Recovery of rapidly decaying source terms from dynamical samples in evolution equations

We analyze the problem of recovering a source term of the form $h(t)=\sum_{j}h_j\phi(t-t_j)\chi_{[t_j, \infty)}(t)$ from space-time samples of the solution $u$ of an initial value problem in a Hilbert space of functions. In the expression of $h$, the terms $h_j$ belong to the Hilbert space, while $\phi$ is a generic real-valued function with exponential decay at $\infty$. The design of the sampling strategy takes into account noise in measurements and the existence of a background source.