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Can we consider an optimal control problem, a convex optimization problem like static optimization problems? If it is possible, under what conditions, will this problem be a convex problem? For example, in the general problem as follows, how can we check the convexity of the problem (if we arrive to a solution, is it optimal or not)?

$$ \max_{u \in \mathcal{U}} Q(u) = \int_{0}^{T} F(x(t),u(t), t) dt + S(x(T),T) \\ \dot{x}(t)= f(x(t),u(t), t) $$

Moreover, is it possible to obtain an explicit form for $Q(u)$? When is $Q(u)$ a concave of $u$?

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I suggest to take a look at "Foundations of Optimization" written by Osman Guler and edited by Springer in 2010. The 3rd chapter is wholly dedicated to Variational Principles and in 4.5 "Optimization on Convex Sets" is stated that:

One of the most important and basic proprieties of convex functions is the fact that any local minimizer on a convex set is a global one.

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