Linear DAE
In this chapter, we will explore linear implicit differential equations, also known as Differential Algebraic Equations (DAEs), characterized by the following form:
\bm{E}\dot{\bm{x}}(t) = \bm{A}\bm{x}(t) + \bm{f}(t) \tag{1}
In the simplest scenario, the matrices \bm{E} and \bm{A} are square constant matrices belonging to \Bbb{R}^{n \times n}, and \bm{f}(t):\Bbb{R} \rightarrow \Bbb{R}^{n} represents an external perturbation that is independent of \bm{x}(t). Such equations frequently arise when linearizing autonomous nonlinear problems around constant (or critical) solutions, where \bm{f}(t) acts as a perturbation that displaces the solution from the nominal trajectory. Other applications include modeling linear electrical circuits, simple mechanical systems, and, more generally, linear systems with additional algebraic constraints.
Learning objectives
After this part, you should be able to distinguish an ODE from a singular linear DAE, reduce simple linear DAEs by row operations, and identify the consistency conditions imposed on the initial state.