This paper studies the global output convergence of a class of recurrent neural networks with globally Lipschitz continuous and monotone nondecreasing activation functions and locally Lipschitz continuous time-varying inputs. We establish two sufficient conditions for global output convergence of this class of neural networks. Symmetry in the connection weight matrix is not required in the pres...

In 2002 F. Wirth has proved that the joint spectral radius of irreducible compact sets of matrices is locally Lipschitz continuous as a function of the matrix set. In the paper, an explicit formula for the related Lipschitz constant is obtained. PACS number 02.10.Ud; 02.10.Yn MSC 2000: 15A18; 15A60

Positive perturbations of generators of locally Lipschitz continuous increasing integrated semigroups on an abstract L space are again generators of locally Lipschitz continuous increasing integrated semigroups. Positive perturbations of generators of positive dual semigroups on a dual abstract L space are generators of semigroups that are weakly∗ right continuous. These results are reformulate...

This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity. We also show that, under so...

We establish optimal local regularity results for vector-valued extremals and minimizers of variational integrals whose integrand is the squared distance function to a compact setK in matrix space MN×n. The optimality is illustrated by explicit examples showing that, in the nonconvex case, minimizers need not be locally Lipschitz. This is in contrast to the case when the set K is suitably conve...

It is well-known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If – in addition – the coefficients grow at most linearly, then this flow ha...

ẋ = f (t, x) + g(t, x)u (1.1) x ∈ Rn, t 0, u ∈ U where f (t, x) and g(t, x) are C0 mappings on R+ × Rn , locally Lipschitz with respect to x ∈ Rn , with f (t, 0) = 0 for all t 0 and U ⊆ Rm is a convex set that contains 0 ∈ Rm . Our objective is to give necessary and sufficient conditions for the existence of a C0 function k : R+ × Rn → U , with k(·, 0) = 0, k(t, x) being locally Lipschitz with ...

This paper develops a new derivative-free method for solving linearly constrained nonsmooth optimization problems. The objective functions in these problems are, in general, non-regular locally Lipschitz continuous function. The computation of generalized subgradients of such functions is difficult task. In this paper we suggest an algorithm for the computation of subgradients of a broad class ...

This paper2 considers feedback design for nonlinear, multi-input affine control systems with disturbances. It studies the problem of assigning, by choice of feedback, a desirable upper bound to a given control Lyapunov function (clf) candidate's derivative along closed-loop trajectories. Specific choices for the upper bound are motivated by C2 and C, disturbance attenuation problems. The main r...

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. ...