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control variables has revived the interest in the old subject of inequalities
PART ACASE a
PART BCASE b
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adjoint equations admissible control assume assumption belongs bound vector boundary conditions brachistochrone problem calculus of variations comp cone conic neighborhood consider const constant constraints continuous function control function control variable convex sets corner condition Corollary corresponding cos2 curve defined definition denote derivatives determined differential equations Euclidean space Euler equations exists extremaloid finite follows given grad gradient Hence implies inequality initial conditions integral interior point interval Lagrange multipliers Lemma limiting surface linear Math matrix maximum principle method minimizing multiplier rule necessary condition neighborhood nonzero normal obtain open ball optimal control optimal control problem optimal solution optimal trajectory optimum problems orbital parameter piecewise continuous plane Pontryagin's maximum principle positive number problem of Mayer proof properties Proposition real numbers refraction region relation result satisfied scalar separating hyperplane singular subarc space subset tangent terminal Theorem theory thrust vanishes vector valued function zero