In control theory, a distributed-parameter system (as opposed to a lumped-parameter system) is a system whose state space is infinite-dimensional. Such systems are therefore also known as infinite-dimensional systems. Typical examples are systems described by partial differential equations or by delay differential equations.

With U, X and Y Hilbert spaces and ${\displaystyle A\,}$ ∈ L(X), ${\displaystyle B\,}$ ∈ L(U, X), ${\displaystyle C\,}$ ∈ L(X, Y) and ${\displaystyle D\,}$ ∈ L(U, Y) the following difference equations determine a discrete-time linear time-invariant system:

with ${\displaystyle x\,}$ (the state) a sequence with values in X, ${\displaystyle u\,}$ (the input or control) a sequence with values in U and ${\displaystyle y\,}$ (the output) a sequence with values in Y.

The continuous-time case is similar to the discrete-time case but now one considers differential equations instead of difference equations:

An added complication now however is that to include interesting physical examples such as partial differential equations and delay differential equations into this abstract framework, one is forced to consider unbounded operators. Usually A is assumed to generate a strongly continuous semigroup on the state space X. Assuming B, C and D to be bounded operators then already allows for the inclusion of many interesting physical examples, but the inclusion of many other interesting physical examples forces unboundedness of B and C as well.

The partial differential equation with ${\displaystyle t>0}$ and ${\displaystyle \xi \in [0,1]}$ given by

fits into the abstract evolution equation framework described above as follows. The input space U and the output space Y are both chosen to be the set of complex numbers. The state space X is chosen to be L²(0, 1). The operator A is defined as

It can be shown that A generates a strongly continuous semigroup on X. The bounded operators B, C and D are defined as