Controlsupport

Controlsupport refers to the region in which a control input actively influences a dynamical system. In continuous-time models, a control signal u(t) defined on an interval [0, T] has a support, commonly denoted supp(u), defined as the closure of the set { t ∈ [0, T] : u(t) ≠ 0 }. Outside this region the input is zero, so the system evolves under its natural dynamics without active control. While the idea is general, the term is most often used with respect to time-domain actuation; spatial notions of active actuator regions may also be described under similar language.

In applications, controlsupport is relevant to sparse or energy-efficient control. Minimizing the measure or duration of supp(u) can reduce energy usage, actuator wear, and communication needs in networked control. In bang-bang or time-optimal problems, the active interval can be compact, concentrating actuation to short time windows.

In discrete-time models, supp(u) is the set of time steps k for which the control input u_k is nonzero. The concept also underpins numerical methods that enforce sparsity or respect actuator constraints, where keeping the support small simplifies implementation.

Mathematically, controlsupport is related to the broader notion of the support of a function. A control with compact support is nonzero only on a finite subinterval, which can simplify analysis of reachability, stability, and the effect of actuators.

See also: control theory, optimal control, sparse control, actuation, bang-bang control, model predictive control, and the mathematical notion of support.