I am interested in optimal control problems with state constraints in the form of ordinary differential equations given by

$\dot{x} = f(x,p), \quad t \in [0,T], \quad x(0) = x_{0}, \quad p \in K$,

where $f:R^n\times R^m \rightarrow R^n$ is non-linear and globally Lipschitz continuous in $x$ and continuous in $p$, and $K$ is a compact set. $p$ is in this case a vector of parameters (i.e., a constant control). It can also be shown for my $f$ that every trajectory of the system is bounded. From this it follows from Lemma 4.2 on p.257 in Convex Analysis and Variational Problems by Ekeland and Temam that the set of trajectories form a relatively compact subset of $C([0,T],\mathbb{R}^n)$ (continuous functions from $[0,T]$ to $R^n$). From Proposition 4.1 in the same book I gather that it is in fact compact in $C([0,T],R^n)$ if the set $A(x) = $ {$f(x,p) \ | \ (x,p)\in R^n \times K$} is convex for all $t\in[0,T]$ and $x\in\mathbb{R}^n$.

In general, it appears difficult to show that $A(x)$ is convex (I looked at "Convexity of Images of Convex Sets under Smooth Maps" and "Convexity of nonlinear image of a small ball with applications to optimization", but was not satisfied with the sufficient criteria found therein). For this reason I wonder if there is a way to show that the trajectories of my system form a compact subset of $C([0,T],\mathbb{R}^n)$ without having to show convexity of $A(x)$?

Kind regards


  • $\begingroup$ Well, as stated, without prescribing e.g. initial conditions, it's not true that trajectories form a compact set in $C^0$, because of course there are solutions with any $x(0)\in\mathbb{R}^n$. On the other hand, any subset of solutions with $\|x(t_0)\|\le C$ for some $t_0$ and $C$ is relatively compact by the Ascoli-Arzelà theorem. So, aren't you missing some constraint or condition on the trajectories you are studying? $\endgroup$ – Pietro Majer Apr 20 '13 at 9:42
  • $\begingroup$ Yes, I forgot the initial conditions. Thanks for pointing that out! I've edited the question. $\endgroup$ – user12400 Apr 20 '13 at 12:01
  • $\begingroup$ Then the set of solutions, being closed bounded and equi-Lipschitz, is a compact set, by the Ascoli-Arzelà theorem. $\endgroup$ – Pietro Majer Aug 12 '13 at 10:37

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