31 research outputs found
Gap functions and error bounds for variational-hemivariational inequalities
In this paper we investigate the gap functions and regularized gap functions for a class of variational–hemivariational inequalities of elliptic type. First, based on regularized gap functions introduced by Yamashita and Fukushima, we establish some regularized gap functions for the variational–hemivariational inequalities. Then, the global error bounds for such inequalities in terms of regularized gap functions are derived by using the properties of the Clarke generalized gradient. Finally, an application to a stationary nonsmooth semipermeability problem is given to illustrate our main results
Existence of solution to a new class of coupled variational-hemivariational inequalities
The objective of this paper is to introduce and study a complicated nonlinear
system, called coupled variational-hemivariational inequalities, which is
described by a highly nonlinear coupled system of inequalities on Banach
spaces. We establish the nonemptiness and compactness of the solution set to
the system. We apply a new method of proof based on a multivalued version of
the Tychonoff fixed point principle in a Banach space combined with the
generalized monotonicity arguments, and elements of the nonsmooth analysis. Our
results improve and generalize some earlier theorems obtained for a very
particular form of the system.Comment: 17
Optimal Control of Convective FitzHugh-Nagumo Equation
We investigate smooth and sparse optimal control problems for convective
FitzHugh-Nagumo equation with travelling wave solutions in moving excitable
media. The cost function includes distributed space-time and terminal
observations or targets. The state and adjoint equations are discretized in
space by symmetric interior point Galerkin (SIPG) method and by backward Euler
method in time. Several numerical results are presented for the control of the
travelling waves. We also show numerically the validity of the second order
optimality conditions for the local solutions of the sparse optimal control
problem for vanishing Tikhonov regularization parameter. Further, we estimate
the distance between the discrete control and associated local optima
numerically by the help of the perturbation method and the smallest eigenvalue
of the reduced Hessian
On the Tykhonov Well-posedness of an Antiplane Shear Problem
We consider a boundary value problem which describes the frictional antiplane
shear of an elastic body. The process is static and friction is modeled with a
slip-dependent version of Coulomb's law of dry friction. The weak formulation
of the problem is in the form of a quasivariational inequality for the
displacement field, denoted by \cP. We associated to problem \cP a boundary
optimal control problem, denoted by \cQ. For Problem \cP we introduce the
concept of well-posedness and for Problem \cQ we introduce the concept of
weakly and weakly generalized well-posedness, both associated to appropriate
Tykhonov triples. Our main result are Theorems \ref{t1} and \ref{t2}. Theorem
\ref{t1} provides the well-posedness of Problem \cP and, as a consequence,
the continuous dependence of the solution with respect to the data. Theorem
\ref{t2} provides the weakly generalized well-posedness of Problem \cQ and,
under additional hypothesis, its weakly well posedness. The proofs of these
theorems are based on arguments of compactness, lower semicontinuity,
monotonicity and various estimates. Moreover, we provide the mechanical
interpretation of our well-posedness results.Comment: 21 page
Duality Arguments in the Analysis of a Viscoelastic Contact Problem
We consider a mathematical model which describes the quasistatic frictionless
contact of a viscoelastic body with a rigid-plastic foundation. We describe the
mechanical assumptions, list the hypotheses on the data and provide three
different variational formulations of the model in which the unknowns are the
displacement field, the stress field and the strain field, respectively. These
formulations have a different structure. Nevertheless, we prove that they are
pairwise dual of each other. Then, we deduce the unique weak solvability of the
contact problem as well as the Lipschitz continuity of its weak solution with
respect to the data. The proofs are based on recent results on
history-dependent variational inequalities and inclusions. Finally, we present
numerical simulations in the study of the contact problem, together with the
corresponding mechanical interpretations.Comment: 25 pages, 4 figure
Equilibrium Problems with Equilibrium Constraints via Multiobjective Optimization
The paper concerns a new class of optimization-related problems called Equilibrium Problems with Equilibrium Constraints (EPECs). One may treat them as two level hierarchical problems, which involve equilibria at both lower and upper levels. Such problems naturally appear in various applications providing an equilibrium counterpart (at the upper level) of Mathematical Programs with Equilibrium Constraints (MPECs). We develop a unified approach to both EPECs and MPECs from the viewpoint of multiobjective optimization subject to equilibrium constraints. The problems of this type are intrinsically nonsmooth and require the use of generalized differentiation for their analysis and applications. This paper presents necessary optimality conditions for EPECs in finite-dimensional spaces based an advanced generalized variational tools of variational analysis. The optimality conditions are derived in normal form under certain qualification requirements, which can be regarded as proper analogs of the classical Mangasarian-Fromovitz constraint qualification in the general settings under consideration
Multiobjective Optimization Problems with Equilibrium Constraints
The paper is devoted to new applications of advanced tools of modern variational analysis and generalized differentiation to the study of broad classes of multiobjective optimization problems subject to equilibrium constraints in both finite-dimensional and infinite-dimensional settings. Performance criteria in multiobjectivejvector optimization are defined by general preference relationships satisfying natural requirements, while equilibrium constraints are described by parameterized generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and are handled in this paper via appropriate normal/coderivativejsubdifferential constructions that exhibit full calculi. Most of the results obtained are new even in finite dimensions, while the case of infinite-dimensional spaces is significantly more involved requiring in addition certain sequential normal compactness properties of sets and mappings that are preserved under a broad spectrum of operations