mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, a positively (or positive) invariant set is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

with the following properties: Suppose

\dot=f(x)

is a

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

x(t,x_0)

is a trajectory, and

x_0

is the initial point. Let

\mathcal := \left \lbrace x \in \mathbb^n\mid \varphi (x) = 0 \right \rbrace

where

\varphi

is a real-valued

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

. The set

\mathcal

is said to be positively invariant if

x_0 \in \mathcal

implies that

x(t,x_0) \in \mathcal \ \forall \ t \ge 0

In other words, once a trajectory of the system enters

\mathcal

, it will never leave it again.

References

*Dr. Francesco Borrell

* A. Benzaouia. book of "Saturated Switching Systems". chapter I, Definition I, Springer 2012.

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