learning rate for backtracking line-search [@armijo96]. In each iteration of SLQ, the line-search parameter controls the maximum step to move along the feedforward component of the control law update. This parameter is chosen by starting with a full step in the direction of the update ($\alpha=1$), and then iteratively shrinking the step size until the cost associated to the updated controller rollout is lower than the current nominal trajectories cost. Before proceeding to the main theorem and its proof, we state the following lemma which will be used in Theorem 1.

$H: \mathbb{R}^n \rightarrow \mathbb{R}$ and $\vy: \mathbb{R} \rightarrow \mathbb{R}^n$ as twice continuously differentiable functions. If $|\tau-\bar{\tau}|<\varepsilon_1$ and $|\delta\!\vy|<\varepsilon_2$, for small enough $\varepsilon_1$ and $\varepsilon_2$, $H(\vy(\tau)+\delta\!\vy)$ can be approximated as $$\begin{aligned}
H(\vy(\tau)+\delta\!\vy) &\simeq H(\bar{\vy}) + \delta\!\vy^\top \nabla\!H(\bar{\vy}) + \partial\bar{\vy}^\top \nabla\!H(\bar{\vy}) (\tau-\bar{\tau}) \notag \\
& + \frac{1}{2} \delta\!\vy^\top \nabla^2\!H(\bar{\vy}) \delta\!\vy + \delta\!\vy^\top \nabla^2\!H(\bar{\vy}) \partial\bar{\vy} (\tau-\bar{\tau})  \notag \\
& + \frac{1}{2} \Big( \partial\bar{\vy}^\top \nabla^2\!H(\bar{\vy}) \partial\bar{\vy} + \partial^2\bar{\vy}^\top \nabla\!H(\bar{\vy}) \Big) (\tau-\bar{\tau})^2 , \notag\end{aligned}$$ where $\bar{\vy} = \vy(\bar{\tau})$, $\partial\bar{\vy} = \frac{d\vy}{d\tau}|_{\bar{\tau}}$, and $\partial^2\bar{\vy} = \frac{d^2\vy}{d\tau^2}|_{\bar{\tau}}$. Furthermore, $\nabla\!H$ and $\nabla^2\!H$ are the gradient and the Hessian of $H$ respectively.

This lemma can be easily proven using the Taylor series expansion of $H$ and few steps 