  We provide a control-theoretic perspective on optimal tensor algorithms for
minimizing a convex function in a finite-dimensional Euclidean space. Given a
function $\Phi: \mathbb{R}^d \rightarrow \mathbb{R}$ that is convex and twice
continuously differentiable, we study a closed-loop control system that is
governed by the operators $\nabla \Phi$ and $\nabla^2 \Phi$ together with a
feedback control law $\lambda(\cdot)$ satisfying the algebraic equation
$(\lambda(t))^p\|\nabla\Phi(x(t))\|^{p-1} = \theta$ for some $\theta \in (0,
1)$. Our first contribution is to prove the existence and uniqueness of a local
solution to this system via the Banach fixed-point theorem. We present a simple
yet nontrivial Lyapunov function that allows us to establish the existence and
uniqueness of a global solution under certain regularity conditions and analyze
the convergence properties of trajectories. The rate of convergence is
$O(1/t^{(3p+1)/2})$ in terms of objective function gap and $O(1/t^{3p})$ in
terms of squared gradient norm. Our second contribution is to provide two
algorithmic frameworks obtained from discretization of our continuous-time
system, one of which generalizes the large-step A-HPE framework and the other
of which leads to a new optimal $p$-th order tensor algorithm. While our
discrete-time analysis can be seen as a simplification and generalization
of~\citet{Monteiro-2013-Accelerated}, it is largely motivated by the
aforementioned continuous-time analysis, demonstrating the fundamental role
that the feedback control plays in optimal acceleration and the clear advantage
that the continuous-time perspective brings to algorithmic design. A highlight
of our analysis is that we show that all of the $p$-th order optimal tensor
algorithms that we discuss minimize the squared gradient norm at a rate of
$O(k^{-3p})$, which complements the recent analysis.
