Lyapunov functions and Stability

Autonomous Equations

Suppose now $U \subseteq \R^d$ is open, and $f \in C^1(U, \R^d)$, consider the equation $$\label{e:D}\tag{D} \dot x = f(x) $$ If $f(x_0) = 0$, then clearly $x(t) = x_0$ is a solution (called an equilibrium). Our aim is to understand the stability of this equilibrium.

Definition 1. An equilibrium point $x_0$ is:

  1. Stable if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that if $\abs{x(0) - x_0} < \delta$, then the solution exists and satisfies $\abs{x(t) - x_0} < \varepsilon$ for all $t \geq 0$.
  2. Asymptotically stable if it is stable and there exists a $\delta_0 > 0$ such that if $\abs{x(0) - x_0} < \delta_0$, then $x(t) \to x_0$ as $t \to \infty$.
  3. Unstable if it is not stable.

Lyapunov Functions

Definition 2. Let $x_0$ be an equilibrium of \eqref{e:D}. A function $V \in C^1(U, \R)$ is a Lyapunov function for $x_0$ if there exists a neighborhood $W \subseteq U$ of $x_0$ such that:

  1. $V(x_0) = 0$ and $V(x) > 0$ for all $x \in W - \set{x_0}$.
  2. $\grad V \cdot f \leq 0$ in $W$.

If instead $\grad V \cdot f < 0$ in $W - \set{x_0}$, then $V$ is called a strict Lyapunov function.

Theorem 3 (Lyapunov’s Stability Theorem). Let $x_0$ be an equilibrium of \eqref{e:D}.

  1. If there exists a Lyapunov function for $x_0$, then $x_0$ is stable.
  2. If there exists a strict Lyapunov function for $x_0$, then $x_0$ is asymptotically stable.

Proof sketch. First check $\partial_t V(x(t)) = \grad V(x(t)) \cdot f(x(t)) \leq 0$ so $V(x(t))$ is decreasing. Then find $c, \delta > 0$ such that $W \supseteq \bar B(x_0, \delta)$, and $\set{V \leq c} \cap B(x_0, \delta)$ contains a neighborhood of $x_0$.

For asymptotic stability, $V(x(t))$ is strictly decreasing and bounded below by $0$, so it must converge to some limit $L \geq 0$. If $L > 0$, show there exists $\delta > 0$ such that $\dot V(x) \leq -\delta_0 < 0$, and use this to find a contradiction.

Linearized Stability

Theorem 4. Suppose $f(x_0) = 0$ and $\lambda_1$, …, $\lambda_d$ are the eigenvalues of $Df_{x_0}$. If $\Re(\lambda_j) < 0$ for all $j \in \set{1, \dots, d}$, then the equilibrium $x_0$ is asymptotically stable.

Lemma 5. Let $A$ be a $d \times d$ matrix and suppose all eigenvalues of $A$ have strictly negative real part. There exists a positive-definite symmetric matrix $P$ such that $$ A^T P + P A = -I \,. $$

Proof sketch. Option 1: Choose $$ P \defeq \int_0^\infty e^{t A^T} e^{t A} \, dt \,. $$ Option 2: Let $L(P) = A^T P + P A^T$. Compute eigenvalues of $L(P)$ and check that $L$ is an invertible operator on the space of $d \times d$ symmetric matrices. Hence, $L(P) = Q$ has a solution for any symmetric matrix $Q$.

Proof sketch of Theorem 4. Without loss of generality, assume $x_0 = 0$. Write $f(x) = A x + R(x)$, where $A = Df_0$ and $\abs{R(x)} / \abs{x} \to 0$ as $x \to 0$.

Define $$ V(x) = x^T P x\,. $$ Check $$ \partial_t V(x(t)) = -\abs{x}^2 + 2 x^T P R(x) \,, $$ and verify that $V$ is a Lyapunov function.

Problem 6. Consider the second-order mechanical system with friction: $$ \ddot{x} + \gamma \dot{x} + \nabla V(x) = 0 $$ where $V \in C^2(U, \R)$, $\gamma > 0$.

  1. Rewrite this as a first-order system for $(x, v) \in \mathbb{R}^d \times \mathbb{R}^d$, where $v = \dot x$.
  2. If $V$ has a strict, non-degenerate, local minimum at $x_0$, then show that $(x_0, 0)$ is an asymptotically stable equilibrium.

Problem 7. Let $V \in C^2(\R^d, \R)$, and consider the system $$ \dot{x} = -\nabla V(x) + \ip{\nabla V(x), x } x $$

  1. Show that the unit sphere $S^{d-1} \subseteq \R^d$ is forward invariant.
  2. Prove that equilibria of this system correspond to points where $\grad V(x)$ is parallel to $x$.

Instability

Theorem 8 (Chetaev’s instability theorem). Let $x_0$ be an equilibrium of \eqref{e:D}. Suppose there exists a neighborhood $W \subseteq U$ of $x_0$, a $C^1$ function $V \colon W \to \R$, and an open set $U_1 \subseteq W$ such that:

  1. $x_0 \in \partial U_1$.
  2. $V(x) > 0$ for all $x \in U_1$, and $V(x) = 0$ for all $x \in \partial U_1 \cap W$.
  3. $\grad V(x) \cdot f(x) > 0$ for all $x \in U_1$.

Then $x_0$ is unstable.

Proof sketch. Pick any $r > 0$ such that $\bar B(x_0, r) \subseteq W$. If $x(0) \in U_1 \cap B(x_0, r)$, then $\dot(V(x(t))) \geq 0$ and so $V(x(t))$ increases with time. Hence the trajectory $x(t)$ can only leave $B(x_0, r) \cap U_1$ through $\partial U_1$.

If it stays $B(x_0, r) \cap U_1$ forever, then show that $\partial_t V(x(t)) \geq \delta_0 > 0$ for some $\delta_0 > 0$, and produce a contradiction. This means $x(t)$ must exit $B(x_0, r)$ in finite time, proving instability.

Theorem 9 (Linear instability). Suppose $f(x_0) = 0$ and $\lambda_1$, …, $\lambda_d$ are the eigenvalues of $Df_{x_0}$. If there exists $j \in \set{1, \dots, d}$ such that $\Re(\lambda_j) > 0$, then the equilibrium $x_0$ is unstable.

Lemma 10. Let $A$ be a $d \times d$ matrix with eigenvalues $\lambda_1$, …, $\lambda_d$.

  1. If $\Re(\lambda_i) > 0$ for all $i$, then there exists a positive-definite symmetric matrix $P$ such that $A^T P + PA = I$.
  2. If $\Re(\lambda_i) \leq 0$ for all $i$, then for any $\epsilon > 0$ there exists a positive-definite symmetric matrix $P$ such that $$ x \cdot (A^T P + PA) x \leq \epsilon \abs{x}^2 \,. $$

Proof sketch of Theorem 9. Without loss of generality, assume $x_0 = 0$. Write $f(x) = A x + R(x)$ with $A = Df_0$ and $\abs{R(x)}/\abs{x} \to 0$ as $x \to 0$.

Decompose $\R^d = E^u \oplus E^{cs}$ into the unstable subspace $E^u$ (corresponding to $\Re(\lambda) > 0$) and center-stable subspace $E^{cs}$ ($\Re(\lambda) \le 0$). Let $A_u$, $A_{cs}$ be the restrictions of $A$ to $E^u$ and $E^{cs}$ respectively, and write any point $x \in \R^d$ as $x = (u, v) \in E^u \oplus E^{cs}$.

Let $\epsilon > 0$ (to be chosen shortly). Use the lemma to find positive transformations $P_u$, $P_{cs}$ such that $A_u^T P_u + P_u A_u = I$, and $(A_{cs}^T P_{cs} + P_{cs} A_{cs}) \leq \epsilon I_{cs}$. Now define $$ V(x) = \frac{1}{2}\norm{u}_{P_u}^2 - \frac{1}{2}\norm{v}_{P_{cs}}^2 \,, $$ and the region $U_1 = \set{x \in B(0, r) : V(x) > 0}$. Show that when $\epsilon$ is small enough Theorem 8 applies and finish the proof.

Remark 11. Suppose $f(x_0) = 0$ and $\lambda_1$, …, $\lambda_d$ are the eigenvalues of $Df_{x_0}$. Theorem 4 shows that if $\Re(\lambda_j) < 0$ for all $j$, then $x_0$ is asymptotically stable. Theorem 9 shows that if even one eigenvalue has $\Re(\lambda_j) > 0$, then $x_0$ is unstable. If $\Re(\lambda_j) \leq 0$ for all $j$ and at least one $\Re(\lambda_j) = 0$, then linear stability is inconclusive, and the stability of the equilibrium is governed by nonlinear effects.

Problem 12. Let $f \in C^1(U, \R^d)$, $x_0 \in U$ such that $f(x_0) = 0$, and $\lambda_1, \dots, \lambda_d$ be the eigenvalues of $Df_{x_0}$. Suppose $\Re(\lambda_j) \leq 0$ for all $j \in \set{1, \dots, d}$. Find examples showing that the equilibrium $x_0$ may be stable, asymptotically stable, or unstable.

Problem 13. Let $f \in C^1(U, \R^d)$ with $f(0) = 0$. Suppose there exists $v_0 \in \R^d - 0$, $r > 0$ such that $$ v_0 \cdot f(x) > 0 \quad \text{for all } x \in B(0, r) \text{ with } x \cdot v_0 > 0 \,. $$ Show that the equilibrium $x_0 = 0$ is unstable.

Proposition 14 (Gradient flows). Let $V \in C^2(U, \R)$, $x_0 \in U$ be such that $V(x_0) = 0$, and consider the gradient flow $$\label{e:G}\tag{G} \dot x = - \grad V(x) \,. $$

  1. If $V$ attains a strict local minimum at $x_0$, then show that $x_0$ is a stable equilibrium for \eqref{e:G}.
  2. If $V$ attains a strict, non-degenerate, local minimum at $x_0$, then show that $x_0$ is an asymptotically stable equilibrium for \eqref{e:G}.
  3. If $x_0$ is an isolated critical point of $V$, but not a local minimum of $V$, then show that $x_0$ is an unstable equilibrium of \eqref{e:G}