In
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
, a distributed-parameter system (as opposed to a
lumped-parameter system) is a
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
whose
state space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory.
For instance, the toy ...
is infinite-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
. Such systems are therefore also known as infinite-dimensional systems. Typical examples are systems described by
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s or by
delay differential equation
In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.
DDEs are also called tim ...
s.
Linear time-invariant distributed-parameter systems
Abstract evolution equations
Discrete-time
With ''U'', ''X'' and ''Y''
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s and ''
'' ∈ ''L''(''X''), ''
'' ∈ ''L''(''U'', ''X''), ''
'' ∈ ''L''(''X'', ''Y'') and ''
'' ∈ ''L''(''U'', ''Y'') the following
difference equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s determine a discrete-time
linear time-invariant system:
:
:
with ''
'' (the state) a sequence with values in ''X'', ''
'' (the input or control) a sequence with values in ''U'' and ''
'' (the output) a sequence with values in ''Y''.
Continuous-time
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 operator In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.
The ter ...
s. 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.
Example: a partial differential equation
The partial differential equation with
and