Can you please suggest a Lyapunov function to prove the stability of the following system:
\begin{equation} \dot x=-\frac{\partial f(x)}{\partial x}-a \lambda - \lambda P u\\ \dot \lambda=(a+Pu)^\top x-b\\ \dot u=\lambda g(u) P^\top x \end{equation}
where $a \in \mathbb{R}^n, P \in \mathbb{R}^{n \times n}, b \in \mathbb{R}, u \in \mathbb{R}^{n}, x \in \mathbb{R}^n, \lambda \in \mathbb{R}, g(u)$ is a scalar function of $u$ and $f(x)$ is a scalar function of $x$.
Would appreciate any help.
Without further specification of $f$ and $g$, it will likely be impossible to determine if this is stable or not.
If $\nabla f$ is invertible and $g(0) = 0$, then $(x^*, \lambda^*,0)$ is an equilibrium point with $\lambda^*$ solving $a^{\sf T}[\nabla f]^{-1}(-a\lambda^{*}) = b$ and $x^*$ being recovered as $x^* = [\nabla f]^{-1}(-a\lambda^{*})$.
If you can determine some further properties of this equilibrium point (for example, via fixed point analysis), you may be able to perform a linear stability analysis.