Necessary and sufficient conditions for viability for semilinear differential inclusions

Ovidiu Cârjă, Mihai Necula, Ioan I. Vrabie
2008 Transactions of the American Mathematical Society  
Given a set K in a Banach space X, we define: the tangent set, and the quasi-tangent set to K at ξ ∈ K, concepts more general than the one of tangent vector introduced by Bouligand (1930) and Severi (1931) . Both notions prove very suitable in the study of viability problems referring to differential inclusions. Namely, we establish several new necessary, and even necessary and sufficient conditions for viability referring to both differential inclusions and semilinear evolution inclusions,
more » ... ion inclusions, conditions expressed in terms of the tangency concepts introduced. Reverts to public domain 28 years from publication OVIDIU CÂRJȂ, MIHAI NECULA, AND IOAN I. VRABIE We have to notice that both (T C2) and (T C3) reduce to (T C1) when A = 0. We also emphasize that the tangency condition (T C3) is necessary and sufficient for viability without assuming the compactness of F (ξ) and is suitable in establishing the viability of an epigraph, that, in its turn, leads to a controllability result. See the application in the last Section 18. The outline of the paper is as follows. In Section 2 we collect some notation, basic definitions and preliminary results in Functional Analysis. In Section 3 we introduce the notion of a tangent set, we recall the definition of tangent vector and clarify the relationship between them. Sections 4 ∼ 7 are devoted to the proof of necessary and also necessary and sufficient conditions for the viability of K with respect to F . The nonautonomous case is considered in Section 8 while the existence of global solutions of (1.1) is studied in Section 9. In Section 10 we introduce the new concepts of A-tangent set and A-quasi-tangent set to K at ξ ∈ K, which are very appropriate for studying the viability of K with respect to A + F . Necessary and even necessary and sufficient conditions for this kind of viability are given in Sections 11 ∼ 15. The quasi-autonomous case of (1.2) is considered in Section 16, while the existence of a global solution is studied in Section 17. Finally, in Section 18, we give an interesting application of one of our viability results in obtaining a sufficient condition of null controllability for a semilinear evolution equation. Preliminaries Brezis-Browder ordering principle. To begin with, let us recall some definitions and notation. Let S be a nonempty set. A binary relation ⊆ S × S is a preorder on S if it is reflexive, i.e., ξ ξ for each ξ ∈ S, and transitive, i.e., ξ η and η ζ imply ξ ζ. Definition 2.1. Let S be a nonempty set, ⊆ S × S a preorder on S, and let N : S → R ∪ {+∞} be an increasing function. An N-maximal element is an element ξ ∈ S satisfying N(ξ) = N(ξ), for every ξ ∈ S with ξ ξ. We may now proceed to the statement of the main result in this section, i.e., the Brezis-Browder ordering principle : Theorem 2.1. Let S be a nonempty set, ⊆ S × S a preorder on S and let N : S → R ∪ {+∞} be a given function. Suppose that : (i) each increasing sequence in S is bounded from above ; (ii) the function N is increasing. Then, for each ξ 0 ∈ S, there exists an N-maximal element ξ ∈ S satisfying ξ 0 ξ. See for the proof of Theorem 2.1 in case N is bounded from above and Cârjȃ-Ursescu [7] for the extension to the general case. Excursion to functional analysis. In what follows, X denotes a real Banach space X with the norm · . In this section we gather some results of functional analysis which will prove useful in the sequel. Proposition 2.1. If lim n x n = x weakly in X, then there exists a sequence (y n ) n , with y n ∈ conv {x k ; k ≥ n} and such that lim n y n = x. See Hille-Phillips [16], Corollary to Theorem 2.9.3, p. 36. License or copyright restrictions may apply to redistribution; see VIABILITY FOR SEMILINEAR DIFFERENTIAL INCLUSIONS 347 Definition 2.2. A function x : Ω → X is called: (i) countably-valued if there exist {Ω n ; n ∈ N} ⊆ Σ and {x n ; n ∈ N} ⊆ X, with Ω k ∩ Ω p = ∅ for each k = p, Ω = n≥0 Ω n , and such that x(θ) = x n for all θ ∈ Ω n ; (ii) measurable if there exists a sequence of countably-valued functions convergent to x µ-a.e. on Ω. Theorem 2.2. A function x : Ω → X is measurable if and only if there exists a sequence of countably-valued functions from Ω to X which is uniformly µ-a.e. convergent on Ω to x. Theorem 2.6. Let (Ω, Σ, µ) be a finite measure space and let X be a Banach space. Let F ⊆ L 1 (Ω, µ; X) be bounded and uniformly integrable. If for each ε > 0 there exist a weakly compact subset C ε ⊆ X and a measurable subset E ε ∈ Σ with µ(Ω \ E ε ) ≤ ε and f (E ε ) ⊆ C ε for all f ∈ F, then F is weakly relatively compact in L 1 (Ω, µ; X). See Diestel [11], or Diestel-Uhl [12], p. 117. Corollary 2.1. If C ⊆ X is weakly compact, then {f ∈ L 1 (τ, T ; X); f (t) ∈ C a.e. for t ∈ [ τ, T ]} is weakly relatively compact in L 1 (τ, T ; X). The next result is an infinite-dimensional version of the Arze là-Ascoli Theorem. Theorem 2.7. Let X be a Banach space. A subset F in C([ τ, T ]; X) is relatively compact if and only if: (i) F is equicontinuous on [ τ, T ]; (ii) there exists a dense subset D in [ τ, T ] such that, for each t ∈ D, F(t) = {f (t); f ∈ F} is relatively compact in X.
doi:10.1090/s0002-9947-08-04668-0 fatcat:mwbuptnlyfcqpcriqeeysjfehi