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The application of a universal separating vector lemma to optimal sampled-data control problems with nonsmooth Mayer cost function

• In this paper we provide a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. Our investigation leads us to consider, in a first place, a general issue on convex sets separation. Precisely, thanks to the classical Fan's minimax theorem, we establish the existence of a universal separating vector which belongs to the convex envelope of a given set of separating vectors of the singletons of a given compact convex set. This so-called universal separating vector lemma is used, together with packages of convex control perturbations, to derive a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. As an illustrative application of our main result we solve a simple example by implementing an indirect numerical method.

Mathematics Subject Classification: Primary: 34H05, 49K15, 93C57.

 Citation:

• Figure 1.  A nonempty convex set not containing the origin $0_{{\mathbb{R}}^2}$ with no separating vector

Figure 2.  Here $v_1$ and $v_2$ are respectively separating vectors of the nonempty closed convex sets $C_1$ and $C_2$ not containing the origin $0_{{\mathbb{R}}^2}$. Does there exist a separating vector of the closed convex set $C : = C_1 \cup C_2$ which is a convex combination of $v_1$ and $v_2$?

Figure 3.  Two-dimensional unbounded closed counterexample from Remark 2.5

Figure 4.  The optimal trajectory $x$ and optimal sampled-data control $u$ for Problem (Ex)

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