Parametric sensitivity functions for hybrid discrete/continuous systems
This paper is concerned with the study of the parameter sensitivity functions of some class of dynamical systems described either by ordinary differential equations (ODEs) or differential-algebraic equations (DAEs). The systems under consideration are assumed to be described by a state space \( S = \bigcup_k S_k\), so that the time evolution in each mode \( S_k\) from \(t_0^k\) to \(t_f^k\) is governed by a set of differential--algebraic equations \( f_k(\dot x^k, x^k, y^k, u^k, p^k, t)=0,\) where \(x^k\) and \(y^k\) are the differential and algebraic variables respectively, \(u^k = u^k(p^k,t)\) the controls and \(p^k\) the parameters. These equations are supplemented with some specification of the parameters \(p^k\) and consistent initial conditions. Further the transition conditions between two modes are properly defined. In the first part of the paper the equations which describe the parameter sensitivity are derived and different types of transitions between modes are given. Then in the case of systems described by pure ODEs the authors give some sufficient conditions which imply the continuity of the sensitivity functions \(\partial x^k/\partial p^k\). These results follow from well known properties about the dependence of the solutions of the differential equations on parameters. Next the authors extend these results to constant coefficients linear DAEs with index one. Further results are given for the case that the coefficients are parameter dependent. The paper ends presenting the numerical results obtained for some particular examples. Here the authors combine a numerical integrator with a state event location algorithm to model a complete evolution of the systems and point out some difficulties that may be found in the transitions.
scientific article; zbMATH DE number 1367758