# 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
$$\frac{\mathrm{d}}{\mathrm{d}t} V(\mathbf{x}, t) \leqslant -\gamma V(\mathbf{x}, t)$$
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.

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Why the analytic solution is not available?

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 upperbound 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.