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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 $x(T) = x_\star$.

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  • $\begingroup$ 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? $\endgroup$ – Arash Jul 27 '18 at 0:23
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Isn't your problem in the standard form for the Pontryagin maximum principle? Look at, e.g., http://liberzon.csl.illinois.edu/teaching/cvoc.pdf for details in the standard smooth setting. Theorem 22.26 of Francis Clarke's 2013 book "Functional Analysis, Calculus of Variations and Optimal Control" takes care of the non-smooth setting.

Solving the two-point boundary value problems that the PMP provides, usually carried out by standard shooting methods, leads to the optimal control trajectories.

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