On the Intersection of a Clarke Cone with a Boltyanskii Cone

We provide an example of two closed sets S1, S2 ⊂ IR such that S1∩S2 = {0}. Yet, at the origin, a Boltyanskii tangent cone C1 to S1 and the Clarke tangent cone C2 to S2 are strongly transversal. This settles a question originally proposed by H. Sussmann. 1 Introduction Let t 7→ x∗(t) and t 7→ u∗(t) be respectively a trajectory and an optimal control for the Mayer problem    minimize φ ( x(T ) ) subject to ẋ(t) = f ( t, x(t), u(t) ) and x(0) = x̄ , ξ(T ) ∈ S . Here x ∈ IR is the state variable, while u ∈ U ⊆ IR is the control variable. A set of necessary conditions for optimality of the pair (x∗, u∗) is provided by the celebrated Pontryagin Maximum Principle (PMP), which has been extensively discussed in the literature on the mathematical theory of control [4]. In essence, these necessary conditions can be traced to a separation property. On one hand we have a set S1 of ”reachable points”, i.e. points x ( T, u) that can be reached at the terminal time T by means of admissible controls t 7→ u(t) ∈ U . On the other hand, we can consider a set S2 of ”profitable points”, i.e. points that satisfy the terminal constraint x ∈ S and achieve a lower cost: φ(x) ≤ φ(x∗(T )), with equality holding only if x = x∗(T ). Necessary conditions for optimality are typically obtained by constructing a tangent cone C1 to the set S1 at the terminal point x∗(T ), and a tangent cone C2 to S2 still at x∗(T ). If these cones are transversal, under suitable assumptions one can conclude that the intersection S1 ∩ S2 is non-trivial, i.e. it contains points other than x∗(T ). Hence, the pair (x∗, u∗) is not optimal. Reversing the argument, the optimality of the trajectory-control pair (x∗, u∗) implies that the tangent cones C1 and C2 are weakly separated. This provides an alternative way to state the PMP. For a deeper discussion of the PMP and for various extensions of this optimality principle we refer to the papers [1–3] and [5–9]. As previously remarked, results on intersections of tangent cones play a crucial role in deriving necessary optimality conditions. In this direction, a question posed by H. Sussmann (see Conjecture 3.6.4 in [7]) is the following: Let n ≥ 1 and let S1, S2 be closed subsets of IR such that 0 ∈ S1 ∩ S2. Assume that 1 • C1 is a Boltyanskii approximating cone to S1 at 0; • C2 is the Clarke tangent cone to S2 at 0; • C1 and C2 are strongly transversal. Does this imply that 0 belongs to the closure of (S1 ∩ S2) \ {0} ? The positive results, in dimension n ≤ 3, follow from standard topological arguments. Aim of the present paper is to prove that this question has a negative answer, in every space dimension n ≥ 4. In Section 2 we review the basic definitions, and discuss an example in space dimension n = 3. Our main counterexample is then given in Section 3, where we construct two sets S1, S2 ⊂ R. At the origin, the Clarke tangent cone to S1 and a Boltyanskii tangent cone to S2 are strongly transversal. Yet, the two sets have trivial intersection, namely S1 ∩ S2 = {0}. Based on the present counterexample, the forthcoming paper by H. Sussmann [10] exhibits an optimal control problem and an optimal trajectory for which the usual conclusions of the PMP are not true, if the Clarke tangent cone to the terminal set is used instead of a Boltyanskii approximating cone. 2 Preliminary analysis For reader’s convenience, we first recall some basic definitions. Definition 1. A nonempty set C ⊆ IR is a cone if whenever v ∈ C and r ≥ 0 it follows that rv ∈ C . Definition 2. Let S ⊆ IR. The Clarke tangent cone to a point x ∈ S is the set of all vectors v ∈ IR such that the following holds. For every sequence xk → x with xk ∈ S for all k ≥ 1, there exist a sequence vk → v such that lim inf h→0+ d ( xk + hvk , S )