Differential Algebraic Equations

Abstract

AbstractLet H be a Hilbert space and $$\nu \in \mathbb {R}$$ 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 $$\displaystyle \begin{aligned} \begin {cases} \left (\partial _{t,\nu }M_{0}+M_{1}+A\right )U=0 & \text{ on }\left (0,\infty \right ),\\ M_{0}U(0{\scriptstyle {+}})=M_{0}U_{0} \end {cases} \end{aligned} $$ ∂ t , ν M 0 + M 1 + A U = 0 on 0 , ∞ , M 0 U ( 0 + ) = M 0 U 0 for U0 ∈ H, M0, M1 ∈ L(H) and $$A\colon \operatorname {dom}(A)\subseteq H\to H$$ A : dom ( A ) ⊆ H → H skew-selfadjoint; that is, we have considered material laws of the form $$\displaystyle M(z)\mathrel{\mathop:}= M_{0}+z^{-1}M_{1}\quad (z\in \mathbb {C}\setminus \{0\}). $$ M ( z ) : = M 0 + z − 1 M 1 ( z ∈ ℂ ∖ { 0 } ) .