Criteria for the trivial solution of differential algebraic equations with small nonlinearities to be asymptotically stable

Differential algebraic equations consisting of a constant coefficient linear part and a small nonlinearity are considered. Conditions that enable linearizations to work well are discussed. In particular, for index-2 differential algebraic equations there results a kind of Perron-Theorem that sounds as clear as its classical model except for the expensive proofs

Canonical Subspaces of Linear Time-Varying Differential-Algebraic Equations and Their Usefulness for Formulating Accurate Initial Conditions

Accurate initial conditions have the task of precisely capturing and fixing the free integration constants of the flow considered. This is trivial for regular ordinary differential equations, but a complex problem for differential-algebraic equations (DAEs) because, for the latter, these free constants are hidden in the flow. We deal with linear time-varying DAEs and obtain an accurate initial condition by means of applying both a reduction technique and a projector based analysis. The highlighting of two canonical subspaces plays a special role. In order to be able to apply different DAE concepts simultaneously, we first show that the very different looking rank conditions on which the regularity notions of the different concepts (elimination of unknowns, reduction, dissection, strangeness, and tractability) are based are de facto consistent. This allows an understanding of regularity independent of the methods

Transfer of boundary conditions for DAEs ofindex 1

In this paper, the concept of Abramov’s method for transferring boundary conditions posed for regular ordinary differential equations is applied to index 1 DAEs. Having discussed the reduction of inhomogeneous problems to homogeneous ones and analyzed the underlying ideas of Abramov’s method, we consider boundary value problems for index 1 linear DAEs both with constant and varying leading matrix. We describe the relations defining the subspaces of solutions satisfying the prescribed boundary conditions at one end of the interval. The index 1 DAEs that realize the transfer are given and their properties are studied. The results are reformulated for inhomogeneous index 1 DAEs, as well

Algebro-Differentialgleichungen mit properem Hauptterm und Traktabilitätsindex

1. Algebro-Differntialgleichungen mit properem Hauptterm 2. Traktabilitätsindex 3. Linear-quadratische Optimalsteuerun

Nonlinear differential-algebraic equations with properly formulated leading term

Nonlinear differential-algebraic equations with properly stated leading term of index one and two are characterized. Linearization and the solvability of perturbed problems are considered

On linear differential-algebraic equations and linearizations

On the background of a careful analysis of linear DAEs, linearizations of nonlinear index-2 systems are considered. Finding appropriate function spaces and their topologies allows to apply the standard Implicit Function Theorem again. Both, solvability statements as well as the local convergence of the Newton-Kantorovich method (quasilinearization) result immediately. In particular, this applies also to fully implicit index 1 systems whose leading nullspace is allowed to vary with all its arguments

Numerical stability criteria for differential-algebraic systems

In this paper we transfer classical results concerning Lyapunov stability of stationary solutions x* to the classes of DAEs being most interesting for circuit simulation, thereby keeping smoothness as low as possible. We formulate all criteria in terms of the original equation. Those simple matrix criteria for checking regularity, Lyapunov stability etc. are easily realized numerically