The behavioral approach to systems theory and control theory was initiated in the late-1970s by J. C. Willems as a result of resolving inconsistencies present in classical approaches based on state-space, transfer function, and convolution representations. This approach is also motivated by the aim of obtaining a general framework for system analysis and control that respects the underlying physics.

The main object in the behavioral setting is the behavior – the set of all signals compatible with the system. An important feature of the behavioral approach is that it does not distinguish a priority between input and output variables. Apart from putting system theory and control on a rigorous basis, the behavioral approach unified the existing approaches and brought new results on controllability for nD systems, control via interconnection, and system identification.

In the behavioral setting, a dynamical system is a triple

where

${\displaystyle w\in {\mathcal {B}}}$ means that ${\displaystyle w}$ is a trajectory of the system, while ${\displaystyle w\notin {\mathcal {B}}}$ means that the laws of the system forbid the trajectory ${\displaystyle w}$ to happen. Before the phenomenon is modeled, every signal in ${\displaystyle \mathbb {W} ^{\mathbb {T} }}$ is deemed possible, while after modeling, only the outcomes in ${\displaystyle {\mathcal {B}}}$ remain as possibilities.

Special cases:

System properties are defined in terms of the behavior. The system ${\displaystyle \Sigma =(\mathbb {T} ,\mathbb {W} ,{\mathcal {B}})}$ is said to be

where ${\displaystyle \sigma ^{t}}$ denotes the ${\displaystyle t}$-shift, defined by