Consider the following dynamical system \begin{align} \dot{x}_1&=-ax_1 + w^T(t)x_2,\quad x_1\in\mathbb{R}^1 \\ \dot{x}_2 &= -w(t)x_1, \quad x_2\in\mathbb{R}^n \end{align} where $a>0$ and $w(t):\mathbb{R}\rightarrow \mathbb{R}^{n}$. It is well know that the zero equilibrium $(x_1^*,x_2^*)=(\mathbb{0},\mathbb{0})$ is uniformly stable if both $w(t)$ and $\dot{w}(t)$ are bounded and there exist positive constants $T$, $\alpha_1$, and $\alpha_2$ such that $$\alpha_2 I\succeq\int\limits_{t}^{t+T}{w(\tau)w^T(\tau)d\tau}\succeq\alpha_1 I,\quad \text{for all } t\ge 0$$ where I is the identity matrix.

Is there any simple approximation of the rate of convergence of $z=[x_1,x_2^T]^T$?I am interested in exponential convergence, that is finding positive constants $b$ and $\lambda$ $$ \Vert z(t) \Vert\le b \mathbf{e}^{-\lambda t}\Vert z(0)\Vert $$

  • $\begingroup$ Is anything known about $w(t)$? For example when it is periodic it can be shown. $\endgroup$ – Kwin van der Veen Aug 29 '18 at 23:24

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