2.6 Contractions and expansions

In this section, we consider the system

 

x' = Ax (2.8)

when it is either a sink or a source and characterize how fast the solutions of (2.8) contract to or expand from the origin. The following result implies that solutions of (2.8) contract exponentially to a sink.

Theorem 2.6.1:     If system (2.8) is a sink, then there exists M, k > 0 such that

$$\mid e^{At} \mid \, \leq Me^{-kt} \mid x \mid \, , \quad \for... ...t \in R_{+} \quad \mbox{and} \quad \forall \; x \in R^{n} \, .$$

Proof: (omitted)

Although Theorem 2.6.1 guarantees that any initial state is attracted at an exponential rate in time to a sink, it does not imply that the distance from 0, i.e. $\mid e^{At} x \mid$ decreases monotonically with t. In other words as the orbits approach 0, they do not necessarily cut the spheres $\mid x \mid \, = \gamma$ in the inward direction.

However, from intuition, at least in R³, one might expect to find a family of concentric ellipsoids such that as the orbits approach 0, they intersect the ellipsoids inward. This is indeed the case.

Theorem 2.6.2:     If the origin is a sink of (2.8), then there exists a positive definite matrix Q such that A^T Q +QA = - I and the quadratic function

V(x) = x^T Qx

satisfies

$$\frac{dV(x(t))}{dt} = \nabla V(x) \cdot Ax = - \mid x (t) \mid^{2}, \quad t \geq 0 \, ,$$

for any solution x(t) of (2.8).

Proof: (omitted)

Remark:     In practice, to find Q one usually solves the matrix equation (2.10) which is often called the Lyapunov equation.

Example 2.6.1:     Consider (2.8) with $A = \left( \begin{array}{rr} -1 & 3 \\ -6 & -2 \end{array} \right)$. Then A^T Q + QA = -I, $Q = \left( \begin{array}{cc} \frac{1}{2} & 0 \\ 0 & \frac{1}{4} \end{array} \right)$ and $V(x) = \frac{1}{2} x_{1}^{2} + \frac{1}{4} x_{2}^{2}$.

$$\frac{dV}{dt} = \nabla V (x) \cdot Ax = - \mid x \mid^{2} \, .$$

Correspondingly, we have the following results for a source of (2.8).

Theorem 2.6.3:     If the origin is a source of (2.8), then

1.

$\exists \; \alpha$, L > 0 such that

$$\mid e^{At} x \mid \, \geq L e^{\alpha t} \mid x \mid \, ,\qu... ...all \quad t \in R_{+} \,, \quad \forall \quad x \in R^{n} \, .$$

2.

$\exists$ a positive definite matrix Q such that

A^T Q + QA = I

and for V(x) = x^T Qx and all solutions x(t) of (2.8)

$$\frac{dV(x(t))}{dt} = \mid x(t) \mid^{2} \, , \quad t \geq 0 \, .$$

The proof is left as an exercise.