Abstract: The regularity and stability of optimal controls of nonstationary Navier-Stokes equations are investigated. Under suitable assumptions every control satisfying first-order necessary conditions is shown to be a continuous function in both space and time. Moreover, the behaviour of a locally optimal control under certain perturbations of the cost functional and the state equation is inv...

This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable, but not Gâteaux-differentiable. Two types of necessary optimality conditions are derived, the first one by means of regularization, the second one by using the directional differentiability ...

We show that there exists a strong uniform embedding from any proper metric space into any Banach space without cotype. Then we prove a result concerning the Lipschitz embedding of locally finite subsets of Lp-spaces. We use this locally finite result to construct a coarse bi-Lipschitz embedding for proper subsets of any Lp-space into any Banach space X containing the l n p ’s. Finally using an...

In this paper we study nonlinear programming problems with equality, inequality, and abstract constraints where some of the functions are Fréchet differentiable at the optimal solution, some of the functions are Lipschitz near the optimal solution, and the abstract constraint set may be nonconvex. We derive Fritz John type and Karush–Kuhn–Tucker (KKT) type first order necessary optimality condi...

This paper presents an inexact version of exponential iterative method designed for solving nonlinear equations F(x)=0, where the function F is only locally Lipschitz continuous. The proposed algorithm completely new as essential extension nonsmooth equations. with backtracking globally and superlinearly convergent under some mild assumptions imposed on F. presented results numerical computatio...

We give a very simple proof of the caracterization of Lipschitz regularity of a function by the size of its Haar coefficients. It is well known that given a real function I periodic with period 27r satisfying a Lipschitz 0: condition for 0 < 0: ::::; 1, its kth Fourier coefficient is bounded by Ikl-a . More precisely, the following result holds (see for example Chapter 12 of [9]). (A) Let I be ...

The present paper is concerned with Lipschitz properties of convex mappings. One considers the general context of mappings defined on an open convex subset Ω of a locally convex space X and taking values in a locally convex space Y ordered by a normal cone. One proves also equi-Lipschitz properties for pointwise bounded families of continuous convex mappings, provided the source space X is barr...

We develop a constructive completion method in general Minkowski spaces, which successfully extends a completion procedure due to Bückner in twoand three-dimensional Euclidean spaces. We prove that this generalized Bückner completion is locally Lipschitz continuous, thus solving the problem of finding a continuous selection of the diametric completion mapping in finite dimensional normed spaces...

We show the existence result of viable solutions to the differential inclusion ẋ(t) ∈ G(x(t)) + F (t, x(t)) x(t) ∈ S on [0, T ], where F : [0, T ] × H → H (T > 0) is a continuous set-valued mapping, G : H → H is a Hausdorff upper semi-continuous set-valued mapping such that G(x) ⊂ ∂g(x), where g : H → R is a regular and locally Lipschitz function and S is a ball, compact subset in a separable H...

We obtain a Viterbo-Hofer-Zehnder type result for Hamiltonian inclusions. Let H : : IR2N -~ IR be a locally Lipschitz function and c E IR. Suppose that E := {aE IR2~ ~ ~(a-) = c} is a nonempty compact subset of IR2N and 0 ~ ~H(x) for x E E. Then for any 03B4 > 0 the Hamiltonian inclusion x E has a conservative periodic solution x(t) such that H (~(t)~ = c’ E (c ~ , c + 8) for all t.