# On the decay rate for degenerate gradient flows subject to persistent excitation

### Dario Prandi, Yacine Chitour, Paolo Mason

**Abstract:**
In this paper, we study the worst rate of exponential decay for degenerate gradient flows in $\mathbb R^n$ of the form $\dot x(t)=-c(t)c(t)^\top x(t)$, issued from adaptative control theory, under a persistent excitation (PE) condition. That is, there exists $a,b,T>0$ such that, for every $t\geq 0$ it holds $a I_n\leq \smallint_t^{t+T}c(s)c(s)^\top ds\leq b I_n$. Our main result is an upper bound of the form $\frac{a}{(1+b)^2T}$, to be compared with the well-known lower bounds of the form $\frac{a}{(1+nb^2)T}$. As a byproduct, we also provide necessary conditions for the exponential convergence of these systems under a more general (PE) condition.
Our techniques relate the worst rate of exponential decay to an optimal control problem that we study in detail.