As control systems become more integrated with high-end engineering systems as well as consumer products, they are expected to achieve specifications that may include logic rules, safety constraints, startup procedures, and so forth. Control design for such complex specifications is a relatively unexplored research area. One possible design approach is based on partitioning the state space into polytopic regions, and then formulating a certain control problem on each polytope, with the intention that the set of all controllers so obtained would collectively achieve the specification. The control problem which must be solved for each polytope is called the reach control problem, and it has been identified as turnkey to the further development of this approach. The reach control problem (RCP) is to find a state feedback to make the closed-loop trajectories of an affine (or linear) control system defined on a polytope reach and exit a prescribed facet of the polytope in finite time. This dissertation studies a number of aspects of the reach control problem, and it uses tools from convex analysis, nonsmooth analysis, and computational geometry for this study. The dissertation has three main themes. First, we formulate and solve a variant of RCP in which trajectories exit the polytope in a monotonic sense; this provides a triangulation-independent solution of RCP. Second, we develop a Lyapunov-like theory for verifying if RCP is solved using a given candidate controller. This involves the introduction of the notion of generalized flow functions, a LaSalle Principle for RCP, and several converse theorems on existence of generalized flow functions. Third, we study the relationship between affine feedbacks and continuous state feedbacks for RCP on simplices. Although the two feedback classes have been shown to be equivalent under an assumption on the triangulation of the state space, we show by a counterexample that the equivalence is no longer true under arbitrary triangulations. Then we provide for single-input systems a constructive method for the synthesis of multi-affine feedbacks for RCP on simplices.

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Submitted by Maria Cristina (library@lncc.br) on 2017-08-14T18:34:35Z No. of bitstreams: 1 Welerson_Dissertação.pdf: 2360212 bytes, checksum: 58a44b5d4fff215888d80a0a417700de (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2017-08-14T18:34:46Z (GMT) No. of bitstreams: 1 Welerson_Dissertação.pdf: 2360212 bytes, checksum: 58a44b5d4fff215888d80a0a417700de (MD5) / Made available in DSpace on 2017-08-14T18:34:57Z (GMT). No. of bitstreams: 1 Welerson_Dissertação.pdf: 2360212 bytes, checksum: 58a44b5d4fff215888d80a0a417700de (MD5) Previous issue date: 2016-11-04 / In this work the optimal control problem with respect to a set of pointwise sources is studied. In particular, the control is given by a finite linear combination of Dirac mass and the state is solution to the associated elliptic boundary value problem. The basic idea consists in minimizing a functional which measures the distance between the state and a target function, with respect to the number, intensities and locations of pointwise loads. The sensitivity of the cost functional with respect to a number of pointwise sources in the set of admissible solutions is derived in its explicit form with help of auxiliaries boundary value problems. The obtained result is then used to devise a non-iterative second order reconstruction algorithm, independent of any initial guess and without introducing regularization techniques. Finally, the devised reconstruction algorithm is applied for numerically solving a set of control and inverse reconstruction problems. / Neste trabalho o problema de controle ótimo com respeito a um conjunto de fontes puntuais é estudado. Em particular, o controle é dado por uma combinação linear finita de massas de Dirac e o estado é solução de um problema de valor de contorno elíptico. Objetiva-se, portanto, minimizar um funcional, que mede a distância entre o estado e uma função alvo, com respeito ao número, intensidades e localizações das cargas puntuais. A sensibilidade do funcional de custo, em relação a um certo número de fontes puntuais no conjunto de soluções admissíveis, é analisada na sua forma explícita com o auxílio de problemas de valor de contorno auxiliares. O resultado obtido é então utilizado para conceber um algoritmo de reconstrução de segunda ordem não iterativo, independente de qualquer chute inicial e sem a introdução de técnicas de regularização. Finalmente, o algoritmo de reconstrução elaborado é aplicado para resolver numericamente um conjunto de problemas de controle e de problemas inversos de reconstrução de fontes.

Cette thèse porte sur l'étude d'équation de Hamilton-Jacobi-Bellman associées à des problèmes de contrôle optimal et de jeux à champ moyen avec la particularité qu'on se place sur un réseau (c'est-à-dire, des ensembles constitués d'arêtes connectées par des jonctions) dans les deux problèmes, pour lesquels on autorise différentes dynamiques et différents coûts dans chaque bord d'un réseau. Dans la première partie de cette thèse, on considère un problème de contrôle optimal sur les réseaux dans l'esprit des travaux d'Achdou, Camilli, Cutrì & Tchou (2013) et Imbert, Moneau & Zidani (2013). La principale nouveauté est qu'on rajoute des coûts d'entrée (ou de sortie) aux sommets du réseau conduisant à une éventuelle discontinuité de la fonction valeur. Celle-ci est caractérisée comme l'unique solution de viscosité d'une équation Hamilton-Jacobi pour laquelle une condition de jonction adéquate est établie. L'unicité est une conséquence d'un principe de comparaison pour lequel nous donnons deux preuves différentes, l'une avec des arguments tirés de la théorie du contrôle optimal, inspirée par Achdou, Oudet & Tchou (2015) et l'autre basée sur les équations aux dérivées partielles, d'après Lions & Souganidis (2017). La deuxième partie concerne les jeux à champ moyen stochastiques sur les réseaux. Dans le cas ergodique, ils sont décrits par un système couplant une équation de Hamilton-Jacobi-Bellman et une équation de Fokker- Planck, dont les inconnues sont la densité m de la mesure invariante qui représente la distribution des joueurs, la fonction valeur v qui provient d'un problème de contrôle optimal "moyen" et la constante ergodique ρ. La fonction valeur v est continue et satisfait dans notre problème des conditions de Kirchhoff aux sommets très générales. La fonction m satisfait deux conditions de transmission aux sommets. En particulier, due à la généralité des conditions de Kirchhoff, m est en général discontinue aux sommets. L'existence et l'unicité d'une solution faible sont prouvées pour des Hamiltoniens sous-quadratiques et des hypothèses très générales sur le couplage. Enfin, dans la dernière partie, nous étudions les jeux à champ moyen stochastiques non stationnaires sur les réseaux. Les conditions de transition pour la fonction de valeur v et la densité m sont similaires à celles données dans la deuxième partie. Là aussi, nous prouvons l'existence et l'unicité d'une solution faible pour des Hamiltoniens sous-linéaires et des couplages et dans le cas d'un couplage non-local régularisant et borné inférieurement. La principale difficulté supplémentaire par rapport au cas stationnaire, qui nous impose des hypothèses plus restrictives, est d'établir la régularité des solutions du système posé sur un réseau. Notre approche consiste à étudier la solution de l'équation de Hamilton-Jacobi dérivée pour gagner de la régularité sur la solution de l'équation initiale. / The dissertation focuses on the study of Hamilton-Jacobi-Bellman equations associated with optimal control problems and mean field games problems in the case when the state space is a network. Different dynamics and running costs are allowed in each edge of the network. In the first part of this thesis, we consider an optimal control on networks in the spirit of the works of Achdou, Camilli, Cutrì & Tchou (2013) and Imbert, Monneau & Zidani (2013). The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. The value function is characterized as the unique viscosity solution of a Hamilton-Jacobi equation for which an adequate junction condition is established. The uniqueness is a consequence of a comparison principle for which we give two different proofs. One uses some arguments from the theory of optimal control and is inspired by Achdou, Oudet & Tchou (2015). The other one is based on partial differential equations techniques and is inspired by a recent work of Lions & Souganidis (2017). The second part is about stochastic mean field games for which the state space is a network. In the ergodic case, they are described by a system coupling a Hamilton- Jacobi-Bellman equation and a Fokker-Planck equation, whose unknowns are the density m of the invariant measure which represents the distribution of the players, the value function v which comes from an "average" optimal control problem and the ergodic constant ρ. The function v is continuous and satisfies general Kirchhoff conditions at the vertices. The density m satisfies dual transmission conditions. In particular, due to the generality of Kirchhoff’s conditions, m is in general discontinuous at the vertices. Existence and uniqueness are proven for subquadratic Hamiltonian and very general assumptions about the coupling term. Finally, in the last part, we study non-stationary stochastic mean field games on networks. The transition conditions for value function v and the density m are similar to the ones given in second part. Here again, we prove the existence and uniqueness of a weak solution for sublinear Hamiltonian and bounded non-local regularizing coupling term. The main additional difficulty compared to the stationary case, which imposes us more restrictive hypotheses, is to establish the regularity of the solutions of the system placed on a network. Our approach is to study the solution of the derived Hamilton-Jacobi equation to gain regularity over the initial equation.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / Optimal Control Problems (OCP), also known as Dynamic Optimization Problems, consist of an Objective Function to be maximized or minimized, associated with a set of differential and algebraic equations which include equality and inequality constraints in the state or control variables and characterize a system of Differential-Algebraic Equations (DAE). The differential-algebraic approach of numerical solution widely used in process simulation due the guarantee of attendance of the implicit algebraic constraints in the original formulation and the elimination of the necessary manipulations to transform the original problem into a purely differential system,was extended to OCP characterizing the called Differential-Algebraic Optimal Control Problem (DAOCP). A category of DAOCP of special interest includes inequality constraints, due the necessity of previous knowledge of the activations and deactivations sequence of these constraints along the trajectory and also of the instants where they occur, named Events. This DAOCPs with inequality constraints is equivalent to a class of hybrid dynamic optimization problems, where continuous and discrete behaviors are associated (FEEHERY, 1998). A particular type of hybrid OCP is that one where continuous state does not present jumps in the Events, called Switched OCP, for which Xu e Antsaklis (2004) considers a solution methodology based on the parameterization of Events with a previous specification of active subsystems sequence, resulting in the solution of a two-point boundary value differential-algebraic problem, formed by the state, co-state and stationarity equations, boundary and continuity conditions and its differentiations, called sensitivity equations. In this work, this indirect approach for Switched OCP was extended for DAOCP with inequality constraints, with the objective to estimate the Events, along the control, state and adjoint variables. The developed approach for Switched OCP described by Xu e Antsaklis (2004) was implemented in a specific code using Maple 9.5, called EVENTS, with the objective to symbolically generate the equations based on the parameterization of Events. This code was incorporated in a interface named OpCol, that collect characterization tools of DAE systems and generation of the optimality conditions extended Pontryagin s Principle for PCOAD of different types. The characterization tools are the INDEX of Murata (1996) that symbolically identifies the index, the resolubility and the consistency of initial conditions and the ACIG of Cunha e Murata (1999) that implements the Gear s algorithm for the index reduction and the index 1 equivalent system generation. The OTIMA (GOMES, 2000; LOBATO, 2004) generates the Euler-Lagrange equations for DAOCP. These tools had been implemented initially in different versions of Maple and all had been update to 9.5 version using the Maplets package that allows the data entry through interactive windows with the user, demanding a little knowledge of the Maple syntax. The OpCol interface was tested for four cases and for each tool a example data bank with typical problems of literature was created to assist the user in its use. Moreover, the direct method implemented in DIRCOL code was extended for multi-phases formulation with estimates of Events and the indirect method with Events Parameterization and differential-algebraic approach implemented in a Matlab code had been used for the numerical solution of three cases: a switched OCP and 2 DAOCP of batch reactors where the control variable is the feed rate of the component B - the first one has parallel reactions and selectivity constraints with 3 phases of index 1, 3 and 1 and the second a safety constraint with 2 phases of index 2 and 1 respectively and had been described by Srinivasan et al. (2003). The methodology used by this authors was applied to attained analytical expressions for the control variable in each phase necessary in indirect method, composing the called Switching Functions, from the optimality conditions based in the Pontryagin s Principle - specifically from the stationarity condition and the active constraint identification that will allow the control variable determination - and of the physical analysis of the problem in order to discard not appropriate activations/deactivations sequences. The results obtained by indirect and direct methods are compared for the 3 cited problems, showing the viability as much of the multiphase formulation using the DIRCOL and also the satisfactory performance of the indirect method with estimates of Events, beyond the utility of the tools of characterization of EADs, of attainment of optimality conditions and parameterization of Events available in Opcol interface. / Os Problemas de Controle Ótimo, também chamados Problemas de Otimização Dinâmica, são formados por uma Função Objetivo a ser maximizada ou minimizada, associada a conjuntos de equações algébricas e diferenciais que incluem restrições de igualdade e de desigualdade nas variáveis de estado e de controle que caracterizam um sistema de Equações Algébrico-Diferenciais (EADs). A extensão do ponto de vista algébricodiferencial de solução numérica aos PCOs, já amplamente utilizado na simulação de processos devido à garantia de atendimento às restrições algébricas originais e implícitas na formulação e à eliminação das manipulações necessárias para transformar o problema original num sistema de equações puramente diferenciais, caracteriza o chamado Problema de Controle Ótimo Algébrico-Diferencial (PCOAD). Uma categoria de PCOAD de especial interesse é a dos que incluem restrições de desigualdade, devido à necessidade de conhecimento prévio da seqüência de ativações e desativações destas restrições ao longo da trajetória e também dos instantes em que elas ocorrem, chamados Eventos. As ativações/desativações das restrições causam flutuações no índice diferencial e no número de graus de liberdade dinâmicos do PCOAD, exigindo técnicas especiais de redução deste índice até um e o emprego de métodos numéricos eficientes que garantam a convergência e estabilidade da solução. Estes PCOADs com restrições de desigualdade são equivalentes a uma classe de problemas de otimização dinâmica híbridos, que associam comportamentos contínuos e discretos (FEEHERY, 1998). Um tipo particular de PCO híbrido é aquele cujo estado contínuo não apresenta saltos nos Eventos, chamado PCO Chaveado, para o qual Xu e Antsaklis (2004) propõem uma metodologia de solução baseada na parametrização dos Eventos com a especificação prévia da seqüência de subsistemas ativos, resultando na solução de um problema de valor no contorno algébrico-diferencial em dois pontos, formado pelas equações de estado, co-estado e de estacionariedade, condições de contorno e de continuidade e suas diferenciações, chamadas equações de sensibilidade. Neste trabalho, esta abordagem indireta empregada para PCO Chaveados foi estendida para PCOAD com restrições de desigualdade, com o objetivo de estimar também os Eventos, além das variáveis de controle, de estado e adjuntas. A abordagem desenvolvida por Xu e Antsaklis (2004) para PCO Chaveados foi implementada num código específico utilizando o Maple 9.5, chamado EVENTS, com o objetivo de gerar simbolicamente as equações baseadas na parametrização dos Eventos. Este código foi incorporado a uma interface chamada OpCol, que reúne ferramentas de caracterização de sistemas de EAD e de geração das condições de otimalidade segundo o Princípio de Pontryagin estendidas para PCOAD de diferentes classes. As ferramentas de caracterização são o INDEX de Murata (1996) que identifica simbolicamente o índice, a resolubilidade e a consistência das condições iniciais e o ACIG de Cunha e Murata (1999) que implementa o algoritmo de Gear para a redução do índice e geração do sistema equivalente de índice 1. O OTIMA (GOMES, 2000; LOBATO, 2004) gera as equações de Euler-Lagrange para PCOAD. Estas ferramentas foram inicialmente implementadas em diferentes versões do Maple e todas foram atualizadas para a versão 9.5 utilizando o pacote Maplets que permite a entrada de dados através de janelas interativas com o usuário, exigindo dele pouco conhecimento da sintaxe Maple. A interface OpCol foi testada para quatro casos e para cada ferramenta foi criado um banco de exemplos com problemas típicos da literatura que auxiliam o usuário na sua utilização. Além disto, o método direto implementado no código DIRCOL estendido para formulações multifásicas com estimativa dos Eventos e o método indireto com Parametrização dos Eventos e abordagem algébrico-diferencial implementado num código MATLAB foram utilizados na solução numérica de três estudos de casos: um PCO chaveado e 2 PCOAD de reatores batelada onde a variável de controle é a taxa de alimentação do componente B: o primeiro tem reações paralelas e restrições de seletividade com 3 fases de índices 1, 3 e 1 e o segundo restrições de segurança com 2 fases de índices 2 e 1 e respectivamente e foram descritos por Srinivasan et al. (2003). A mesma metodologia utilizada por estes autores foi empregada na obtenção de expressões analíticas para a variável de controle em cada fase necessárias no método indireto, compondo as chamadas Funções Identificadoras de Fase (FIF), a partir das condições de otimalidade baseadas no Princípio de Pontryagin - especificamente a partir da condição de estacionariedade e da identificação da restrição ativa que permitirá a determinação da variável de controle - e da análise física do problema de modo a descartar seqüências de ativação/desativação não apropriadas. Os resultados obtidos pelo método indireto e pelo método direto são comparados entre si para os 3 problemas citados, mostrando a viabilidade tanto da formulação multifásica empregando o DIRCOL quanto o desempenho satisfatório do método indireto com estimativa de Eventos, além da utilidade das ferramentas de caracterização de EADs, de obtenção das condições de otimalidade e de parametrização dos eventos disponibilizadas na interface Opcol. / Mestre em Engenharia Química

Mathematical theory of partial differential equations (PDEs) is a pretty old classical area with wide range of applications to almost every branch of science and engineering. With the advanced development of functional analysis and operator theory in the last century, it became a topic of analysis. The theory of homogenization of partial differential equations is a relatively new area of research which helps to understand the multi-scale phenomena which has tremendous applications in a variety of physical and engineering models, like in composite materials, porous media, thin structures, rapidly oscillating boundaries and so on. Hence, it has emerged as one of the most interesting and useful subject to study for the last few decades both as a theoretical and applied topic. In this thesis, we study asymptotic analysis (homogenization) of second-order partial differential equations posed on an oscillating domain. We consider a two dimensional oscillating domain (comb shape type) consisting of a fixed bottom region and an oscillatory (rugose) upper region. We introduce optimal control problems for the Laplace equation. There are mainly two types of optimal control problems; namely distributed control and boundary control. For distributed control problems in the oscillating domain, one can apply control on the oscillating part or on the fixed part and similarly for boundary control problem (control on the oscillating boundary or on the fixed part the boundary). We consider all the four cases, namely distributed and boundary controls both on the oscillating part and away from the oscillating part. The present thesis consists of 8 chapters. In Chapter 1, a brief introduction to homogenization and optimal control is given with relevant references. In Chapter 2, we introduce the oscillatory domain and define the basic unfolding operators which will be used throughout the thesis. Summary of the thesis is given in Chapter 3 and future plan in Chapter 8. Our main contribution is contained in Chapters 4-7. In chapters 4 and 5, we study the asymptotic analysis of optimal control problems namely distributed and boundary controls, respectively, where the controls act away from the oscillating part of the domain. We consider both L2 cost functional as well as Dirichlet (gradient type) cost functional. We derive homogenized problem and introduce the limit optimal control problems with appropriate cost functional. Finally, we show convergence of the optimal solution, optimal state and associate adjoint solution. Also convergence of cost-functional. In Chapter 6, we consider the periodic controls on the oscillatory part together with Neumann condition on the oscillating boundary. One of the main contributions is the characterization of the optimal control using unfolding operator. This characterization is new and also will be used to study the limiting analysis of the optimality system. Chapter 7 deals with the boundary optimal control problem, where the control is applied through Neumann boundary condition on the oscillating boundary with a suitable scaling parameter. To characterize the optimal control, we introduce boundary unfolding operators which we consider as a novel approach. This characterization is used in the limiting analysis. In the limit, we obtain two limit problems according to the scaling parameters. In one of the limit optimal control problem, we observe that it contains three controls namely; a distributed control, a boundary control and an interface control.

Mathematical theory of homogenization of partial differential equations is relatively a new area of research (30-40 years or so) though the physical and engineering applications were well known. It has tremendous applications in various branches of engineering and science like : material science ,porous media, study of vibrations of thin structures, composite materials to name a few. There are at present various methods to study homogenization problems (basically asymptotic analysis) and there is a vast amount of literature in various directions. Homogenization arise in problems with oscillatory coefficients, domain with large number of perforations, domain with rough boundary and so on. The latter one has applications in fluid flow which is categorized as oscillating boundaries. In fact ,in this thesis, we consider domains with oscillating boundaries. We plan to study to homogenization of certain optimal control problems with oscillating boundaries. This thesis contains 6 chapters including an introductory Chapter 1 and future proposal Chapter 6. Our main contribution contained in chapters 2-5. The oscillatory domain under consideration is a 3-dimensional cuboid (for simplicity) with a large number of pillars of length O(1) attached on one side, but with a small cross sectional area of order ε2 .As ε0, this gives a geometrical domain with oscillating boundary. We also consider 2-dimensional oscillatory domain which is a cross section of the above 3-dimensional domain. In chapters 2 and 3, we consider the optimal control problem described by the Δ operator with two types of cost functionals, namely L2-cost functional and Dirichlet cost functional. We consider both distributed and boundary controls. The limit analysis was carried by considering the associated optimality system in which the adjoint states are introduced. But the main contribution in all the different cases(L2 and Dirichlet cost functionals, distributed and boundary controls) is the derivation of error estimates what is known as correctors in homogenization literature. Though there is a basic test function, one need to introduce different test functions to obtain correctors. Introducing correctors in homogenization is an important aspect of study which is indeed useful in the analysis, but important in numerical study as well. The setup is the same in Chapter 4 as well. But here we consider Stokes’ Problem and study asymptotic analysis as well as corrector results. We obtain corrector results for velocity and pressure terms and also for its adjoint velocity and adjoint pressure. In Chapter 5, we consider a time dependent Kirchhoff-Love equation with the same domain with oscillating boundaries with a distributed control. The state equation is a fourth order hyperbolic type equation with associated L2-cost functional. We do not have corrector results in this chapter, but the limit cost functional is different and new. In the earlier chapters the limit cost functional were of the same type.

Les systèmes implicites (dits aussi « descripteurs ») peuvent décrire des processus régis à la fois par des équations dynamiques et statiques et permettent de préserver la structure des systèmes physiques. Ils comportent trois types de modes : dynamiques finis, infinis (réponse temporelle impulsive (en cas continu) ou acausale (en cas discret)) et statiques. Dans le cadre du formalisme descripteur, les contributions de cette thèse sont triples : i) revisiter des résultats existants pour les systèmes d’état, ii) étendre certains résultats classiques au cas des systèmes implicites, iii) résoudre rigoureusement des problèmes de commande non standard. Ainsi, le présent mémoire commence par revisiter les résultats concernant la caractérisation LMI stricte de la dissipativité, les caractérisations de l’admissibilité et des performances H2 ou H∞ par LMI étendues et les équations de Sylvester et de Riccati généralisées. Il aborde dans un deuxième temps, le problème de stabilisation simultanée, avec ou sans critère H∞, à travers l’extension de certains résultats récents au cas des systèmes implicites. La solution proposée s’appuie sur la résolution combinée d’une équation algébrique de Riccati généralisée (GARE) et d’un problème de faisabilité sous contrainte LMI stricte. Il traite enfin des problèmes H2 et H∞ non standards : i) en présence de pondérations instables voire impropres, ii) sous contraintes de régulation; dans le cas des systèmes implicites. Ces dernières contributions permettent désormais de traiter rigoureusement, sans approximations ou transformations, de nombreux problèmes H2 ou H∞ formalisant des problèmes pratiques de commande, dont ceux faisant intervenir une pénalisation haute fréquence de la commande ou un modèle interne instable des signaux exogènes. / The descriptor systems have been attracting the attention of many researchers over recent decades due to their capacity to preserve the structure of physical systems and to describe static constraints and impulsive behaviors. Within the descriptor framework, the contributions of this dissertation are threefold: i) review of existing results for state-space systems, ii) generalization of classical results to descriptor systems, iii) exact and analytical solutions to non standard control problems. A realization independent Kalman-Yakubovich-Popov (KYP) lemma and dilated LMI characterizations are deduced for descriptor systems. The solvability and corresponding numerical algorithms of generalized Sylvester equations and generalized algebraic Riccati equations (GARE) associated with descriptor systems are provided. In addition, the simultaneous H∞ control problem is considered through extending recently reported results. A sufficient condition is proposed through a combination of a generalized algebraic Riccati equation and a set of LMIs. Moreover, the nonstandard H2 and H∞ control problems with unstable and/or nonproper weighting functions or subject to regulation constraints are addressed. These contributions allow, without approximation or transformation, dealing with many practical problems defined within H2 or H∞ control methodologies, where the control signals are penalized at high frequency or unstable internal models specified by external signals is involved.

This thesis focuses on matrix generalizations of Hurwitz polynomials. A real polynomial with all its roots in the open left half plane of the complex plane is called a Hurwitz polynomial. The study of these Hurwitz polynomials has a long and abundant history, which is associated with the names of Hermite, Routh, Hurwitz, Liénard, Chipart, Wall, Gantmacher et al. The direct matricial generalization of Hurwitz polynomials is naturally defined as follows: A p by p matrix polynomial F is called a Hurwitz matrix polynomial if the determinant of F is a Hurwitz polynomial. Recently, Choque Rivero followed another line of matricial extensions of the classical Hurwitz polynomial, called matrix Hurwitz type polynomials. However, the notion “matrix Hurwitz type polynomial” is still irrelative to “Hurwitz matrix polynomial” due to the totally unclear zero location of the former notion. So the main goal of this thesis is to discover the relation between the two notions “matrix Hurwitz-type polynomials” and “Hurwitz matrix polynomials' and provide some criteria to identify Hurwitz matrix polynomials. The central idea is to determine the inertia triple of matrix polynomials in terms of some related matrix sequences. Suppose that F is a p by p matrix-valued polynomial of degree n. We split F into the odd part and the even part, which allow us to introduce an essential rational matrix function of right type G. From the matrix coefficients of the Laurent series of G we construct the (n-1)-th extended sequence of right Markov parameters (SRMP) of F. Then we show that the inertia triple of F can be characterized by a combination of the inertia triples of two block Hankel matrices generated by the (n-1)-th SRMP of F and the number of zeros (counting for multiplicities) of greatest right common divisors of the even part and the odd part of F lying on the left half of the real axis. By an analogous approach we also obtain the dual results for the inertia triple of F in terms of the SLMP of F. Then we demonstrate that F is a Hurwitz matrix polynomial of degree n if and only if the (n − 1)-th SRMP (resp. SLMP) of F is a Stieltjes positive definite sequence. On this account, the two notions “Hurwitz matrix polynomials” and “matrix Hurwitz type polynomials” are equivalent. In addition, we investigate quasi-stable matrix polynomials appearing in the theory of stability, which contain Hurwitz matrix polynomials as a special case. We seek a correspondence between quasi-stable matrix polynomials, Stieltjes moment problems and multiple Nevanlinna-Pick interpolation in the Stieltjes class. Accordingly, we prove that F is a quasi-stable matrix polynomial if and only if the (n − 1)-th SRMP (resp. SLMP) of F is a Stieltjes non-negative definite extendable sequence and the zeros of right (resp. left) greatest common divisors of the even part and the odd part of F are located on the left half of the real axis.:1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Matrix polynomials and greatest common divisors. . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Greatest common divisors of matrix polynomials . . . . . . . . . . . . . . . . . . . . . 8 3 Matrix sequences and their connection to truncated matricial moment problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Matrix fraction description and some related topics . . . . . . . . . . . . . . . . . . 19 4.1 Realization of Matrix fraction description from Markov parameters . . . . . . . 19 4.2 The interrelation between Hermitian transfer function matrices and monic orthogonal system of matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . .27 5 The Bezoutian of matrix polynomials and the inertia problem of matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 5.2 The Anderson-Jury Bezoutian matrices in connection to special transfer function matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 6 Para-Hermitian strictly proper transfer function matrices and their related monic Hurwitz matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7 Solution of matricial Routh-Hurwitz problems in terms of the Markov pa- rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8 Matrix Hurwitz type polynomials and some related topics . . . . . . . . . . . . . . 67 9 Hurwitz matrix polynomials and some related topics . . . . . . . . . . . . . . . . . . 77 9.1 Hurwitz matrix polynomials, Stieltjes positive definite sequences and matrix Hurwitz type polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9.2 S -system of Hurwitz matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 82 10 Quasi-stable matrix polynomials and some related topics . . . . . . . . . . . . 95 10.1 Particular monic quasi-stable matrix polynomials and Stieltjes moment problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95 10.2 Particular monic quasi-stable matrix polynomials and multiple Nevanlinna- Pick interpolation in the Stieltjes class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 10.3 General description of monic quasi-stable matrix polynomials . . . . . . . . .104 List of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 List of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Selbständigkeitserklärung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

This cumulative dissertation covers four papers on the management and governance of family firms. The first paper provides a systematic review of the literature on family-related determinants of the board of directors’ tasks, composition and processes in family firms. The review clusters and synthesizes the literature into six major determinants, details the methods used, and provides recommendations for future research in the field. The second paper develops a contingency approach to board task needs of family firms. The paper identifies five contingency factors and demonstrates how board task needs typically evolve over the ownership stages of family businesses. The third paper constitutes a qualitative empirical study on the role of board control in controlling owner family businesses. Based on a multiple case study approach, the study shows that controlling owners frequently use board control as a self-governing mechanism to mitigate self-control problems. Additionally, the study provides insights on favorable board processes and board composition in the controlling owner setting. Overall, the dissertation underlines the importance of factoring in the influence of family firm heterogeneity on the board of directors. The fourth paper concludes the dissertation with a teaching case study on a small family firm that is exposed to the threat of a disruptive innovation in its industry.:1 Introduction 1.1 Research Objective 1.2 Summary of the Research Papers 1.3 References 2 Determinants of Boards in Family Firms: A Systematic Literature Review 2.1 Abstract 2.2 Introduction 2.3 Boards of Directors in Family Firms 2.4 Research Method 2.5 Findings on Family-Related Determinants 2.6 Synthesis of Results 2.7 Future Research 2.8 Conclusion 2.9 Appendix 2.10 References 3 The Board of Directors in Family Firms: One Size Fits Forever? 3.1 Abstract 3.2 Introduction 3.3 Board Tasks in Family Firms 3.4 Development of a Conceptual Model 3.5 Limitations of the Model 3.6 Conclusion 3.7 References 4 Self-Control Through Board Control: Formalized Governance in Controlling Owner Family Businesses 4.1 Abstract 4.2 Introduction 4.3 Theoretical Foundations 4.4 Research Method 4.5 Findings 4.6 Discussion 4.7 Conclusion 4.8 References 5 Teaching Case Study ATB: Digital Disruption in the Manufacturing Industry 5.1 Abstract 5.2 Case Manuscript 5.3 Teaching Note 5.4 References 6 Conclusion 6.1 Contribution of the Dissertation 6.2 Limitations and Avenues for Future Research 6.3 References