Consistency Condition

Assume a hard problem (Differential Index > 1)

$$\begin{array}{rl}\dot{x}& =f(x,z)\\ 0& =c(x,z)\to 0=c^{(n-1)}(x,z)\text{ such that }\frac{\partial c^{(n-1)}}{\partial z}\text{ is full rank}\end{array}$$

How to ensure that $c^{(n-1)}=0$ implies that $c=0$?

• Need to impose

$$\begin{array}{rl}c(t=0)& =0\\ \dot{c}(t=0)& =0\\ \ddot{c}(t=0)& =0\\ ⋮& \\ c^{(n-2)}(t=0)& =0\\ c^{(n-1)}(t)& =0\end{array}$$