This question has been asked here but there is no answer: https://math.stackexchange.com/questions/1585400/stabilize-the-vector-field-of-y-f-y-hthht-1h-y-of-ode-y

Consider autonomous ODE $y' = f(y)\quad (1)$ which has an invariant set $M$ defined by the equations $$h (y) = 0 \qquad (2) $$ i.e., assuming that the initial conditions satisfy $h (y (0)) = 0$, the solution of the ODE satisties $h ( y ( t )) = 0$ for all later times $t\geq 0$. Defined the Jacobian matrix ($y,h\in \mathbb{R}^n$) $$H(y)=\frac{\partial h}{\partial y}$$ and assume that it has full row rank for all $t$ (in particular, there are no more equations in (2) than in (1)). Next we stabilize the vector field, replacing the autonomous (1) by $$y' = f (y) - \gamma H^T(HH^T)^{-1}h( y )\quad(3) $$ Show that if there is a constant $\gamma_0$ such that $$|Hf(y)|_2 \leq \gamma_0 |h(y)|_2$$ for all $y$ in the neighborhood of the invariant set $M$ then $M$ becomes asymptotically stable, i.e. $|h(y(t))|$ decreases in $t$ for trajectories of (3) starting near $M$ , provided that $\gamma\geq \gamma_0$.

I have no clue to prove the claim. Can anyone help me? Thank you in advance !