On differential algebraic equations with discontinuities
Many of the differential algebraic equations (DAEs) that arise in control problems take the form \(A(z,u)z'=f_ 1(z,u)\), \(0=f_ 2(z,u)\), \(z(0)=z_ 0\) where \(A\) is singular but has constant rank. This paper examines what the response of the state should be if the control \(u\) has a jump discontinuity at a time \(t_ 0\). This is done by defining a class of regularizations, that is, continuous controls which approach \(u\) in a neighborhood of \(t_ 0\). A ``genuine initial value'' is then defined in terms of a limit involving the regularization. The existence of genuine initial values is shown to be characterized by the equations defining the DAE arising from potentials.
scientific article; zbMATH DE number 63821