Nonsmooth Term Research Articles

Maximal monotone model with delay term of convolution

Mechanical models are governed either by partial differential equations with boundary conditions and initial conditions (e.g., in the frame of continuum mechanics) or by ordinary differential equations (e.g., after discretization via Galerkin procedure or directly from the model description) with the initial conditions. In order to study dynamical behavior of mechanical systems with a finite number of degrees of freedom including nonsmooth terms (e.g., friction), we consider here problems governed by differential inclusions. To describe effects of particular constitutive laws, we add a delay term. In contrast to previous papers, we introduce delay via a Volterra kernel. We provide existence and uniqueness results by using an Euler implicit numerical scheme; then convergence with its order is established. A few numerical examples are given.

Approximate Penalty Method in Optimal Control Problems for Nonsmooth Singular Systems

We consider an abstract optimal control problem with additional constraints and nonsmooth terms, but without the requirement that both the state equation on the set of admissible controls and the extremum problem be solvable. We use the approximate penalty method proposed here to find an approximate (in the weak sense) solution of the problem. As an example, we consider the optimal control problem for a singular nonlinear elliptic type equation.

A New Exact Penalty Function

For constrained smooth or nonsmooth optimization problems, new continuously differentiable penalty functions are derived. They are proved exact in the sense that under some nondegeneracy assumption, local optimizers of a nonlinear program are precisely the optimizers of the associated penalty function. This is achieved by augmenting the dimension of the program by a variable that controls both the weight of the penalty terms and the regularization of the nonsmooth terms.

Local Strong Homogeneity of a Regularized Estimator

This paper deals with regularized pointwise estimation of discrete signals which contain large strongly homogeneous zones, where typically they are constant, or linear, or more generally satisfy a linear equation. The estimate is defined as the minimizer of an objective function combining a quadratic data-fidelity term and a regularization prior term. The latter term is the sum of the values obtained by applying a potential function (PF) to each component, called a difference, of a linear transform of the signal. Minimizers of functions of this form arise in various settings in statistics and optimization. The features exhibited by such an estimate are closely related to the shape of the PF. Our goal is to determine estimators providing solutions which involve large strongly homogeneous zones--where more precisely the differences are null--in spite of the noise corrupting the data. To this end, we require that the strongly homogeneous zones, recovered by the estimator, be insensitive to any variation of the data inside a small open ball. More generally, this requirement is addressed to any local or global minimizer of the objective function whose local behavior with respect to the data gives rise to a locally continuous minimizer function. On the one hand, we show that if the PF is smooth at zero, then all the data, yielding minimizers with large, strongly homogeneous zones, are contained in a closed, negligible set. The chance that noisy data generate such minimizers is null. In contrast, if the PF is nonsmooth at zero, then for almost all data, the strongly homogeneous zones recovered by a minimizer function are preserved constant under any small perturbation of the data. The data domain is thus organized into volumes whose elements yield minimizers which share the same strongly homogeneous zones. This explains why the solutions, obtained using nonsmooth-at- zero PFs, exhibit strongly homogeneous zones. These theoretical results are illustrated using a numerical example. Our analysis can be extended to general functions combining smooth and nonsmooth terms.

A Perturbation Result for a Double Eigenvalue Hemivariational Inequality with Constraints and Applications

In this paper we prove a perturbation result for a new type of eigenvalue problem introduced by D. Motreanu and P.D. Panagiotopoulos (1998). The perturbation is made in the nonsmooth and nonconvex term of a double eigenvalue problem on a spherlike type manifold considered in ’Multiple solutions for a double eigenvalue hemivariational inequality on a spherelike type manifold‘ (to appear in Nonlinear Analysis). For our aim we use some techniques related to the Lusternik-Schnirelman theory (including Krasnoselski‘s genus) and results proved by J.N. Corvellec et al. (1993), M. Degiovanni and S. Lancelotti (1995), and V.D. Radulescu and P.D. Panagiotopoulos (1998). We apply these results in the study of some problems arising in Nonsmooth Mechanics.

Asymptotic behaviour of the solution of the singularly perturbed parabolic problem in the critical case

A zeroth-order boundary-layer asymptotic expansion is constructed and proved for solving a mixed boundary-value problem for a parabolic equation in the critical case. A smoothing procedure is applied to the nonsmooth terms of the asymptotic expansion when constructing the solution.

Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case

The structure of the global discretization error is studied for the implicit midpoint and trapezoidal rules applied to nonlinearstiff initial value problems. The point is that, in general, the global error contains nonsmooth (oscillating) terms at the dominanth 2-level. However, it is shown in the present paper that for special classes of stiff problems these nonsmooth terms contain an additional factor ? (where-1/? is the magnitude of the stiff eigenvalues). In these cases a "full" asymptotic error expansion exists in thestrongly stiff case (? sufficiently small compared to the stepsizeh). The general case (where the oscillating error components areO(h 2) and notO(?h 2)) and applications of our results (extrapolation and defect correction algorithims) will be studied in separate papers.