Lyapunov function

Theorem (Lyapunov's Direct Method for local stability)

(locally) stable in the sense of Lyapunov (i.s.L.)

Given system $\dot{\mathbf{x}} = f(\mathbf{x})$ , with $f$ continuous, for some region $\mathcal{D}$ around the origin (specifically open subset of $\mathbb{R}^n$ containing the origin), if there exists (can produce) a scalar, continuously-differentiable function $V(\mathbf{x})$ such that

\begin{gathered} V(\mathbf{x}) > 0, \forall \mathbf{x} \in {\cal D} \setminus \{0\} \quad V(0) = 0, \text{ and} \\ \dot{V}(\mathbf{x}) = \frac{\partial V}{\partial \mathbf{x}} f(\mathbf{x}) \le 0, \forall \mathbf{x} \in {\cal D} \setminus \{0\} \quad \dot{V}(0) = 0 \end{gathered}

then the origin ( $\mathbf{x} = 0$ ) is stable in the sense of Lyapunov (i.s.L.).

(i.e. if there exists a Lyapunov function $V(\mathbf{x},t)$ in some neighborhood $\mathcal{D}$ of the origin, then the equilibrium state at the origin is stable)

(locally) asymptotically stable

If additionally,

\dot{V}(\mathbf{x}) = \frac{\partial V}{\partial \mathbf{x}} f(\mathbf{x}) < 0, \forall \mathbf{x} \in {\cal D} \setminus \{0\}

then origin is (locally) asymptotically stable.

(i.e. if $\dot{V}(\mathbf{x},t)$ is negative definite in neighborhood $\mathcal{D}$ then equilibrium state at origin is asymptotically stable)

(locally) exponentially stable

Additionally if

\dot{V}(\mathbf{x}) = \frac{\partial V}{\partial \mathbf{x}} f(\mathbf{x}) \le -\alpha V(x), \forall \mathbf{x} \in {\cal D} \setminus \{0\}

for some $\alpha > 0$ , then origin is (locally) exponentially stable.

Theorem (Lyapunov analysis for global stability)

globally asymptotically stable

Given system $\dot{\mathbf{x}} = f(\mathbf{x})$ , with $f$ continuous, if there exists (can produce) a scalar, continuously-differentiable function $V(\mathbf{x})$ such that

\begin{gathered} V(\mathbf{x}) \succ 0, \ \dot{V}(\mathbf{x}) = \frac{\partial V}{\partial \mathbf{x}} f(\mathbf{x}) \prec 0, \text{ and} \\ V(\mathbf{x}) \rightarrow \infty \text{ whenever } ||\mathbf{x}||\rightarrow \infty, \end{gathered}

(note: $\succ$ denoting PSD) then the origin ( $\mathbf{x} = 0$ ) is globally asymptotically stable (G.A.S.).

globally exponentially stable

If additionally,

\dot{V}(\mathbf{x}) \preceq -\alpha V(\mathbf{x})

for some $\alpha > 0$ , then the origin is globally exponentially stable.

Theorem (Lyapunov analysis for stable linear systems)

Suppose linear system $\dot{\mathbf{x}} = \mathbf{A}\mathbf{x}$ , where a Lyapunov function can be found,

V(\mathbf{x}) = \mathbf{x}^T {\bf P} \mathbf{x}, \quad {\bf P} ={\bf P^T} \succ 0

where

\dot{V}(\mathbf{x}) = \mathbf{x}^T {\mathbf{PA}} \mathbf{x} + \mathbf{x}^T {\bf A}^T {\bf P}\mathbf{x} \prec 0

then the origin is globally exponentially stable.

Lyapunov functions for fixed linear systems

For unforced fixed linear system $\dot{\mathbf{x}} = A\mathbf{x}(t)$ , can choose Lyapunov function $V(\mathbf{x}) = \mathbf{x}^T(t) M \mathbf{x}(t)$ , where $M$ is a symmetric positive definite matrix.

Then, $\dot{V}(\mathbf{x}(t))$ #incomplete

References

https://underactuated.mit.edu/lyapunov.html
P. E. Sarachik, Principles of Linear Systems, Cambridge Press, 1996, pp. 193-202.
https://math24.net/stability-theory-basic-concepts.html
https://ecs-pw-facweb.ecs.csus.edu/~fbelkhou/LSsummary1.pdf
https://www.cds.caltech.edu/~murray/courses/cds101/fa02/caltech/mls93-lyap.pdf
T. L. Vincent and J. S. Brown, Evolutionary game theory, natural selection, and darwinian dynamics. Cambridge: Cambridge university press, 2005, p. 51.