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For functions whose Laplace transforms converge on a common right half-plane, the causal convolution $(f\ast g)(t)={\int }_0^{t}f(s)g(t-s)ds$ satisfies $\mathcal{L}(f\ast g)=\mathcal{L}(f)\mathcal{L}(g)$. This converts a transformed boundary forcing into a known time-domain integral representation.

For a finite-dimensional continuous-time linear time-invariant system $\dot{x}=Ax+Bu$ on ${\mathbb{R}}^n$, with constant matrices $A,B$ over the real numbers, no state constraints, and every piecewise continuous function $u:[0,T]\to {\mathbb{R}}^m$ allowed as a control input without an amplitude bound, controllability in any fixed $T>0$ is equivalent to the controllability matrix having rank of a matrix $n$. The same equivalence holds with all square-integrable functions as inputs. A stronger useful statement iswhere $e^{AT}$ is a matrix exponential and $\mathrm{im}\mathcal{C}$ is the column space. Thus a rank of a matrix $r<n$ leaves only an $r$-dimensional affine subspace reachable at that time. ^[ref]

The Kalman rank criterion allows arbitrarily large commands as $T$ shrinks. It does not assert controllability with bounded control inputs, nor test a general nonlinear dynamical system just by counting its inputs. For the double integrator, $B=(0,1{)}^T$ and $AB=(1,0{)}^T$ give two independent directions, despite there being only one control input.

The Laplace transform is $\hat{f}(p)={\int }_0^{\infty }{e}^{-pt}f(t)dt$ and converts differentiation into multiplication by $p$ plus initial data.

The Poisson arrivals see time averages property says that an external Poisson process independent of a stationary system sees the system-state distribution at its arrival times. Thus the fraction of arrivals that accept a state-dependent admission rule $p(n)$ is ${\sum }_n{\pi }_{n}p(n)$.