Completions of nonlinear DAE flows based on index reduction techniques and their stabilization (Q1034656)

The authors study nonlinear differential-algebraic equations (DAEs) \(F(t,x,\dot{x})= 0\), where \(F\) is a sufficiently smooth \(n\)-vector function of the scalar \(t\) on some interval and the vector \(x\in{\mathbb R^n}\) and its derivative \(\dot{x}\) on some domain in \(\mathbb R^{n\times{n}}\). Especially a linear DAE with variable coefficients \(E(t)\dot{x}=A(t)x+f(t)\) is considered. A completion of a DAE is an ordinary differential equation whose solutions include the solutions of that DAE. A custom approach to the numerical solution of DAEs of higher differentiation index is based on reducing the given DAE to some of its completions. The authors investigate the so-called ``least squares completion'' based on a usage of the derivative array equation. Such a completion has as a rule a larger solution set than the origin DAE. The problem developed in the paper is the generating of completions for which the additional dynamics have the desired behavior and can be stabilized. The authors exploit a complex algebraic techniques and results (with rank-type properties and pseudoinversion) presented by \textit{P. Kunkel} and \textit{V. Mehrmann} [Differential-algebraic equations. Zürich, Switzerland: EMS Publishing House (2006; Zbl 1095.34004)]. They create completions which provide that solutions which are not satisfying the given values of first integrals and tend to the prescribed values exponentially. A numerical example is presented. Alternative simpler numerical methods of variational type which don't use the notion of index and are applicable to general DAEs of any index as well as to problems which are not reducible to the normal form are presented in recent papers by \textit{V. Gorbunov} and \textit{V. Sviridov} [Appl. Numer. Math. 59, No. 3--4, 656--670 (2009; Zbl 1160.65045)], and by \textit{V. Gorbunov, I. Lutoshkin} and \textit{Yu. Martynenko} [Appl. Numer. Math. 59, No. 3--4, 639--655 (2009; Zbl 1160.65033)].