Editing Sliding mode control (section)

===Existence of closed-loop solutions===
Note that because the control law is not [[continuous function|continuous]], it is certainly not locally [[Lipschitz continuous]], and so existence and uniqueness of solutions to the [[Closed-loop transfer function|closed-loop system]] is ''not'' guaranteed by the [[Picard–Lindelöf theorem]]. Thus the solutions are to be understood in the [[Aleksei Fedorovich Filippov|Filippov]] sense.<ref name="Zinober1990"/><ref name="Filippov88">{{Cite book
 | last       = Filippov
 | first      = A.F.
 | title      = Differential Equations with Discontinuous Right-hand Sides
 | publisher  = Kluwer
 | year       = 1988
 | isbn       = 978-90-277-2699-5
}}</ref> Roughly speaking, the resulting closed-loop system moving along <math>\sigma(\mathbf{x}) = \mathbf{0}</math> is approximated by the smooth [[dynamic system|dynamics]] <math>\dot{\sigma}(\mathbf{x}) = \mathbf{0};</math> however, this smooth behavior may not be truly realizable. Similarly, high-speed [[pulse-width modulation]] or [[delta-sigma modulation]] produces outputs that only assume two states, but the effective output swings through a continuous range of motion. These complications can be avoided by using a different [[nonlinear control]] design method that produces a continuous controller. In some cases, sliding-mode control designs can be approximated by other continuous control designs.<ref name="Khalil02"/>