Let E be a closed set in R, and suppose that there is a k ≥ 1 such that every x, y ∈ E can be connected by a rectifiable path in E with length ≤ k |x−y|. This condition is satisfied by chord-arc curves, Lipschitz manifolds of any dimension, and fractals like Sierpinski gaskets and carpets. Note that length-minimizing paths in E are chord-arc curves with constant k. A basic feature of this condi...

This paper provides a converse Liapunov theorem for uniformly locally exponentially stable, locally Lipschitz, non-linear, time-varying, possibly non-smooth systems that admit Carathéodory solutions. The main result proves that a critical point of such a system is uniformly locally exponentially stable if and only if the system admits a local (possibly non-smooth, timevarying) Liapunov function.

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...

This paper presents a Riemannian trust region algorithm for unconstrained optimization problems with locally Lipschitz objective functions defined on complete Riemannian manifolds. To this end we define a function Φ : TM → R on the tangent bundle TM , and at k-th iteration, using the restricted function Φ|TxkM where TxkM is the tangent space at xk, a local model function Qk that carries both fi...

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...

We consider optimization problems with objective and constraint functions that may be nonconvex and nonsmooth. Problems of this type arise in important applications, many having solutions at points of nondifferentiability of the problem functions. We present a line search algorithm for situations when the objective and constraint functions are locally Lipschitz and continuously differentiable o...

Strong convergence results on tamed Euler schemes, which approximate stochastic differential equations with superlinearly growing drift coefficients that are locally one-sided Lipschitz continuous, are presented in this article. The diffusion coefficients are assumed to be locally Lipschitz continuous and have at most linear growth. Furthermore, the classical rate of convergence, i.e. one–half,...