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Keywords:
second-order derivative; $C^{1,1}$ function; $\ell $-stable function; $\tilde {\ell }$-stability
second-order derivative; $C^{1,1}$ function; $\ell $-stable function; $\tilde {\ell }$-stability
Summary:
The notion of $\tilde {\ell }$-stability is defined using the lower Dini directional derivatives and was introduced by the authors in their previous papers. In this paper we prove that the class of $\tilde {\ell }$-stable functions coincides with the class of C$^{1,1}$ functions. This also solves the question posed by the authors in SIAM J. Control Optim. 45 (1) (2006), pp.\ 383--387.
References:
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[2] Bednařík, D., Pastor, K.: Errata: Elimination of strict convergence in optimization. SIAM J. Control Optim. 45 (2006), 383-387. DOI 10.1137/050636309 | MR 2225311
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[4] Cannarsa, P., Sinestrari, C.: Semiconave Functions, Hamilton-Jacobi Equations, and Optimal Control. Progress in Nonlinear Differential Equations 58, Birkäuser, Boston, MA (2004). MR 2041617
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