Fr echet diierentiability of the solution of the heat equation with respect to a nonlinear boundary condition Arnd RR osch Abstract. Consider the heat equation with a nonlinear function in the boundary condition which depends only on the boundary values x of the solution u of the initial-boundary value problem. The function belongs to a set of admissible diierentiable or uniformly Lipschitz con...

In this paper we study the class of differential variational inequality(DVI) in a finite-dimension Euclidean space <n, which is the following form ẋ(t) = f(t, x(t)) + B(t, x(t))u(t) , x(0) = x0 ∈ <n 0 ≤ (ũ− u(t))T [G(t, x(t)) + F (u(t))] for almost all ũ ∈ K u(t) ∈ K We study stability and perturbation of the DVI under the OSL condition. Besides, we establish a Prior Bound Theorem, which is a u...

We extend the classic convergence rate theory for subgradient methods to apply to non-Lipschitz functions. For the deterministic projected subgradient method, we present a global O(1/ √ T ) convergence rate for any convex function which is locally Lipschitz around its minimizers. This approach is based on Shor’s classic subgradient analysis and implies generalizations of the standard convergenc...

This paper is concerned with the stability of nonlinear Lipschitz systems subject to bounded process and measurement noises when transmission from sensor to controller is subject to distortion due to quantization. A stabilizing technique and a sufficient condition relating transmission rate to Lipschitz coefficients are presented for almost sure asymptotic bounded stability of nonlinear uncerta...

We study the global attractors of abstract semilinear parabolic equations and their projections to finite-dimensional planes. It is well-known that the attractor can be embedded into the finite-dimensional inertial manifold if the so-called spectral gap condition is satisfied. We show that in the case when the spectral gap condition is violated, it is possible to construct the nonlinearity in s...

In the paper some known and new extensions of the famous theorem of Filippov (1967) and a theorem of Pliś (1965) for differential inclusions are presented. We replace the Lipschitz condition on the set-valued map in the right-hand side by a weaker onesided Lipschitz (OSL), one-sided Kamke (OSK) or a continuity-like condition (CLC). We prove new Filippov-type theorems for singularly perturbed an...

Approximate optimality conditions for a class of nonconvex semi-infinite programs involving support functions are given. The objective function and the constraint functions are locally Lipschitz functions on n . By using a Karush-Kuhn-Tucker KKT condition, we deduce a necessary optimality condition for local approximate solutions. Then, generalized KKT conditions for the problems are proposed. ...

Consider an o-minimal expansion of the real field. We show that definable Lipschitz continuous maps can be definably fine approximated by definable continuously differentiable Lipschitz maps whose Lipschitz constant is close to that of the original map.

For a linear-quadratic state constrained optimal control problem, it is proved in [11] that under an independence condition for the active constraints, the optimal control is Lipschitz continuous. We now give a new proof of this result based on an analysis of the Euler discretization given in [9]. There we exploit the Lipschitz continuity of the control to estimate the error in the Euler discre...

R.P. Boas has found necessary and sufficient conditions of belonging of function to Lipschitz class. From his findings it turned out, that the conditions on sine and cosine coefficients for belonging of function to Lip α (0 < α < 1) are the same, but for Lip 1 are different. Later his results were generalized by many authors in the viewpoint of generalization of condition on the majorant of mod...