Definitions of Order and Junction Conditions in Singular Optimal Control Problems

Abstract

The generalized Legendre-Clebsch higher order tests for optimality of singular arcs in optimal control problems depend upon the orders of the arcs involved. To date three distinct definitions of order have been given but many authors do not distinguish among them. The features of each definition are discussed with special reference to the applicability of the higher order tests and of the conditions at junctions between singular and nonsingular arcs; only in terms of one of the definitions are the junction conditions generally valid. An illustrative example is presented.

A Hybrid Direct–Indirect Approach for Solving the Singular Optimal Control Problems of Finite and Infinite Order

“…Note that, d dt rðx; k; tÞ is explicitly a function of x, k, _ x, _ k and t. By substituting _ x and _ k from (1b) and (3), d dt rðx; k; tÞ can be expressed as a function of x, k and t. It is easy to show that the control function u does not appear in d dt r (Lewis 1980). By repeating this manner, d j dt j rðx; k; tÞ can be expressed as a function of x, k, t and maybe u.…”

“…By repeating this manner, d j dt j rðx; k; tÞ can be expressed as a function of x, k, t and maybe u. Furthermore, if u appears in d j dt j r, then it appears linearly (Lewis 1980). It is possible that the control u does not appear in d j dt j r for any j.…”

“…It is possible that the control u does not appear in d j dt j r for any j. However, if w be the first integer number, in which u appears in d w dt w r, then w is always even (Lewis 1980;Lamnabhi-Lagarrigue 1987). In the former case, the order of SOCP is defined to be infinite and in the latter case, the integer number j ¼ w 2 is called order of the singular problem.…”

A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems

“…is explicitly a function of x, λ,ẋ,λ and t. By replacingẋ andλ with (1b) and (3), one can express d dt σ(x, λ, t) as a function of x, λ and t. According to Lewis (1980), the control function u does not appear in…”

“…Furthermore, if u appears in d j dt j σ, then it appears linearly (Lewis, 1980). It is possible that the control u does not appear in (Lamnabhi-Lagarrigue, 1987;Lewis, 1980).…”