Projected dynamical systems is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of optimization and equilibrium problems and the dynamical world of ordinary differential equations. A projected dynamical system is given by the flow to the projected differential equation

${\displaystyle {\frac {dx(t)}{dt}}=\Pi _{K}(x(t),-F(x(t)))}$

where K is our constraint set. Differential equations of this form are notable for having a discontinuous vector field.