Free final-time optimal control of descriptor systems

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Abstract
Linear time-invariant systems of the form E dx/dt=Ax + Bu are considered, where E is a square matrix that may be singular and x is a column vector called the descriptor vector of the system. It is assumed that for any admissible initial descriptor vector x(0—), any control vector u(t) yields one and only one descriptor vector x(t). The problem is this: find a control vector u(t) that will drive the descriptor vector of the system from a fixed vector x(0—) to a (not necessarily fixed) final descriptor vector x(tf) while, together with some (not a priori fixed) final time tf, minimizing a cost functional J = 1/2 1∫0 ( xT Qx + uT Ru) dt. Using elementary matrix and variational techniques, necessary conditions are derived for the existence of minima of J; the problem of finding sufficient conditions for the existence of minima of J is not considered. The general results are applied to the special case E = diag { In−m, 0}, B = [o, Im]. The problem of choosing the matrix Q of the cost functional is investigated. A simple illustrative example is discussed.