Differential algebraic equations

Abstract

Let H be a Hilbert space and ν∈ ℝ. We saw in the previous chapter how initial value problems can be formulated within the framework of evolutionary equations. More precisely, we have studied problems of the form {(∂t,νM0+M1+A)U=0on(0,∞),M0U(0+)=M0U0 $$\displaystyle \begin{aligned} \begin {cases} \left (\partial _{t,\nu }M_{0}+M_{1}+A\right )U=0 & \text{ on }\left (0,\infty \right ),\\ M_{0}U(0{\scriptstyle {+}})=M_{0}U_{0} \end {cases} \end{aligned} $$ for U0 ∈ H, M0, M1 ∈ L(H) and A: dom (A) ⊆ H→ H skew-selfadjoint; that is, we have considered material laws of the form M(z): =M0+z−1M1(z∈ℂ∖{0}). $$\displaystyle M(z)\mathrel{\mathop:}= M_{0}+z^{-1}M_{1}\quad (z\in \mathbb {C}\setminus \{0\}). $$