An existence result for elliptic partial differential–algebraic equations arising in semiconductor modeling

Index-2 elliptic partial differential-algebraic models for circuits and devices

“…Moreover, Lemma 5 applies, because passivity is given (Lemma 6), and the uniquely defined solution x satisfies the estimates (5.2)-(5.4), and thus, it satisfies the estimate (5.10). Hence, the coupled system (5.11)-(5.14) defines a map from M to itself, The proof from [2] applies also to our index-2 case. Hence, by Schauder's fixed point theorem, T admits a fixed point (x, Φ, φ), which solves the original problem (4.1)-(4.4).…”

“…This lemma was proven in [2]. The dissipativity condition (5.1) is equivalent to the usual passivity condition, since…”

“…The proof of Theorem 1 is based on an iteration map argument and it is an extension of the proof used in [2], where an index-1 coupled network-device system is studied. The derivation requires several steps: passivity of the semiconductor device, a priori estimates and an iteration map.…”

“…in Ω, ∀t ∈ [t 0 , t 1 ] 2 and consider boundary data e * D ∈ C([t 0 , t 1 ], R m D ). In fact, Lemma 5 can be extended to this setting[2]. If x satisfies the a priori estimate (5.2)-(5.4), then the applied potentials e D = S x are bounded:e D (t) ≤ C D (t 0 , t 1 ), ∀t ∈ [t 0 , t 1 ],(5.10) where C D (t 0 , t 1 ) depends on the time interval [t 0 , t 1 ].…”

Estimation and Global Stability Analysis of PDAEs with Singular Time Derivative Matrix and Wetland Conservation Application

“…where j ∈ R denotes the th eigenvalue and (z) denotes the corresponding orthogonal eigenfunction. Obviously, the eigenfunctions { (z)} ∞ =1 form an orthogonal basis for eigenvalue problem (8). Furthermore, in the next subsection, we show that (D) has discrete spectrum consisting only of real eigenvalues with at most a finite number of positive eigenvalues which is a generalization of [17].…”