Trajectorieswithout

Trajectorieswithout is a neologism in the field of dynamical systems and trajectory planning, used to denote a class of system trajectories that avoid a specified exclusion set. The concept emphasizes constraint satisfaction along the evolution of a system, rather than optimization of an objective alone.

Formally, consider a dynamical system x' = f(x,t) on a state space X, with a constraint or forbidden set C ⊆ X. A trajectory x(t) defined on an interval I is a trajectorieswithout with respect to C if x(t) ∉ C for all t in I, and typically x(0) ∉ C as well. If C is time-dependent, C = C(t) and the condition becomes x(t) ∉ C(t) for all t ∈ I.

Variants include trajectories that avoid multiple sets, or that avoid C while satisfying additional goals such as terminal conditions or optimization criteria. In viability theory, such trajectories are related to the concept of viability kernels and controlled invariance, which characterize initial states from which trajectorieswithout are guaranteed to exist under given dynamics and controls.

Computationally, trajectorieswithout arise in obstacle-avoiding motion planning, barrier-method optimization, and reachability analyses. Algorithms such as sampling-based planners, convex barrier formulations, and model-predictive control can produce feasible trajectories that remain outside C.

Applications span robotics, autonomous vehicles, and safety-critical systems where remaining outside unsafe or forbidden regions is essential. The term is not universally standardized and may be used informally to describe any path that respects exclusion constraints, rather than a formal, widely adopted mathematical definition.