On some analogs of Sturm's and Kneser's theorems for nonlinear systems

Analogs of certain conjugate point properties in the calculus of variations are developed for optimal control problems. The main result in this direction is concerned with the characterization of a parameterized family of extremals going through the first backward conjugate point, t_c. A corollary of this result is that for the linear quadratic problem (LQP) there exists at least a one-parameter family of extremals going though the conjugate point which gives the same cost as the candidate extremal, i.e., the extremal control is optimal but nonunique on t_c, t_f]. An analysis of the effect on the conjugate point of employing penalty functions for terminal equality constraints in the LQP is presented, also. It is shown that the sequence of approximate conjugate points is always conservative, and it converges to the conjugate point of the constrained problem. Furthermore, it is proved that the addition of terminal constraints has the effect of causing the conjugate point to move backward (or remain the same).