The subject of the thesis is the study of operator functions in the setting of symmetric operator spaces. In this latter setting, it is of great importance to analyze the properties of so-called operator functions A --> f(A), where the variable A is a self-adjoint operator and f is a complex-valued Borel function on the real line. The thesis study the question of differentiability of this type of operator functions. The latter question is intimately related to the study of commutators. Text not only extends existing results to the setting of unbounded self-adjoint linear operators, but it is also shown that this can be obtained via a unified approach utilizing the left regular representation of von Neumann algebras.

A fault in the system operation is deemed to occur when the system practically experiences an abnormal condition, such as a malfunction in the actuators/sensors. Hence, detection and isolation of the faulty components is crucial in control applications. Effective control and monitoring of a system requires accurate information of internal behaviour of the system. This internal behaviour can be analyzed by system's states. Practically, in many real systems, state space variables are not fully available for measurements. The two critical problems stated have motivated significant research work in the area of robust state and fault estimation. Fault reconstruction and estimation is regarded as a stronger extension to fault detection and isolation (FDI) since accurate fault estimation automatically implies fault detection. It is well known that two promising control strategies to cope with uncertain control processes are H_infinity Control and Sliding Mode Control. Therefore, in this PhD thesis, we employ these tools and we propose observer based robust fault reconstruction (RFR) by integrating H_infinity filtering and Sliding Mode Control. We also employ adaptive control on the sliding motion to deal with faults with unknown bounds. Another open problem in the context of FDI and RFR is due to systems with multiple faults at different system's components since it is often the case where actuators and also sensors suffer from faults during the course of the system's operation. Both actuators and sensors can suffer from faults either alone, at separate times or simultaneously. The co-existence of unknown fault at both sensor(s) and actuator(s) has not been addressed in any earlier design of fault reconstruction schemes. In this Thesis, inspired by the theory of singular systems, we aim at solving this problem. A New structure for reduced-order unknown input observers (UIOs) with application to chaotic communication and sensor fault reconstruction is also proposed. / Controls

This work is devoted to the equation <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cint_%7BS%7D%0A%5Cfrac%7Bu(y)%20%5C,%20dS(y)%7D%7B%7Cx-y%7C%5E%7BN-1%7D%7D%20=%20f(x)%20%5Ctext%7B,%7D%20%5Cqquad%20%5Cqquad%20x%20%5Cin%20S%20%5Ctext%7B,%7D%0A%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20(1)%0A" /> where S is the graph of a Lipschitz function φ on RN with small Lipschitz constant, and dS is the Euclidian surface measure. The integral in the left-hand side is referred to as a simple layer potential and f is a given function. The main objective is to find a solution u to this equation along with estimates for solutions near points on S. Our analysis is carried out in local Lp-spaces and local Sobolev spaces, and the estimates are given in terms of seminorms. In Paper 1, we consider the case when S is a hyperplane. This gives rise to the classical Riesz potential operator of order one, and we prove uniqueness of solutions in the largest class of functions for which the potential in (1) is defined as an absolutely convergent integral. We also prove an existence result and derive an asymptotic formula for solutions near a point on the surface. Our analysis allows us to obtain optimal results concerning the class of right-hand sides for which a solution to (1) exists. We also apply our results to weighted Lp- and Sobolev spaces, showing that for certain weights, the operator in question is an isomorphism between these spaces. In Paper 2, we present a fixed point theorem for a locally convex space <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathscr%7BX%7D" />, where the topology is given by a family <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5C%7Bp(%20%5C,%20%5Ccdot%20%5C,%20;%20%5Calpha%20)%5C%7D_%7B%5Calpha%20%5Cin%20%5COmega%7D" /> of seminorms. We study the existence and uniqueness of fixed points for a mapping<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathscr%7BK%7D%20%5C,%20:%20%5C;%20%5Cmathscr%7BD_K%7D%20%5Crightarrow%20%5Cmathscr%7BD_K%7D" /> defined on a set <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathscr%7BD_K%7D%20%5Csubset%20%5Cmathscr%7BX%7D" />. It is assumed that there exists a linear and positive operator K, acting on functions defined on the index set Ω, such that for every <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?u,v%20%5Cin%20%5Cmathscr%7BD_K%7D" />,   <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?p(%5Cmathscr%7BK%7D(u)%20-%20%5Cmathscr%7BK%7D(v)%20%5C,%20;%20%5C,%20%5Calpha%20)%20%0A%5Cleq%20K(p(u-v%20%5C,%20;%20%5C,%20%5Ccdot%20%5C,%20))%20(%5Calpha)%20%5Ctext%7B,%7D%20%5Cqquad%20%5Cqquad%20%5Calpha%20%5Cin%20%5COmega%0A%5Ctext%7B.%7D%0A" /> Under some additional assumptions, one of which is the existence of a fixed point for the operator K + p(<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathscr%7BK%7D(0)" /> ; · ), we prove that there exists a fixed point of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathscr%7BK%7D" />. For a class of elements satisfying Kn (p(u ; · ))(α) → 0 as n → ∞, we show that fixed points are unique. This class includes, in particular, the solution we construct in the paper. We give several applications, proving existence and uniqueness of solutions for two types of first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms. In Paper 3, we treat equation (1) in the case when S is a general Lipschitz surface and 1 &lt; p &lt; ∞. Our results are presented in terms of Λ(r), which is the Lipschitz constant of φ on the ball centered at the origin with radius 2r. Estimates of solutions to (1) are provided, which can be used to obtain knowledge about behaviour near a point on S in terms of seminorms. We also show that solutions to (1) are unique if they are subject to certain growth conditions. Examples are given when specific assumptions are placed on Λ. The main tool used for both existence and uniqueness is the fixed point theorem from Paper 2. In Paper 4, we collect some properties and estimates of Riesz potential operators, and also for the operator that was used in Paper 1 and Paper 3 to invert the Riesz potential of order one on RN, for the case when the density function is either radial or has mean value zero on spheres. It turns out that these properties define invariant subspaces of the respective domains of the operators in question.

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Thesis (Ph. D.)--University of Missouri-Columbia, 2008. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on June 18, 2009) Vita. Includes bibliographical references.

Thesis (Ph.D.)--University of Missouri-Columbia, 2005. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (January 25, 2007) Vita. Includes bibliographical references.

BRAGA, José Ederson Melo. Problemas variacionais de fronteira livre com duas fases e resultados do tipo Phragmén-Lindelof regidos por equações elípticas não lineares singulares/degeneradas. 2015. 126 f. Tese (Doutorado em Matemática) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2015. / Submitted by Erivan Almeida (eneiro@bol.com.br) on 2015-11-27T16:20:15Z No. of bitstreams: 1 2015_tese_jembraga.pdf: 26015791 bytes, checksum: 81b3be9d5afc13c4662dde0e0fc555da (MD5) / Approved for entry into archive by Rocilda Sales(rocilda@ufc.br) on 2015-11-27T17:03:58Z (GMT) No. of bitstreams: 1 2015_tese_jembraga.pdf: 26015791 bytes, checksum: 81b3be9d5afc13c4662dde0e0fc555da (MD5) / Made available in DSpace on 2015-11-27T17:03:58Z (GMT). No. of bitstreams: 1 2015_tese_jembraga.pdf: 26015791 bytes, checksum: 81b3be9d5afc13c4662dde0e0fc555da (MD5) Previous issue date: 2015 / In this work of thesis we discuss recents results on the regularity and geometric properties of variational solutions of two phase free boundary problems governed by singular/degenerate nonlinear elliptic equations. We also discuss Phragmén-Lindelof type results for such equations classifying those solutions in half spaces. / Neste trabalho de tese discutimos resultados recentes sobre a regularidade e propriedades geométricas de soluções variacionais de problemas de fronteira livre de duas fases regidos por equações elípticas não lineares degeneradas/singulares. Discutimos também resultados do tipo Phragmém-Lindelof para tais equações classificando essas soluções em semi-espaços.

SENA FILHO, Edvalter da Silva. Finitude para pares de germes de aplicações Bi-K-bi-Lipschitz equivalentes. 2016. 61 f. Tese (Doutorado em Matemática)- Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2016. / Submitted by Rocilda Sales (rocilda@ufc.br) on 2017-01-11T16:36:25Z No. of bitstreams: 1 2016_tese_essenafilho.pdf: 507783 bytes, checksum: 757aea745e363acfffd93d083b635d07 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-01-12T12:51:02Z (GMT) No. of bitstreams: 1 2016_tese_essenafilho.pdf: 507783 bytes, checksum: 757aea745e363acfffd93d083b635d07 (MD5) / Made available in DSpace on 2017-01-12T12:51:02Z (GMT). No. of bitstreams: 1 2016_tese_essenafilho.pdf: 507783 bytes, checksum: 757aea745e363acfffd93d083b635d07 (MD5) Previous issue date: 2016 / In this paper, we analyze the behavior of equivalence classes provided by the relation Bi-K-bi-Lipschitz. We show that when we are working with germs pairs of polynomial applications (f; g) : (Rn; 0) ! (Rp Rq; 0), with degree of f1; :::; fp; g1; :::; gq less than or equal to k 2 N, we have only a fi nite number of equivalence classes. We will also show in this work that the sets of equivalence classes with respect to strongly bi-lipschitz relation is fi nite. / Neste trabalho, iremos analisar o comportamento das classes de equivalência, fornecida pela rela ção Bi-K-bi-Lipschitz. Mostramos que, quando estamos trabalhando com pares de germes de aplica ções polinomiais (f; g) : (Rn; 0) ! (Rp Rq; 0), onde o grau de f1; :::fp; g1; :::; gq s~ao menores ou iguais a k 2 N, temos apenas uma quantidade fi nita de classes de equivalência. Tamb em mostraremos neste trabalho que o conjuntos das classes de equivalência com respeito a rela ção fortemente bi-lipschitz e fi nito.

Cette thèse, constituée de trois grandes parties, a pour objet l’étude générale ducomportement, en temps grands, de l’unique solution du problème de Cauchy-Dirichlet pour deséquations de Hamilton-Jacobi visqueuses de type sur et sous quadratiques. Après un bref rappeldes notions de base de la théorie sur les solutions de viscosité qui constitue le cadre de ce travail, lapremière partie établit des résultats sur l’existence globale en temps et l’unicité de la solution deviscosité dudit problème de Cauchy-Dirichlet. La deuxième partie s’intéresse au comportement decette solution pour des Hamiltoniens sur quadratiques. Sous des hypothèses très générales, nousprouvons que le comportement de la solution dépend du signe de l’unique constante ergodiquec du problème ergodique associé à des conditions aux limites de type contrainte d’état. Lorsquec∗ < 0; nous obtenons (i) une convergence vers l’unique solution du problème stationaire associétandis que lorsque c∗ ≥ 0; nous obtenons (ii) un comportement de type Hamilton-Jacobi (ou detype ergodique) se produit. Dans la troisième partie, consacrée à l’étude pour des Hamiltonienssous-quadratiques, nous montrons qu’il se produit un comportement de type (i) lorsque l’uniqueconstante ergodique c∗; du problème ergodique associé à des conditions aux limites de typeexplosives, est strictement négative et lorsque c∗ > 0 et 3/2< m ≤ 2; un comportement de type(ii) se produit, où m représente l’exposant du terme en gradient. Mais lorsque c∗ = 0 ou c∗ > 0et 1 < m ≤ 3/2; nous prouvons que pour certains domaines, la fonction u(x; t) + c∗t n’est pasminorée où u est la solution des équations de Hamilton-Jacobi visqueuses étudiées, produisantainsi un résultat de non-convergence. / The main goal of this thesis is the general study of the large time behavior of theunique solution of the Cauchy-Dirichlet problem for viscous Hamilton-Jacobi equations of subandsuperquadratic types. This work splits into three parts. After a brief review of basic conceptsof the theory on the viscosity solutions which is the framework of this work, the first part mainlyprovides results on the global in time existence and the uniqueness of the viscosity solution of theabove mentioned Cauchy-Dirichlet problem. The second part studies the large time behavior ofthat solution for superquadratic Hamiltonians. Under rather general assumtions, we prove thatthe behavior of the solution depends on the the sign of the unique ergodic constant c∗ of theergodic problem associated with boundary condition of state constraint-type. When c∗ < 0; weobtain (i) a convergence to the unique solution of the associated stationary problem whereaswhen c∗ ≥ 0; we obtain (ii) a behavior of Hamilton-Jacobi–type (or ergodic-type) happen.In thethird part, devoted to the study for subquadratic Hamiltonians, we prove that a behavior of(i)-type happens when the unique ergodic constant c∗; of the ergodic problem associated withblow-up boundary condition, is non-positve and when c∗ > 0 and 3/2 < m ≤ 2; we obtain abehavior of (ii)-type. But when c∗ = 0 ou c∗ > 0 et 1 < m ≤ 3/2; we prove that for some domains,the function u(x; t)+c∗t is unbounded from below where u is the solution of the studied viscousHamilton-Jacobi, thus providing us with a result of non-convergence.