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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 0 T L(x,u)dt+F(Ex(T)) subject to the implicit dynamical constraint f(x,Ex ' ,u)=0 and Ex(0)=Ex 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:
49K15Optimal control problems with ODE (optimality conditions)
34A99General theory of ODE
93C15Control systems governed by ODE
49J15Optimal control problems with ODE (existence)
93C05Linear control systems