Index preserving polynomial representation of nonlinear differential-algebraic systems

Recently in (9) a procedure was presented that allows to reformulate nonlinear ordinary differential equations in a way that all the nonlinearities become polynomial on the cost of increasing the dimension of the system. We generalize this procedure (called `polynomialization') to systems of differential-algebraic equations (DAEs). In particular, we show that if the original nonlinear DAE is regular and strangeness-free (i.e., it has differentiation index one) then this property is preserved by the polynomial representation. For systems which are not strangeness-free, i.e., where the solution depends on derivatives of the coefficients and inhomogeneities, we also show that the index is preserved for arbitrary strangeness index. However, to avoid ill-conditioning in the representation one should first perform an index reduction on the nonlinear system and then construct the polynomial representations. Although the analytical properties of the polynomial reformulation are very appealing, care has to be given to the numerical integration of the reformulated system due to additional errors. We illustrate our findings with several examples

Time-varying linear DAEs transferable into standard canonical form

We introduce a solution theoryfor time-varying linear differential-algebraic equations(DAEs) E(t)˙x = A(t)x which can be transformed into standard canonical form (SCF), i.e. the DAE is decoupled into an ODE ˙z1 = J(t)z1 and a pure DAE N(t)˙z1 = z1 , where N is pointwise strictly lower triangular. This class is a time-varying generalization of time-invariant DAEs where the corresponding matrix pencil is regular. It will be shown in which sense the SCF is a canonical form, that it allows for a transition matrix similar to the one for ODEs, and how this can be exploited to derive a variation of constants formula. Furthermore, we show in which sense the class of systems transferable into SCF is equivalent to DAEs which are analytically solvable, and relate SCF to the derivative array approach, differentiation index and strangeness index. Finally, an algorithm is presented which determines the transformation matrices which put a DAE into SCF

Robustness of stability of time-varying index-1 DAEs

We study exponential stability and its robustness for time-varying linear index-1 differential-algebraic equations. The effect of perturbations in the leading coefficient matrix is investigated. An appropriate class of allowable perturbations is introduced. Robustness of exponential stability with respect to a certain class of perturbations is proved in terms of the Bohl exponent and perturbation operator. Finally, a stability radius involving these perturbations is introduced and investigated. In particular, a lower bound for the stability radius is derived. The results are presented by means of illustrative examples

An algorithm for the reduction of linear DAE

We study linear Differential Algebraic Equations, DAE, with time varying coefficients. Such equations B(t)&(t) = A(t)z(t) + f(t) are intensively studied from a numerical point of view. Canonical forms have been proposed to find conditions under which the equation admits a solution, to find the set of consistent initial conditions and to determine conditions under which there is a unique solution. However, since the situation where the system admits infinitely many solutions for one initial value is not really tractable in a numerician framework, few algorithms may be found in this latter case. Among them, we find the method of P. Kunkel and V. Mehrmann who propose a new set of local characterizing quantities for the treatment of the system. This leads to a generalization of the global index. Nevertheless, these latter characterizing quantities impose too restrictive conditions on the input equations. We propose new definitions for them that lead to a new algorithm which puts the initial system into a reduced form without doing any assumption on it. This allows us to propose a new generalization of the global index and a definition for the singularities of the initial system. The questions of existence and uniqueness of solutions are solved in all interval which does not contain singularity. Finally, since from a practical point of view the general case of analytic functions is difficult to handle, we focus on the polynomial case. We propose an effective algorithm that has been implemented and report some experiments