A note on optimal control of generalized state-space (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, which may be singular, and B is a rectangular matrix having full column rank. It is assumed that for any ‘admissible’ initial state x(0−), any control vector u(t) yields one and only one state vector x(t). The problem is this: find a control vector u(t) that will drive the system from an ‘admissible’ initial state x(0−) to a fixed final state xf , in a fixed time tf while minimizing some cost functional $. Only elementary matrix and variational techniques are used. Necessary conditions are derived for the existence of minima of J; the problem of finding sufficient conditions of the existence of minima of J is not considered. It is shown that in many cases the necessary conditions for the existence of minima of J yield a two-point boundary-value problem consisting of a system of ordinary differential equations containing only elements of, x(t) and the boundary conditions prescribed at 0− and tf . If some vector x ∗ is a solution of the two-point boundary-value problem, a control vector u ∗ such that E dx∗/dt = Ax ∗ + Bu ∗, x ∗(0−) = x(0−) and x ∗(tf ) = xf can be obtained from x ∗, and x ∗ and u ∗ will minimize the cost functional J if J has a minimum.

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