In this paper, we consider a class of nonsmooth, nonconvex constrained optimization problems where the objective function may be not Lipschitz continuous and the feasible set is a general closed convex set. Using the theory of the generalized directional derivative and the Clarke tangent cone, we derive a first order necessary optimality condition for local minimizers of the problem, and define...

We consider a finite string where, at both end points, a homogeneous Dirichlet boundary condition holds. One boundary point is fixed, and the other is moving; hence the length of the string is changing in time. The string is controlled through the movement of this boundary point. We consider movements of the boundary that are Lipschitz continuous. Only movements for which at the given finite te...

The problem of unknown input observer design for non-linear Lipschitz systems is considered. A new dynamic framework which is a generalization of previously used linear unknown input observers is introduced. The additional degrees of freedom offered by this dynamic framework are used to deal with the Lipschitz non-linearity. The necessary and sufficient condition that ensures asymptotic converg...

Our aim in the present paper is three fold. Firstly, we obtain a common fixed point theorem for a pair of self mappings satisfying a Lipschitz type condition employing the property (E.A.) along with a relatively new notion of absorbing pair of maps wherein we never require conditions on the completeness of the space, containment of range of one mapping into the range of other, continuity of the...

This paper provides new conditions under which optimal controls are Lipschitz continuous for dynamic optimization problems with functional inequality constraints, a control constraint expressed in terms of a general closed convex set and a coercive cost function. It is shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by...

(1.1) ∂tb(x, u) + div(F̄ (t, x, u)− k∇u) = f(t, x, u, s), s(t, x) = ∫ t 0 K(t, z)ψ(u(z, x))dz in Ω × (0, T ], T < ∞, Ω ⊂ R is a bounded domain, ∂Ω ∈ C, see [26]. If Ω is convex, then ∂Ω is assumed to be Lipschitz continuous. We consider a Dirichlet boundary condition (1.2) u(t, x) = 0 on I × ∂Ω, I = (0, T ], together with the initial condition (1.3) u(0, x) = u0(x) x ∈ Ω. We assume 0 < ε ≤ ∂sb(x...

This paper concerns observers design for Lipschitz nonlinear systems with sampled output. Using reachability analysis, an upper approximation of the attainable set is given. When this approximation is formulated in terms of a convex combination of linear mappings, a sufficient condition is given in terms of linear matrix inequalities which can be solved employing a linear matrix inequalities so...

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz co...

For a metric space (K, d) the Banach space Lip(K) consists of all scalar-valued bounded Lipschitz functions on K with the norm ‖f‖L = max(‖f‖∞, L(f)), where L(f) is the Lipschitz constant of f . The closed subspace lip(K) of Lip(K) contains all elements of Lip(K) satisfying the lip-condition lim0<d(x,y)→0 |f(x) − f(y)|/d(x, y) = 0. For K = ([0, 1], | · |), 0 < α < 1, we prove that lip(K) is a p...

We consider a nonlinear (possibly) degenerate elliptic operator Lv = − div a(∇v) + b(x, v) where the functions a and b are, unnecessarly strictly, monotonic. For a given boundary datum φ we prove the existence of the maximum and the minimum of the solutions and formulate a Haar-Rado type result, namely a continuity property for these solutions that may follow from the continuity of φ. In the ho...