stochastic calculus in the presence of randomness. Continuous-time processes include Brownian motion and Poisson processes, which evolve in real time without explicit time steps.

Analysis relies on calculus and transform methods, such as the Laplace transform and Fourier transform, together with stability, controllability, and observability concepts. Numerically, Continuetijd models are often simulated by discretizing time with methods like Euler or Runge-Kutta, producing approximations that converge as the time step shrinks.

Applications of Continuetijd span engineering (control systems, electrical circuits, signal processing), physics (mechanics, quantum dynamics), biology (population dynamics, pharmacokinetics), and finance (continuous-time models in economics).

Real-world data are typically obtained at discrete intervals, so discretization and sampling are essential for estimation and implementation, but the underlying theory of Continuetijd remains foundational for understanding system behavior between measurement points.