# Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems

Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz, $$\begin{split} \dot{x}_1 &= x_2 - 314.2 \\ \dot{x}_2 &= 122\sin{x_1}\cos{x_1} - 154.8x_3\sin{x_1} - 0.5x_2 + 201.2 \\ \dot{x}_3 &= 0.64\cos{x_1} - 0.8x_3 + 0.2u \end{split}$$ where $\mathbf{x} = \mathbf{x}(t) = [x_1(t), x_2(t), x_3(t)]^T \in X \subset\mathbb{R}^3$ are the state variables; $t \in \mathbb{R}^{+}$ denotes time and $u = u(\mathbf{x},t) \in U \subset \mathbb{R}$ is the driving variable dependent both on $\mathbf{x}$ and $t$, which is unknown. We mention here that $X=[0.4, 2.1] \times [298.5, 329.9] \times [-2, 2]$, and $U = [-6, 6]$.

We need to solve the Lyapunov stability problem, namely, how to find a bounded control law $u(\mathbf{x},t)\in U$ and the corresponding invariance set $\mathcal{X}_o\subset X$, such that $\forall \mathbf{x}_0 \in \mathcal{X}_o, \forall t>0$, $\mathbf{F}(\mathbf{x}_0, t, u) \in \mathcal{X}_o$?

Alternatively, given a positive-definite Lyapunov function such as $$V(\mathbf{x}) = 0.5(x_2 - 314.2)^2 + 0.5(1.6\sin{x_1}\cos{x_1} - 2x_3\sin{x_1} + 0.6)^2$$ find $u(\mathbf{x},t) \in U$ and the invariant set $\mathcal{X}_o \subset X$, $\forall \mathbf{x}_0 \in \mathcal{X}_o$, $\forall t>0$, such that $$\frac{\mathrm{d}}{\mathrm{d}t} V(\mathbf{x}, t) \leq -\gamma V(\mathbf{x}, t)$$ where $\gamma>0$ is the decay rate corresponding to the convergence speed to the equilibrium. For example, it could be in the range $[0.01,1]$.

The analytical solution might not be available. We may discretize the above problem. Hence, a time series of the control law is preferable.

Any hints or comments are appreciated.

• The autonomous system of the above system is unstable. That is why it need a driving variable u to make it stable. – user2336850 May 20 '13 at 20:04
• Why did you call this "non-autonomous" ? I do not see $t$ there explicitly in the RHS. Hence it does not explicitly depend on its independent variable so it should be autonomous and not non-autonomous? – Nasser Oct 27 '13 at 17:51
• Nasser: $u=u(\mathbf{x},t)$ – Ernesto Iglesias Feb 14 '17 at 15:20

Have you done the time derivative of $V(x)$?
Hint: Once you compute $\frac{d}{dt}V(x)$ (note that it does not depends explicitly on $t$), you can upper bound it with expressions like $\sin x \leq 1$. And for the worst case condition, find such $u$ in order to make the derivative always negative for some compact set $\mathcal{C}=\{x: V(x) \leq c\}$. Actually, $\mathcal{C}$ of interest is given by the bounds of your state vector in the statement of your problem.