Necessary conditions in dynamic optimization by Francis Clarke;

By Francis Clarke;

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11) 0 We view this problem as one in which the choice variables are x, ˙ y˙ ∈ L2 , and n x(0) ∈ R . Any feasible (x, y) has x ∞ 1 ≤ ε, |x(t)| ˙ dt ≤ ε. 0 The infimum in the problem is no less than −1, and the arc (0, t), which is feasible, provides the value εi − 1 to the cost, where εi := |µi − Φ(0)| > 0. 11). It follows that (xi , yi ) converges uniformly to (0, t), and we may suppose that (vi , vi , λi , λi ) converges almost everywhere to (0, 0, 1, 1). We may write x˙ i (t) in the form λi (t)fi (t), where fi (t) ∈ Ft (xi (t)), and where fi converges almost everywhere to 0.

6 with R(t) := 4r0 (t) to deduce that Ft is pseudo-Lipschitz of radius R (t) := 5r0 (t)/4 near (x∗ (t), x˙ ∗ (t)) for x ∈ B(x∗ (t), ε ). 1. 3, where the pseudo-Lipschitz radius R had to be large relative to the pseudo-Lipschitz function k. 2 The results of these sections subsume and extend in a variety of ways (see the Introduction) the necessary conditions for P in the literature, notably those obtained by Clarke [11, 17], Loewen and Rockafellar [43], Ioffe [35], Mordukhovich [47], Smirnov [55], and Vinter [58]; they also answer in the affirmative some questions raised by Ioffe in [35].

BOUNDARY TRAJECTORIES Proof. We may assume that k is positive-valued. Let us define δ > 0 to be the essential lower bound in the statement of the proposition. Set ε0 := min{ε, δ/2}, r0 (t) := ε0 k(t), where the ε is that of the pseudo-Lipschitz condition. Then, by taking x = x∗ (t) in that condition, we deduce that for x ∈ B(x∗ (t), ε0 ), we have Ft (x) ∩ B(x˙ ∗ (t), r0 (t)) = ∅. Since R(t)/r0 (t) ≥ 2, the tempered growth condition is satisfied. We remark that the proposition covers the standard case in which the radius function is identically +∞ and F is pseudo-Lipschitz of infinite radius (that is, actually Lipschitz).

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