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

We analyze the problem of recovering a source term of the form $h(t)={\sum }_j{h}_j\phi (t-t_j){\chi }_{[t_j,\mathrm{\infty })}(t)$ 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 $\mathrm{\infty }$. The design of the sampling strategy takes into account noise in measurements and the existence of a background source.