# how to impose a terminal condition in a minimisation problem?

Consider the problem of minimising $$J(u(.)) \triangleq \int_0^T l(x(t),u(t)) d t + \phi(x(T)), \: T> 0$$ over a space of controls $\mathcal{U}$ with the constraint $$\dot x(t) = f(x(t),u(t)), \: t >0, \: x(0) = x_0.$$ To make it simple, everything is $\mathbb{R}$ valued. We also assume that the functions $l,\phi,f$ and the space of controls $\mathcal{U}$ satisfy appropriate conditions for having a well-posed problem.

Under this form we can employ either Bellman or Pontryagin principles to study the optimal control.

Question. Can we / How to use this framework to impose a terminal value for $x$? say $x(T) = x_\star$.

• Applying a terminal set like $x(T)=x_\star$ or $x(T)\in \mathbb{T}$ is not hard as far as your solver can manage it. For an equality constraint, why not using the Lagrange multipliers? – Arash Jul 27 '18 at 0:23