Sensitivity analysis of distributed-parameter optimal control problems for nonlinear parabolic equations

A sequence of optimal control problems for systems described by nonlinear parabolic equations is considered. It is proved that, under the -convergence of objective functionals, the parabolicG-convergence of operators in the state equations, and the Kuratowski convergence of control constraint sets, a convergent sequence of optimal pairs has a limit which is an optimal pair for the limit control problem. The convergence of minimal values is also obtained.This research was supported in part by the Istituto Nazionale di Alta Matematica F. Severi, Rome, Italy. Part of this research was carried out while the author was visiting the Scuola Normale Superiore, Pisa, Italy.     

Control problems for parabolic and hyperbolic equations via the theory ofG and Γ convergence

Summary A sequence of optimal control problems for systems governed by PDE'sis considered. The parameter (index of an element of the sequence) appears in the cost functionals which have integral form, as well as in the state equations which are of parabolic or hyperbolic type. It is proved that, under some -convergence of the cost functionals and some convergence of the indicator functions of sets of admissible solutions, the optimal solutions exist and converge to an optimal solution of the limit problem.     

Γ-convergence and optimal control problems

In this paper, we give some applications ofG-convergence and -convergence to the study of the asymptotic limits of optimal control problems. More precisely, given a sequence (P _h) of optimal control problems and a control problem (P), we determine some general conditions, involvingG-convergence and -convergence, under which the sequence of the optimal pairs of the problems (P _h) converges to the optimal pair of problem (P).The authors wish to thank Professor E. De Giorgi for many stimulating discussions.     

Γ-Limit of Periodic Obstacles

We compute the -limit of a sequence obstacle functionals in the case of periodic obstacles.     

Convergence of Discrete Dirichlet Forms to Continuous Dirichlet Forms on Fractals

It is well known that a Dirichlet form on a fractal structure can be defined as the limit of an increasing sequence of discrete Dirichlet forms, defined on finite subsets which fill the fractal. The initial form is defined on V ⁽⁰⁾, which is a sort of boundary of the fractal, and we have to require that it is an eigenform, i.e., an eigenvector of a particular nonlinear renormalization map for Dirichlet forms on V ⁽⁰⁾. In this paper, I prove that, provided an eigenform exists, even if the form on V ⁽⁰⁾ is not an eigenform, the corresponding sequence of discrete forms converges to a Dirichlet form on all of the fractal, both pointwise and in the sense of -convergence (but these two limits can be different). The problem of -convergence was first studied by S. Kozlov on the Gasket.     

Variational Convergence of Composed Convex Functions

In this paper we introduce a concept of variational convergence for mappings taking values in order topological vector spaces. This variational convergence notion is shown to be well adapted to the (epi)-convergence of composed convex functions, in the sense that it is preserved after composition with nondecreasing functions. It is proved how this stability result can be applied to the continuity of multipliers under perturbations associated with a family of constrained optimization problems. Other applications are also given.     

The lower algebraicK-theory of virtually infinite cyclic groups

This paper gives a proof of a conjecture of W.-C. Hsiang for the negativeK-theory of integral grouprings , when the group is a subgroup of a uniform lattice in a Lie group. The authors' earlier paper reduced this result to the very special cases where either is finite or is virtually infinite cyclic. The finite case was done much earlier by Carter extending results of Bass and Murthy. The major work of the present paper consists of proving the conjecture when is virtually infinite cyclic.Both authors were supported in part by the National Science Foundation.     

Fiber Bundles and Regular Approximation of Codimension-One Cycles

We derive an approximation of codimension-one integral cycles(and cycles modulo p) in a compact Riemannian manifold bymeans of piecewise regular cycles: we obtain both flat convergence andconvergence of the masses. The theorem is proved by using suitableprincipal bundles with a discrete group. As a byproduct, we give analternative proof of the main results, which does not use the regularitytheory for homology minimizers in a Riemannian manifold. This also givesa result of -convergence.     

Soliton-Stripe Patterns in Charged Langmuir Monolayers

We consider a charged Langmuir monolayer problem where electrostatic interaction forces undulations in the molecular concentration of the monolayer. Using the -convergence theory in singular perturbative variational calculus, we prove the existence of soliton-stripe lamellar patterns as one-dimensional local minimizers of the free energy, which are characterized by sharp domain walls delineating fully segregated dense liquid and dilute gas regions of the monolayer.     

On the Operator Semirings of a Γ-Semiring

In this paper we introduce the notion of operator semirings of a -semiring to study -semirings. It is shown that the lattices of all left (right) ideals (two-sided ideals) of a -semiring and its right (respectively left) operator semiring are isomorphic. This has many applications to characterize various -semirings.AMS Subject Classification (2000): 16Y60, 16Y99     

Study of Noetherian Γ-Semirings via Operator Semirings

In this paper we study Noetherian -semirings and obtain Cohens theorem for a special class of -semirings. Weak primary decomposition theorem for a particular type of -semirings is also obtained.Presently Lecturer in Mathematics, University of Burdwan, GOLAPBAG, W.B. INDIA.     

Variational approximation of anisotropic functionals on partitions

We prove an approximation result in the sense of -convergence for functionals defined on partitions modelling anisotropic multi-phase systems. Mathematics Subject Classification (2000) 49J45     

Limits of nonlinear Dirichlet problems in varying domains

We study the general form of the limit, in the sense of -convergence, of a sequence of nonlinear variational problems in varying domains with Dirichlet boundary conditions. The asymptotic problem is characterized in terms of the limit of suitable nonlinear capacities associated to the domains.     

A convergence result for a sequence of distributed-parameter optimal control problems

In this paper, we consider a sequence of abstract optimal control problems by allowing the cost integrand, the partial differential operator, and the control constraint set all to vary simultaneously. Using the notions of -convergence of functions,G-convergence of operators, and Kuratowski-Mosco convergence of sets, we show that the values of the approximating problems converge to that of the limit problem. Also we show that a convergent sequence of optimal pairs for the approximating problems has a limit which is optimal for the limit problem. A concrete example of parabolic optimal control problems is worked out in detail.This research was supported by NSF Grant No. DMS-88-02688.     

On the boundedness of Cauchy singular operator from the spaceL p toL q,p>q>-1

It is proved that for a Cauchy type singular operator, given by equality (1), to be bounded from the Lebesgue spaceL _p () tol _q (), as = _n=1 ^Ȟ _n , _n ={z:|z|=r _n }, it is necessary and sufficient that either condition (4) or (5) be fulfilled.     

Linear Versus Nonlinear Rules for Mixture Normal Priors

The problem under consideration is the -minimax estimation, under L₂loss, of a multivariate normal mean when the covariance matrix is known. The family of priors is induced by mixing zero mean multivariate normals with covariance matrix I by nonnegative random variables , whose distributions belong to a suitable family G. For a fixed family G, the linear (affine) -minimax rule is compared with the usual -minimax rule in terms of corresponding -minimax risks. It is shown that the linear rule is "good", i.e., the ratio of the risks is close to 1, irrespective of the dimension of the model. We also generalize the above model to the case of nonidentity covariance matrices and show that independence of the dimensionality is lost in this case. Several examples illustrate the behavior of the linear -minimax rule.     

A New Relaxation Space for Obstacles

The space of obstacles (i.e. p-quasi upper semicontinuous functions) is endowed with a distance which is topologically equivalent to the -convergence. We find the metric completion of this space and we give some application for minimization problems of cost functionals depending on obstacles via their level sets. An element of the completion is a decreasing and _p -continuous on the left mapping Rt _t , where _t are positive Borel measures vanishing on sets of zero p-capacity.