The infinite‐order singular problem

Abstract

AbstractIn singular optimal control problems, the optimal control is determined by solving the algebraic equation which results by successively differentiating the switching function until the control appears explicitly. In certain classes of problems, the control never appears, and such problems are termed infinite‐order singular problems. It is shown that this class has many useful properties with respect to the theory and computation of optimal controls. In particular, it is shown that for the time‐invariant, singular, linear‐quadratic problem: (i) the singular order is infinity or less than or equal to the state dimension, (ii) infinite‐order problems can arise only from exact differential type cost functions, (iii) the range of the second‐variation operator (Hessian) is finite‐dimensional, (iv) the computational method converges strongly, and (v) conjugate direction methods converge in a finite number of steps. The latter property is especially useful in the generation of test problems for optimal control computation schemes.