What is meant by forward invariance of a set? Can anyone give me a simple example to make me understand this.

Question

I would like to understand the concept of set invariance in short. Can anyone explain with a short example. If the example can be related to nonlinear control theory (state-space), that would be nice.

Answer 1

Consider a dynamical system with state $x(t)$ at time $t$ starting at some state $x_0$ at $t=0$. A set $S$ is forward invariant if $x_0\in S$ implies that $x(t)\in S$ for all $t\ge0$. That just means that the state trajectory is confined in that set at all times and this may help a lot analyzing its stability or some other properties.

For instance, consider a matrix $A$ that is Metzler (i.e. all the off-diagonal entries are nonnegative), then the set $S=\mathbb{R}^n_{\ge0}$ (nonnegative orthant) is forward invariant. There are more general systems which are cone invariant but not necessarily on the nonnegative orthant. There are similar analogues in the nonlinear case such as Lotka-Volterra models given by

$$ \begin{array}{rcl} \dot{x}_1&=&x_1(a-bx_2)\\ \dot{x}_1&=&x_2(c-dx_2). \end{array} $$