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1
Content available remote A short proof of the separable reduction theorem
Penot J. P.
Demonstratio Mathematica
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2010
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Vol. 43, nr 3
653-663
EN
We present a simple proof of the separable reduction theorem, a crucial result of nonsmooth analysis which allows to extend to Asplund spaces the results known for separable spaces dealing with Fréchet subdifferentials. It relies on elementary results in convex analysis and avoids certain technicalities.
2
Content available remote The union of uniform closed balls conjecture
Nour C., Stern R. J., Takche J.
Control and Cybernetics
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2009
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Vol. 38, no 4B
1525-1534
EN
The exterior sphere condition is compared to proximal smoothness, and examples are provided, which show that the two properties are not necessarily equivalent. Then conditions are given under which equivalence holds, and an open question involving the union of uniform closed balls property is stated in the form of a conjecture.
3
Content available remote The nonsmooth maximum principle
Clarke F. H., Pinho M. R. de
Control and Cybernetics
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2009
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Vol. 38, no 4A
1151-1167
EN
We present a brief survey of the nonsmooth maximum principle of optimal control, focusing, in particular, upon the alternative forms of the adjoint equation. We obtain a new version of the theorem that asserts for the first time the full Weierstrass condition together with the Euler form of the adjoint equation, thereby extending a result of de Pinho and Vinter. The new theorem also features stratified hypotheses and conclusions. Two examples illustrate its use.
4
On strict pseudoconvexity
Ivanov V., I.
Journal of Applied Analysis
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2007
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Vol. 13, nr 2
183-196
EN
The present paper provides first and second-order characterizations of a radilly lower semicontinuous strictly pseudoconvex function ∫ : X → R defined on a convex set X in the real Euclidean space Rn in twerms of the lower Dini-directional derivative. In particular we obtain connections between the strictly pseudoconvex functions, nonlinear programming problem, Stampacchia variational inequality, and strict Minty variational inequality. We extend to the radially continuous functions the characterization due to Diewert, Avriel, Zang [6]. A new implication appears in our conditions. Connections with other classes of functions are also derived
5
Content available remote The maximum principle in optimal control, then and now
Clarke F.
Control and Cybernetics
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2005
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Vol. 34, no 3
709-722
EN
We discuss the evolution of the Pontryagin maximum principle, focusing primarily on the hypotheses required for its validity. We proceed to describe briefly a unifying result giving rise to both classical and new versions, a recent theorem of the author giving necessary conditions for optimal control problems formulated in terms of differential inclusions. We conclude with a new application of this result for the case in which mixed constraints on the state and control are imposed in terms of equalities, inequalities, and unilateral set constraints. In order to lighten the exposition, the discussion is limited to differentiable data, thereby avoiding mention of generalized gradients or normal cones, except, in the technical section on differential inclusions.
6
Second-order characterizations of convex and pseudoconvex functions
Ginchev I., Ivanov V. I.
Journal of Applied Analysis
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2003
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Vol. 9, nr 2
261-273
EN
The present paper gives characterizations of radially u.s.c. convex and pseudoconvex functions f: X —> R defined on a convex subset X of a real linear space E in terms of first and second-order upper Dini-directional derivatives. Observing that the property f radially u.s.c. does not require a topological structure of E, we draw the possibility to state our results for arbitrary real linear spaces. For convex functions we extend a theorem of Huang, Ng [10]. For pseudoconvex functions we generalize results of Diewert, Avriel, Zang [6] and Crouzeix [4]. While some known results on pseudoconvex functions are stated in global concepts (e.g. Komlosi [11]), we succeeded in realizing the task to confine to local concepts only.
7
Content available remote Unilateral Contact Applications Using Fem Software
Stavroulaki M. E., Stavroulakis G. E.
International Journal of Applied Mathematics and Computer Science
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2002
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Vol. 12, no 1
115-125
EN
Nonsmooth analysis, inequality constrained optimization and variational inequalities are involved in the modelling of unilateral contact problems. The corresponding theoretical and algorithmic tools, which are part of the area known as nonsmooth mechanics, are by no means classical. In general purpose software some of these tools (perhaps in a simplified way) are currently available. Two engineering applications, a rubber-coated roller contact problem and a masonry wall, solved with MARC, are briefly presented, together with elements of the underlying theory.
8
An intersection formula for the normal cone associated with the hypertangent cone
Ciligot-Travain M.
Journal of Applied Analysis
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1999
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Vol. 5, nr 2
239-247
EN
The aim of this work is to present a new result about estimation of the hypertangent normal cone of an intersection without using directionally Lipschitz assumptions.
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