Non-existence of Adjoints on Singular Optimal Solutions with the Control in a State-feedback Form

Critical points on optimal singular trajectories have been demonstrated and discussed. The existence of a critical point implies that there is a manifold in the state space on which the induced adjoint equations resulting from state-feedback control representation degenerate, and so gradient optimization based on these adjoints fails. On the other hand, the original adjoints exist and are properly approximated by induced adjoints derived from the prototype adjoint representation. In this way it has been shown that full elimination of adjoint variables from the expression for optimal singular control (though attractive) is not always advantageous in practical optimization.