On the Sequential Normal Compactness Condition and its Restrictiveness in Selected Function Spaces

Abstract

Sequential normal compactness is one of the most important properties in terms of modern variational analysis. It is necessary for the derivation of calculus rules for the computation of generalized normals to set intersections or preimages of sets under transformations. While sequential normal compactness is inherent in finite-dimensional Banach spaces, its presence has to be checked in the infinite-dimensional situation. In this paper, we show that broad classes of sets in Lebesgue and Sobolev spaces which are reasonable in the context of optimal control suffer from an intrinsic lack of sequential normal compactness.

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Acknowledgments

This work is supported by the DFG grant Analysis and Solution Methods for Bilevel Optimal Control Problems within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization).

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Correspondence to Patrick Mehlitz.

Appendix

Appendix

In the lemma below, we show a nearby relationship between the operators lin and conv.

Lemma A.1

Let \(S\subset \mathcal X\) be a nonempty subset of a Banach space \(\mathcal X\) . Then the relation \(\operatorname {lin} S=\operatorname {conv}\bigcup _{\alpha \in \mathbb {R}}\alpha S\) holds true.

Proof

Let us prove the inclusion [⊂]first. Since we have \(S\subset \operatorname {conv}\bigcup _{\alpha \in \mathbb {R}}\alpha S\), it is sufficient toshow that \(L:=\operatorname {conv}\bigcup _{\alpha \in \mathbb {R}}\alpha S\) is a linearspace. Therefore, we choose x,yL and γx,γy arbitrarily. Then we find integers n,m,vectors \({s^{x}_{1}},\dots ,{s^{x}_{n}}, {s^{y}_{1}},\dots ,{s^{y}_{m}}\in S\), scalars\({\alpha ^{x}_{1}},\dots ,{\alpha ^{x}_{n}},{\alpha ^{y}_{1}},\dots ,{\alpha ^{y}_{m}}\in \mathbb {R}\), and scalars \(\mu _{1},\dots ,\mu _{n},\nu _{1},\dots ,\nu _{m}\in \mathbb {R} ^{+}_{0}\) such that

$$x=\sum\limits_{i = 1}^{n}\mu_{i}{\alpha^{x}_{i}}{s^{x}_{i}},\quad y=\sum\limits_{j = 1}^{m}\nu_{j}{\alpha^{y}_{j}}{s^{y}_{j}},\quad\sum\limits_{i = 1}^{n}\mu_{i}= 1,\quad\sum\limits_{j = 1}^{m}\nu_{j}= 1. $$

Thus, we obtain

$$\gamma^{x}x+\gamma^{y}y=\sum\limits_{i = 1}^{n}\mu_{i}\gamma^{x}{\alpha^{x}_{i}}{s^{x}_{i}}+\sum\limits_{j = 1}^{m}\nu_{j}\gamma^{y}{\alpha^{y}_{j}}{s^{y}_{j}}=\sum\limits_{i = 1}^{n}\frac{\mu_{i}}{2}2\gamma^{x}{\alpha^{x}_{i}}{s^{x}_{i}}+\sum\limits_{j = 1}^{m}\frac{\nu_{j}}{2}2\gamma^{y}{\alpha^{y}_{j}}{s^{y}_{j}}\in L$$

since\(\tfrac {\mu _{1}}{2},\dots ,\tfrac {\mu _{n}}{2},\tfrac {\nu _{1}}{2},\dots ,\tfrac {\nu _{m}}{2}\in \mathbb {R} ^{+}_{0}\) are nonnegative scalars which satisfy

$$\sum\limits_{i = 1}^{n}\frac{\mu_{i}}{2}+\sum\limits_{j = 1}^{m}\frac{\nu_{j}}{2}= 1.$$

Consequently, L is a linearspace which contains S, i.e. lin SL.

The converse inclusion [⊃]follows from the obvious fact that αS ⊂ lin S holds for all α.This shows L ⊂ conv lin S = lin S andcompletes the proof. □

In this paper, we exploit the following consequence of Lebesgue’s dominated convergence theorem.

Lemma A.2

Fix a complete andσ-finitemeasure space (Ω,Σ, 𝔪)as well as a parameterp ∈ [1,). LetuLp (𝔪; q)be arbitrarily chosen and letk}k ⊂Σbe a sequence of sets which satisfies 𝔪(Ωk) → 0ask. Then\(\chi _{{\Omega }_{k}}u\to 0\)and\((1-\chi _{{\Omega }_{k}})u\to u\)inholdLp (𝔪; q)ask.

Proof

By construction, the sequences \(\{\chi _{{\Omega }_{k}}u\}_{k\in \mathbb {N} }\) and\(\{(1-\chi _{{\Omega }_{k}})u\}_{k\in \mathbb {N} }\) converge pointwise almost everywhere on Ωto 0 and u, respectively. Moreover, \(\left \{|{\chi _{{\Omega }_{k}}u}{2}\right \}_{k\in \mathbb {N} }\) and\(\left \{|{(1-\chi _{{\Omega }_{k}})u}{2}\right \}_{k\in \mathbb {N} }\) areboth majorized by |u2 ∈ Lp (𝔪), i.e.

$$\forall k\in\mathbb {N}\,\forall\omega\in{\Omega}\colon\quad|{\chi_{{\Omega}_{k}}(\omega)u(\omega)}{2}\leq|{u(\omega)}{2}\quad|{(1-\chi_{{\Omega}_{k}}(\omega))u(\omega)}{2}\leq|{u(\omega)}{2} $$

is valid. Thus, the lemma’s assertion follows from the dominated convergence theorem, see [30, Theorem5.2.2]. □

The following lemma shows that first-order Sobolev regularity is preserved while taking the component- and pointwise minimum in vector-valued function spaces. It is a straightforward consequence of [3, Corollary 5.8.2].

Lemma A.3

Fixp ∈ (1,). For functionsy,zW1,p(Ω)q, the componentwiseminimum function min{y; z}belongs toW1,p(Ω)qas well.For any indicesi ∈{1,…,q}as well asj ∈{1,…,d}, weobtain

$$\tfrac{\partial}{\partial\omega_{j}}\min\{y;z\}_{i}=\chi_{\{\omega\in{\Omega}\,|\,y_{i}(\omega)<z_{i}(\omega)\}}\tfrac{\partial}{\partial\omega_{j}}y_{i}+\chi_{\{\omega\in{\Omega}\,|\,y_{i}(\omega)\geq z_{i}(\omega)\}}\tfrac{\partial}{\partial \omega_{j}}z_{i}. $$

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Mehlitz, P. On the Sequential Normal Compactness Condition and its Restrictiveness in Selected Function Spaces. Set-Valued Var. Anal 27, 763–782 (2019). https://doi.org/10.1007/s11228-018-0475-6

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Keywords

  • Decomposable set
  • Optimal control
  • Sequential normal compactness

Mathematics Subject Classification (2010)

  • 28B05
  • 49J53