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Variational calculus for descriptor problems. (English) Zbl 0644.49019
This note is concerned with the infimization of the usual performance index ${\int }_{0}^{T}L\left(x,u\right)dt+F\left(Ex\left(T\right)\right)$ subject to the implicit dynamical constraint $f\left(x,E{x}^{\text{'}},u\right)=0$ and $Ex\left(0\right)=E{x}_{0}$. The constant matrix E is allowed to be singular. The first order necessary condition for optimality is derived resulting in a Hamiltonian characterization in terms of Ex’ rather than x’. This approach sidesteps the regularity conditions of the Lagrange multiplier theory. The general linear time- invariant case is considered. For nonlinear problems, it is assumed that the dynamical constraint is an index one singular system. Under mild assumptions on the problems considered, it is shown that the necessary condition for optimality is also sufficient and the optimal control exists.
Reviewer: S.Campbell
##### MSC:
 49K15 Optimal control problems with ODE (optimality conditions) 34A99 General theory of ODE 93C15 Control systems governed by ODE 49J15 Optimal control problems with ODE (existence) 93C05 Linear control systems