Abstract
This paper considers the singular optimal control problem for n-dimensional nonlinear systems, for which n control variables appear linearly both in the system dynamics and in the performance index. Necessary conditions (Theorems 1 and 2) and sufficient conditions (Theorems 3 and 4) for singular optimality are derived by using the theory of differential forms and the Stokes' theorem. Theorem 1 includes both the Euler-Lagrange equation in the calculus of variation and the Shima-Sawaragi's condition for the optimality of boundary control. We show that the generalized Legendre-Clebsch condition is derived from Theorem 2 and that the Jacobson's condition is derived from Theorem 4. It is also shown that for time-invariant systems, the necessary condition derived from Theorems 1 and 2 is equiva lent to the necessary condition for the related static optimal control problem.
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