How is an ODE a consistency condition?

Question

I was reading a text on Optimal Control Theory by E. Todorov, when I came accross this passage (on page 10):

An ODE is a consistency condition which singles out specific trajectories without reference to neighbouring trajectories (as would be the case in a PDE).

I had never thought of a differential equation as a "consistency condition" and, in fact, I am not quite sure of the meaning of this.

How is a differential equation a"consistency condition"?
What is the difference between ODE and PDE mentionned in the passage?

The answer to these questions need not be related to Control Theory only.

Answer 1

Consistency conditions can be translated to:

The set of possible solutions have to satisfy the ode/pde. Only trajectories that are described via the ode/pde (+ic's/bc's) are candidates for solutions.

Example:
The following (dynamical) optimization problem requires the measured output $y$ to follow the reference trajectory $y^*$. The consistency condition would be the ode describing its dynamic behavior:

$$ \begin{align} & \min_{u(t)} || y(t) - y^*(t) ||, \: t \in [0, t_f] \\ \text{s.t. } & \dot{x} = f(x,u), \: x(0) = x_0 \\ & y = h(x) \end{align} $$