Applied Mathematics Commons^™

Optimal Control Of Delay Systems With Differential And Algebraic Dynamic Constraints, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

This paper concerns constrained dynamic optimization problems governed by delay control systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential inclusion and a linear constraint link between "slow" and "fast" variables. We pursue a two-hold goal: to study variational stability for this class of control systems with respect to discrete …

Optimal Boundary Control Of Hyperbolic Equations With Pointwise State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond

Mathematics Research Reports

In this paper we consider dynamic optimization problems for hyperbolic systems with boundary controls and pointwise state constraints. In contrast to parabolic dynamics, such systems have not been sufficiently studied in the literature. The reason is the lack of regularity in the case of hyperbolic dynamics. We present necessary optimality conditions for both Neumann and Dirichlet boundary control problems and discuss differences and relationships between them.

Computing Isotypic Projections With The Lanczos Iteration, David K. Maslen, Michael E. Orrison, Daniel N. Rockmore

Dartmouth Scholarship

When the isotypic subspaces of a representation are viewed as the eigenspaces of a symmetric linear transformation, isotypic projections may be achieved as eigenspace projections and computed using the Lanczos iteration. In this paper, we show how this approach gives rise to an efficient isotypic projection method for permutation representations of distance transitive graphs and the symmetric group.

Finite Horizon Riemann Structures And Ergodicity, Victor J. Donnay, Charles Pugh

Mathematics Faculty Research and Scholarship

In this paper we show that any surface in R-3 can be modified by gluing on small 'focusing caps' so that its geodesic flow becomes ergodic. A new concept, finite horizon cap geometry, is what makes the construction work.

A Computational Model For Martensitic Thin Films With Compositional Fluctuation, Pavel Bělík, Mitchell Luskin

Faculty Authored Articles

We develop a computational model for the martensitic first-order structural phase transformation in a single crystal thin film, and we use this model to study the effect of spatial compositional fluctuation, spatial temporal noise, and the loss of stability of the metastable phase at temperatures sufficiently far from the transformation temperature.

Examples Of Cayley 4-Manifolds, Weiqing Gu, Christopher Pries '03

All HMC Faculty Publications and Research

We determine several families of so-called Cayley 4-dimensional manifolds in the real Euclidean 8-space. Such manifolds are of interest because Cayley 4-manifolds are supersymmetric cycles that are candidates for representations of fundamental particles in String Theory. Moreover, some of the examples of Cayley manifolds discovered in this paper may be modified to construct explicit examples in our current search for new holomorphic invariants for Calabi-Yau 4-folds and for the further development of mirror symmetry.

We apply the classic results of Harvey and Lawson to find Cayley manifolds which are graphs of functions from the set of quaternions to itself. We …

Discrete-Time Approximations Of Stochastic Delay Equations: The Milstein Scheme, Yaozhong Hu, Salah-Eldin A. Mohammed, Feng Yan

Articles and Preprints

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDE's). The scheme has convergence order 1. In order to establish the scheme, we prove an infinite-dimensional Itô formula for "tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the non-anticipating nature of the SDDE, the use of anticipating calculus methods appears to be novel.

Semilinear Equations With Discrete Spectrum, Alfonso Castro

All HMC Faculty Publications and Research

This is an overview of the solvability of semilinear equations where the linear part has discrete spectrum. Semilinear elliptic and hyperbolic equations, as well as Hammerstein integral equations, are used as motivating examples. The presentation is intended to be accessible to non experts.

An Existence Result For A Class Of Sublinear Semipositone Systems, Alfonso Castro, C. Maya, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We consider the existence of positive solutions for the system

-Δu_i = λ[f_i(u₁,u₂,...,u_m) - h_i]; Ω

u_i = 0; ∂Ω

where λ > 0 is a parameter, Δ is the Laplacian operator, Ω is a bounded domain in Rⁿ; n ≥ 1 with a smooth boundary ∂Ω, f_i are C¹ functions satisfying f₁(0,0,...,0) = 0, lim z→∞ f_i(z,z,...,z) = ∞ and lim z→∞ f_i(z,z,...,z)/z = 0, and h_i are nonnegative continuous functions in Ω for i = 1,2,...,m. …

On Multivariate Abel-Gontscharoff Interpolation, Tian-Xiao He

Scholarship

By using Gould's annihilation coefficients, we obtain an explicit fundamental polynomials of Multivariate Abel-Gontscharoff Interpolation and its remainder expression.

Integral Transforms, Convolution Products, And First Variations, Bong Jin Kim, Byoung Soo Kim, David Skough

Department of Mathematics: Faculty Publications

We establish the various relationships that exist among the integral transform F_α,βF, the convolution product (F∗G)α, and the first variation δF for a class of functionals defined on K[0,T], the space of complex-valued continuous functions on [0,T] which vanish at zero.

Discrete Approximations And Necessary Optimality Conditions For Functional-Differential Inclusions Of Neutral Type, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

This paper deals with necessary optimality conditions for optimal control systems governed by constrained functional-differential inclusions of neutral type. While some results are available for smooth control systems governed by neutral functional-differential equations, we are not familiar with any results for neutral functional-differential inclusions, even with smooth cost functionals in the absence of endpoint constraints. Developing the method of discrete approximations and employing advanced tools of generalized differentiation, we conduct a variational analysis of neutral functional-differential inclusions and obtain new necessary optimality conditions of both Euler-Lagrange and Hamiltonian types.

On The Empirical Balanced Truncation For Nonlinear Systems, Marissa Condon, Rossen Ivanov

Articles

Novel constructions of empirical controllability and observability gramians for nonlinear systems for subsequent use in a balanced truncation style of model reduction are proposed. The new gramians are based on a generalisation of the fundamental solution for a Linear Time-Varying system. Relationships between the given gramians for nonlinear systems and the standard gramians for both Linear Time-Invariant and Linear Time-Varying systems are established as well as relationships to prior constructions proposed for empirical gramians. Application of the new gramians is illustrated through a sample test-system.

Analysis Of A Multigrid Algorithm For Time Harmonic Maxwell Equations, Jay Gopalakrishnan, Joseph E. Pasciak, Leszek Demkowicz

Mathematics and Statistics Faculty Publications and Presentations

This paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic Maxwell equations. In particular, a "backslash" multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used. The method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator. This idea has previously been used in multigrid analysis of indefinite second order elliptic problems. However, the Maxwell application involves a nonelliptic indefinite operator. With the help of a few new estimates, the …

A Characterization Of Hybridized Mixed Methods For Second Order Elliptic Problems, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we give a new characterization of the approximate solution given by hybridized mixed methods for second order self-adjoint elliptic problems. We apply this characterization to obtain an explicit formula for the entries of the matrix equation for the Lagrange multiplier unknowns resulting from hybridization. We also obtain necessary and sufficient conditions under which the multipliers of the Raviart–Thomas and the Brezzi–Douglas–Marini methods of similar order are identical.

Quasioptimality Of Some Spectral Mixed Methods, Jay Gopalakrishnan, Leszek Demkowicz

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we construct a sequence of projectors into certain polynomial spaces satisfying a commuting diagram property with norm bounds independent of the polynomial degree. Using the projectors, we obtain quasioptimality of some spectralmixed methods, including the Raviart–Thomas method and mixed formulations of Maxwell equations. We also prove some discrete Friedrichs type inequalities involving curl.

Uniqueness Theorems In Bioluminescence Tomography, Ge Wang, Yi Li, Ming Jiang

Mathematics and Statistics Faculty Publications

Motivated by bioluminescent imaging needs for studies on gene therapy and other applications in the mouse models, a bioluminescence tomography (BLT) system is being developed in the University of Iowa. While the forward imaging model is described by the well-known diffusion equation, the inverse problem is to recover an internal bioluminescent source distribution subject to Cauchy data. Our primary goal in this paper is to establish the solution uniqueness for BLT under practical constraints despite the ill-posedness of the inverse problem in the general case. After a review on the inverse source literature, we demonstrate that in the general case …

The Dual Spectral Set Conjecture, Steen Pedersen

Mathematics and Statistics Faculty Publications

Suppose that Λ = (aZ + b) ∪ (cZ + d) where a, b, c, d are real numbers such that a ≠ 0 and c ≠ 0. The union is not assumed to be disjoint. It is shown that the translates Ω + λ, λ is an element of Λ, tile the real line for some bounded measurable set Ω if and only if the exponentials e_λ(x) = e^i2πλx, λ is an element of Λ, form an orthogonal basis for some bounded measurable set Ω'.

Analysis Of Social Aspects Of Migrant Labourers Living With Hiv/Aids Using Fuzzy Theory And Neutrosophic Cognitive Maps, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic logic grew as an alternative to the existing topics and it represents a mathematical model of uncertainty, vagueness, ambiguity, imprecision, undefined-ness, unknown, incompleteness, inconsistency, redundancy and contradiction. Despite various attempts to reorient logic, there has remained an essential need for an alternative system that could infuse into itself a representation of the real world. Out of this need arose the system of neutrosophy and its connected logic, neutrosophic logic. This new logic, which allows also the concept of indeterminacy to play a role in any real-world problem, was introduced first by one of the authors Florentin Smarandache. In this …

A Survey Of Results Involving Transforms And Convolutions In Function Space, David Skough, David Storvick

Department of Mathematics: Faculty Publications

In this paper we survey various results involving Fourier-Wiener transforms, Fourier-Feynman transforms, integral transforms and convolution products of functionals over function space that have been established since Cameron and Martin first introduced Fourier-Wiener transforms in 1945.

Equilibrium Problems With Equilibrium Constraints Via Multiobjective Optimization, Boris S. Mordukhovich

Mathematics Research Reports

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 …

Necessary Conditions In Nonsmooth Minimization Via Lower And Upper Subgradients, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns first-order necessary optimality conditions for problems of minimizing nonsmooth functions under various constraints in infinite-dimensional spaces. Based on advanced tools of variational analysis and generalized differential calculus, we derive general results of two independent types called lower subdifferential and upper subdifferential optimality conditions. The former ones involve basic/limiting subgradients of cost functions, while the latter conditions are expressed via Frechetjregular upper subgradients in fairly general settings. All the upper subdifferential and major lower subdifferential optimality conditions obtained in the paper are new even in finite dimensions. We give applications of general optimality conditions to mathematical programs with …

Optimal Control Of Delayed Differential-Algebraic Inclusions, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

This paper concerns constrained dynamic optimization problems governed by delayed differential-algebraic systems. Dynamic constraints in such systems, which are particularly important for engineering applications, are described by interconnected delay-differential inclusions and algebraic equations. We pursue a two-hold goal: to study variational stability of such control systems with respect to discrete approximations and to derive necessary optimality conditions for both delayed differential-algebraic systems and their finite-difference counterparts using modern tools of variational analysis and generalized differentiation. We are not familiar with any results in these directions for differential-algebraic inclusions even in the delay-free case. In the first part of the paper …

The Approximate Maxium Principle In Constrained Optimal Control, Boris S. Mordukhovich, Ilya Shvartsman

Mathematics Research Reports

The paper concerns optimal control problems for dynamic systems governed by a parametric family of discrete approximations of control systems with continuous time. Discrete approximations play an important role in both qualitative and numerical aspects of optimal control and occupy an intermediate position between discrete-time and continuous-time control systems. The central result in optimal control of discrete approximations is the Approximate Maximum Principle (AMP), which is justified for smooth control problems with endpoint constraints under certain assumptions without imposing any convexity, in contrast to discrete systems with a fixed step. We show that these assumptions are essential for the validity …

Normal Forms For Nonlinear Discrete Time Control Systems, Boumediene Hamzi, Issa Amadou Tall

Miscellaneous (presentations, translations, interviews, etc)

We study the feedback classification of discrete-time control systems whose linear approximation around an equilibrium is controllable. We provide a normal form for systems under investigation.

Global Solutions To The Lake Equations With Isolated Vortex Regions, Chaocheng Huang

Mathematics and Statistics Faculty Publications

The vorticity formulation for the lake equations in R² is studied.

Optimization And Feedback Control Of Constrained Parabolic Systems Under Uncertain Perturbations, Boris S. Mordukhovich, Ilya Shvartsman

Mathematics Research Reports

This paper concerns a minimax control design problem for a class of parabolic systems with nonregular boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We deal with boundary controllers acting through Dirichlet boundary conditions that are the most challenging for the parabolic dynamics.

Neumann Boundary Control Of Hyperbolic Equations With Pointwise State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond

Mathematics Research Reports

We consider optimal control problems for hyperbolic systems with controls in Neumann boundary conditions with pointwise (hard) constraints on control and state functions. Focusing on hyperbolic dynamics governed by the multidimensional wave equation with a nonlinear term, we derive new necessary optimality conditions in the pointwise form of the Pontryagin Maximum Principle for the state-constrained problem under consideration. Our approach is based on modern methods of variational analysis that allows us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional.

Controllability And Local Accessibility—A Normal Form Approach, Wei Kang, Mingqing Xiao, Issa Amadou Tall

Articles and Preprints

Given a system with an uncontrollable linearization at the origin, we study the controllability of the system at equilibria around the origin. If the uncontrollable mode is nonzero, we prove that the system always has other equilibria around the origin. We also prove that these equilibria are linearly controllable provided a coefficient in the normal form is nonzero. Thus, the system is qualitatively changed from being linearly uncontrollable to linearly controllable when the equilibrium point is moved from the origin to a different one. This is called a bifurcation of controllability. As an application of the bifurcation, systems with a …

Dirichlet Boundary Control Of Hyperbolic Equations In The Presence Of State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond

Mathematics Research Reports

We study optimal control problems for hyperbolic equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under hard/pointwise control and state constraints. Imposing appropriate convexity assumptions on the cost integral functional, we establish the existence of optimal control and derive new necessary optimality conditions in the integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained systems.