Asymptotic Behavior
Control Engineering with Python
Symbols
| 🐍 | Code | 🔍 | Worked Example |
| 📈 | Graph | 🧩 | Exercise |
| 🏷️ | Definition | 💻 | Numerical Method |
| 💎 | Theorem | 🧮 | Analytical Method |
| 📝 | Remark | 🧠 | Theory |
| ℹ️ | Information | 🗝️ | Hint |
| ⚠️ | Warning | 🔓 | Solution |
🐍 Imports
🐍 Streamplot Helper
ℹ️ Assumption
From now on, we only deal with well-posed systems.
🏷️ Asymptotic
Asymptotic = Long-Term: when \(t \to + \infty\)
⚠️
Even simple dynamical systems may exhibit
complex asymptotic behaviors.
Lorenz System
\[ \begin{array}{lll} \dot{x} &=& \sigma (y - x) \\ \dot{y} &=& x (\rho - z) \\ \dot{z} &=& xy - \beta z \end{array} \]
Visualized with Fibre
Hadley System
\[ \begin{array}{lll} \dot{x} &=& -y^2 - z^2 - ax + af\\ \dot{y} &=& xy - b xz - y + g \\ \dot{z} &=& bxy + xz - z \end{array} \]
Visualized with Fibre
🏷️ Equilibrium
An equilibrium of system \(\dot{x} = f(x)\) is a state \(x_e\) such that the maximal solution \(x(t)\) such that \(x(0) = x_e\)
is global and,
is \(x(t) = x_e\) for any \(t > 0\).
💎 Equilibrium
The state \(x_e\) is an equilibrium of \(\dot{x} = f(x)\)
\[\Longleftrightarrow\]
\[f(x_e) = 0.\]
Stability
About the long-term behavior of solutions.
“Stability” subtle concept,
“Asymptotic Stability” simpler (and stronger),
“Attractivity” simpler yet, (but often too weak).
Attractivity
Context: system \(\dot{x} = f(x)\) with equilibrium \(x_e\).
🏷️ Global Attractivity
The equilibrium \(x_e\) is globally attractive if for every \(x_0,\) the maximal solution \(x(t)\) such that \(x(0)=x_0\)
is global and,
\(x(t) \to x_e\) when \(t \to +\infty\).
🏷️ Local Attractivity
The equilibrium \(x_e\) is locally attractive if for every \(x_0\) close enough to \(x_e\), the maximal solution \(x(t)\) such that \(x(0)=x_0\)
is global and,
\(x(t) \to x_e\) when \(t \to +\infty\).
🔍 Global Attractivity
The system
\[ \begin{array}{cc} \dot{x} &=& -2x + y \\ \dot{y} &=& -2y + x \end{array} \]
is well-posed,
has an equilibrium at \((0, 0)\).
🐍 Vector field
📈 Stream plot
🔍 Local Attractivity
The system
\[ \begin{array}{cc} \dot{x} &=& -2x + y^3 \\ \dot{y} &=& -2y + x^3 \end{array} \]
is well-posed,
has an equilibrium at \((0, 0)\).
🐍 Vector field
📈 Stream plot
🔍 No Attractivity
The system
\[ \begin{array}{cr} \dot{x} &=& -2x + y \\ \dot{y} &=& 2y - x \end{array} \]
is well-posed,
has a (unique) equilibrium at \((0, 0)\).
🐍 Vector field
📈 Stream plot
🧩 Pendulum
The pendulum is governed by the equation
\[ m \ell^2 \ddot{\theta} + b \dot{\theta} + m g \ell \sin \theta = 0 \]
where \(m>0\), \(\ell>0\), \(g>0\) and \(b\geq0\).
1. 🧮
Compute the equilibria of this system.
2. 🧠
Can any of these equilibria be globally attractive?
3. 📈
Assume that \(m=1\), \(\ell=1\), \(g=9.81\) and \(b=1\).
Make a stream plot of the system.
4. 🔬
Determine which equilibria are locally attractive.
5. 📈 Frictionless Pendulum
Assume now that \(b=0\).
Make a stream plot of the system.
6. 🧮 🧠
Prove that the equilibrium at \((0,0)\) is not locally attractive.
🗝️ Hint. Study how the total mechanical energy \(E\)
\[ E(\theta,\dot{\theta}) := m\ell^2 \dot{\theta}^2 / 2 - m g\ell \cos \theta \]
evolves in time.
🔓 Pendulum
1. 🔓
The 2nd-order differential equations of the pendulum are equivalent to the first order system
\[ \left| \begin{array}{rcl} \dot{\theta} &=& \omega \\ \dot{\omega} &=& (- b / m \ell^2) \omega - (g /\ell) \sin \theta \\ \end{array} \right. \]
Thus, the system state is \(x =(\theta, \omega)\) and is governed by \(\dot{x} = f(x)\) with
\[ f(\theta, \omega) = (\omega, (- b / m \ell^2) \omega - (g /\ell) \sin \theta). \]
Hence, the state \((\theta, \omega)\) is a solution to \(f(\theta, \omega) = 0\) if and only if \(\omega = 0\) and \(\sin \theta = 0\). In other words, the equilibria of the system are characterized by \(\theta = k \pi\) for some \(k \in \mathbb{Z}\) and \(\omega (= \dot{\theta}) = 0\).
2. 🔓
Since there are several equilibria, none of them can be globally attractive.
Indeed let \(x_1\) be a globally attractive equilibrium and assume that \(x_2\) is any other equilibrium. By definition, the maximal solution \(x(t)\) such that \(x(0) = x_2\) is \(x(t) = x_2\) for every \(t\geq0\). On the other hand, since \(x_1\) is globally attractive, it also satisfies \(x(t) \to x_1\) when \(t\to +\infty\), hence there is a contradiction.
Thus, \(x_1\) is the only possible equilibrium.
3. 🔓
4. 🔓
From the streamplot, we see that the equilibria
\[(\theta, \dot{\theta}) = (2k\pi, 0), \; k\in \mathbb{Z}\]
are asymptotically stable, but that the equilibria
\[(\theta, \dot{\theta}) = (2(k+1) \pi, 0),\; k\in \mathbb{Z}\]
are not (they are not locally attractive).
5. 🔓
6. 🔓
\[ \begin{split} \dot{E} &= \frac{d}{dt} \left(m\ell^2 \dot{\theta}^2 / 2 - m g\ell \cos \theta\right) \\ &= m\ell^2 \ddot{\theta}\dot{\theta} + m g \ell (\sin \theta) \dot{\theta} \\ &= \left(m\ell^2 \ddot{\theta} + m g \ell \sin \theta \right)\dot{\theta} \\ &= \left(- b \dot{\theta}\right)\dot{\theta} \\ &= 0 \end{split} \]
Therefore, \(E(t)\) is constant.
On the other hand,
\[ \min \, \{E(\theta, \dot{\theta}) \; | \; (\theta,\dot{\theta}) \in \mathbb{R}^2\} = E(0, 0) = -mgl. \]
Moreover, this minimum is locally strict. Precisely, for any \(0< |\theta| < \pi\), \[ E(0, 0) < E(\theta,\dot{\theta}). \]
If the origin was locally attractive, for any \(\theta(0)\) and \(\dot{\theta}(0)\) small enough, we would have \[ E(\theta(t), \dot{\theta}(t)) \to E(0, 0) \; \mbox{ when } \; t\to +\infty \] (by continuity). But if \(0 < |\theta(0)| < \pi\), we have
\[E(\theta(0), \dot{\theta}(0)) > E(0, 0)\]
and that would contradict that \(E(t)\) is constant.
Hence the origin is not locally attractive.
💎 Attractivity (Low-level)
The equilibrium \(x_e\) is globally attractive iff:
for any state \(x_0\) and for any \(\epsilon > 0\) there is a \(\tau \geq 0\),
such that the maximal solution \(x(t)\) such that \(x(0) = x_0\) is global and,
satisfies:
\[ \|x(t) - x_e\| \leq \epsilon \; \mbox{ when } \; t \geq \tau. \]
⚠️ Warning
Very close values of \(x_0\) could theoretically lead to very different “speed of convergence” of \(x(t)\) towards the equilibrium.
This is not contradictory with the well-posedness assumption: continuity w.r.t. the initial condition only works with finite time spans.
🔍 (Pathological) Example
\[ \begin{array}{lll} \dot{x} &=& x + xy - (x + y)\sqrt{x^2 + y^2} \\ \dot{y} &=& y - x^2 + (x - y) \sqrt{x^2 + y^2} \end{array} \]
Equivalently, in polar coordinates:
\[ \begin{array}{lll} \dot{r} &=& r (1 - r) \\ \dot{\theta} &=& r (1 - \cos \theta) \end{array} \]
🐍 Vector Field
📈 Stream Plot
Asymptotic Stability
Asymptotic stability is a stronger version of attractivity which is by definition robust with respect to the choice of the initial state.
🏷️ Global Asympt. Stability
The equilibrium \(x_e\) is globally asympt. stable iff:
for any state \(x_0\) and for any \(\epsilon > 0\) there is a \(\tau \geq 0\),
and there is a \(r > 0\) such that if \(\|x_0' - x_0\| \leq r\),
such that the maximal solution \(x(t)\) such that \(x(0) = x_0'\) is global and,
satisfies:
\[ \|x(t) - x_e\| \leq \epsilon \; \mbox{ when } \; t \geq \tau. \]
Set of Initial Conditions
Let \(f: \mathbb{R}^n \to \mathbb{R}^n\) and \(X_0 \subset \mathbb{R}^n\).
Let \(X(t)\) be the image of \(X_0\) by the flow at time \(t\):
\[ X(t) := \{x(t) \; | \; \dot{x} = f(x), \; x(0)= x_0, \; x_0 \in X_0 \ \}. \]
💎 Global Asympt. Stability
An equilibrium \(x_e\) is globally asympt. stable iff
for every bounded set \(X_0\) and any \(x_0 \in X_0\) the associated maximal solution \(x(t)\) is global and,
\(X(t) \to \{x_e\}\) when \(t\to +\infty\).
🏷️ Limits of Sets
\[ X(t) \to \{x_e\} \]
to be interpreted as
\[ \sup_{x(t) \in X(t)} \|x(t) - x_e\| \to 0. \]
🏷️ Hausdorff Distance
\[ \sup_{x(t) \in X(t)} \|x(t) - x_e\| = d_H(X(t), \{x_e\}) \]
where \(d_H\) is the Hausdorff distance between sets:
\[ d_H(A, B) := \max \left\{ \sup_{a \in A} d(a, B), \sup_{b\in B} d(A, b) \right\}. \]
🏷️ Local Asymptotic Stability
The equilibrium \(x_e\) is locally asympt. stable iff:
there is a \(r>0\) such that for any \(\epsilon > 0\),
there is a \(\tau \geq 0\) such that,
if \(\|x_0 - x_e\| \leq r\), the maximal solution \(x(t)\) such that \(x(0) = x_0\) is global and satisfies:
\[ \|x(t) - x_e\| \leq \epsilon \; \mbox{ when } \; t \geq \tau. \]
💎 Local Asympt. Stability
An equilibrium \(x_e\) is locally asympt. stable iff:
There is a \(r>0\) such that for every set \(X_0\) such that
\[ X_0 \subset \{x \; | \; \|x - x_e\| \leq r \}, \]
and for any \(x_0 \in X_0\), the associated maximal solution \(x(t)\) is global and
\[ X(t) \to \{x_e\} \mbox{ when } t\to +\infty. \]
🏷️ Stability
An equilibrium \(x_e\) is stable iff:
for any \(r>0\),
there is a \(\rho \leq r\) such that if \(|x(0) - x_e| \leq \rho\), then
the solution \(x(t)\) is global,
for any \(t\geq 0\), \(|x(t) - x_e| \leq r\).
🧩 Vinograd System
Consider the system:
\[ \begin{array}{rcl} \dot{x} &=& (x^2 (y-x) +y^5) / (x^2 + y^2 (1 + (x^2 + y^2)^2 )) \\ \dot{y} &=& y^2 (y - 2x) / (x^2 + y^2 (1 + (x^2 + y^2)^2 )) \end{array} \]
🐍 Vector field
📈 Stream plot
1. 🧮
Show that the origin \((0, 0)\) is the unique equilibrium.
2. 📈🔬
Does this equilibrium seem to be attractive (graphically) ?
3. 🧠
Show that for any equilibrium of a well-posed system:
💎 (locally) asymptotically stable \(\Rightarrow\) stable
4. 🧪📈
Does the origin seem to be stable (experimentally ?)
Conclude accordingly.
🔓 Vinograd System
1. 🔓
\((x,y)\) is an equilibrium of the Vinograd system iff
\[ \begin{array}{rcl} (x^2 (y-x) +y^5) / (x^2 + y^2 (1 + (x^2 + y^2)^2 )) &=& 0 \\ y^2 (y - 2x) / (x^2 + y^2 (1 + (x^2 + y^2)^2 )) &=& 0 \end{array} \]
or equivalently
\[ y^2 (y - 2x) = 0 \; \mbox{ and } \; x^2 (y-x) +y^5 = 0. \]
If we assume that \(y=0\), then:
\(y^2 (y - 2x) = 0\) is satisfied and
\(x^2 (y-x) +y^5 = 0 \, \Leftrightarrow \, -x^3 = 0 \, \Leftrightarrow \, x=0.\)
Under this assumption, \((0,0)\) is the only equilibrium.
Otherwise (if \(y\neq 0\)),
\(y^2 (y - 2x) = 0\) yields \(y = 2x\),
The substitution of \(y\) by \(2x\) in \(x^2 (y-x) +y^5 = 0\) yields \[x^3(1+32x^2)=0\] and therefore \(x=0\).
\(y^2 (y - 2x) = 0\) becomes \(y^3=0\) and thus \(y=0\).
The initial assumption cannot hold.
Conclusion:
The Vinograd system has a single equilibrium: \((0, 0)\).
2. 🔓
Yes, the origin seems to be (globally) attractive.
As far as we can tell, the streamplot displays trajectories that ultimately all converge towards the origin.
3. 🔓
Let’s assume that \(x_e\) is a (locally) asymptotically stable of a well-posed system.
Let \(r>0\) such that this property is satisfied and let
\[ B := \{x \in \mathbb{R}^n \; | \; \|x - x_e\| \leq r \} \subset \mathrm{dom} \, f. \]
The set \(x(t, B)\) is defined for any \(t\geq 0\) and since \(B\) is a neighbourhood of \(x_e\), there is \(\tau \geq 0\) such that for any \(t \geq \tau\), the image of \(B\) by \(x(t, \cdot)\) is included in \(B\).
\[ t\geq \tau \, \Rightarrow \, x(t, B) \subset B. \]
Additionally, the system is well-posed.
Hence there is a \(r' > 0\) such that for any \(x_0\) in the closed ball \(B'\) of radius \(r'\) centered at \(x_e\) and any \(t \in [0, \tau]\), we have
\[ \|x(x_0, t) - x(x_e, t) \| \leq r. \]
Since \(x_e\) is an equilibrium, \(x(t, x_e) = x_e\), thus \(\|x(x_0, t) - x_e \| \leq r.\) Equivalently,
\[ 0\leq t \leq \tau \, \Rightarrow \, x(t, B') \subset B. \]
Note that since \(x(0, B') = B'\), this inclusion yields \(B' \subset B\). Thus, for any \(t \geq 0\), either \(t\in [0, \tau]\) and \(x(t, B') \subset B\) or \(t\geq \tau\) and since \(B' \subset B\),
\[x(t, B') \subset x(t, B) \subset B.\]
Conclusion: we have established that there is a \(r > 0\) such that \(B \subset \mathrm{dom} \, f\) and a \(r'>0\) such that \(r' \leq r\) and
\[ t\geq 0 \, \Rightarrow x(t, B') \subset B. \]
In other words, the system is stable! 🎉
4. 🔓
No! We can pick initial states \((0, \varepsilon)\), with \(\varepsilon >0\) which are just above the origin and still the distance of their trajectory to the origin will exceed \(1.0\) at some point: