No / Given only the traffic correlations of counts and intervals, a Batch Renewal Arrival Process (BRAP) is completely determined, as the least biased choice and thus, it provides the analytic means to construct suitable traffic models for the study of queueing systems independently of any other traffic characteristics. In this context, the BRAP and the Batch Markovian Arrival Process (BMAP) are employed in the continuous time domain towards the analysis of the stable BRAP/GE/1 and BMAP/GE/1 queues with infinite capacity, single servers and generalized exponential (GE) service times. Novel closed form expressions for the steady state probabilities of these queues are obtained, based on the embedded Markov chains (EMCs) technique and the matrix-geometric (M-G) method, respectively. Moreover, the stable GEsGGeo/GE/1 queue with GE-type service times and a GEsGGeo BRAP consisting of bursty GE-type batch interarrival times and a shifted generalized geometric (sGGeo) batch size distribution is adopted to assess analytically the combined adverse effects of varying degrees of correlation of intervals between individual arrivals and the burstiness of service times upon the typical quality of service (QoS) measure of the mean queue length (MQL). Moreover, a comprehensive experimental study is carried out to investigate numerically the relative impact of count and interval traffic correlations as well as other traffic characteristics upon the performance of stable BRAP/GE/1 and BMAP/GE/1 queues. It is suggested via a conjecture that the BRAP/GE/1 queue is likely to yield pessimistic performance metrics in comparison to those of the stable BMAP/GE/1 queues under the worst case scenario (i.e., a worst case scenario) of the same positive count and interval traffic correlations arising from long sojourn in each phase.

對一些退休基金或是壽險基金來說，因為它們的金額都相當龐大，進而影響的層面也相當廣泛；它們影響著金融市場的發展、有價證券的價格和市場的報酬。 所以，對現今市場投資在這樣長期資產的投資策略而言，以下我們要討論的議題將是非常重要。 以前的退休基金管理計畫是建立在單期的假設當中。根據目前所持有所有資產的部位、目前市場的狀況與對未來報酬的期望，基金管理人將尋求對下期的最適投資決策。傳統的方法都是用期望值-變異數方式(Mean-Variance approach)去極大化投資的報酬，以求取最適部位。但是單期的期望值-變異數方式(Mean-Variance approach)面對了二個問題： 一、 集合各個單期最適決策用多期的眼光來看不一定是最適。 二、 單期最適決策並不能同時處理投資面與集資面的問題。例如：退休基金同時間有每月的收入與每月的投資面。 不像單期的投資方式，使用多期的投資方法比較能符合這樣的投資問題與要求，也比較具有合理性。 投資在長期資產的部位與報酬率，最容易受到利率變動的影響。換句話說，利率變化是影響投資在長期性資產的最大變因。因此，我們將討論的問題：在利率是隨機變動時，以Vasicek (1977)的利率模型為主，加入投資人風險偏好的共同基金的分配原則。這樣的理論下，我們將利用風險中立的方法求出最適的投資組合，以滿足在長期投資觀點下避險與套利的需求。其中投資人的風險偏好是以Merton (1973)提出的常數相關風險偏好(Constant Relative Risk Aversion；CRRA)的效用函數去討論；在文章最後，我們將針對投資人的風險偏好做一些討論，包括：改變CRRA的參數、自然對數的效用函數(Logarithmic utility function)。 以往的研究都採用動態程式設計(Dynamic programming approach)的方法來解決這樣多期投資的問題，但是這樣的方法運用的計算相當複雜，也不一定求的出最適部位解；而利用Cox and Huang (1989)提出的風險中立方法(Martingale approach)將完全的解決以上遇到的問題。 / In this study, we investigate the dynamic mutual fund separation theorem applied to portfolio management for constant relative risk averse investors where, in particular, the interest rate risks are incorporated. Within this economy, the real interest rates and stock prices are assumed to follow the Markovian processes whose drifts and diffusion parameters are driven by certain state variables. Our approach involves the use of the martingale methodology developed by Cox and Huang (1989) as proposed in the work of Deelstra et al. (2003). Following their framework, we consider the economy of the investors that consists of cash, bond fund and stock indices. Adding to the previous works, we investigate the obtained optimal strategies through numerical examples in order to be compared to the allocations of popular advice and clarify the hedge and arbitrage demands in financial decision from long-term perspective. Finally, certain mutual funds are constructed to validate the validity of the popular advice.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

En Europe, la présence de grands carnivores dans des paysages anthropisés entraîne une forte compétition avec l'homme et alimente d'importantes polémiques concernant leur protection légale. La perception antagoniste de ces espèces à la fois emblématiques pour certains et sources de conflits pour d'autres, rend la gestion de leurs populations très délicate. Depuis la recolonisation spontanée du loup (Canis lupus) dans les Alpes françaises au début des années 1990, la population s'est accrue numériquement et spatialement. Parallèlement, les dégâts occasionnés par le loup sur la filière élevage ont suivi la même tendance. L'Etat met en place aujourd'hui un contrôle raisonné de la population, sous réserve que les objectifs de conservation, exigés par la Directive Habitat, soient respectés. En s'inspirant du cas d'étude du loup en France, nous proposons dans cette thèse un cadre de prise de décision structurée adapté pour la gestion et la conservation d'espèces rares et difficilement observables, protégées par des accords législatifs mais qui, dans un contexte social conflictuel, peuvent être régulées. La modélisation séquentielle du processus décisionnel s'est déroulée dans un contexte de forte incertitude selon plusieurs étapes : 1) appréhender les objectifs de conservation et/ou contrôle du loup en France pour les formaliser sous forme mathématique via une fonction d'utilité, 2) suivre la population par une méthode non-invasive pour définir des indicateurs de gestion fiables et évaluer le statut de conservation de la population, 3) coupler les mesures létales adoptées actuellement à un modèle démographique décrivant la dynamique du loup et intégrant sa structure sociale, 4) et déterminer la décision. Cette dernière étape est réalisée à l'aide d'une méthode d'optimisation qui calcule la stratégie optimale de gestion en fonction de la structure sociale de la population et des différentes sources d'incertitude accumulées à chaque étape du processus décisionnel. Nous avons choisi comme indicateur de gestion le taux de croissance, à partir duquel nous avons défini l'utilité. Cet indicateur était robuste à l'incertitude d'échantillonnage émergeant de la détection partielle et hétérogène des individus. Des analyses de sensibilité de la décision ont montré une forte influence de la fonction d'utilité sur la stratégie optimale, soulignant ainsi l'importance de définir correctement les objectifs. Nous avons également montré que la stratégie optimale était sensible aux variations des paramètres démographiques, montrant ainsi l'intérêt des méthodes de capture-marquage-recapture pour les estimer correctement. Nous discutons enfin de l'extension de notre approche à un cadre décisionnel de gestion adaptative pour traiter des problèmes de conservation dans un contexte conflictuel. / Large carnivore management in Europe is controversial because of conflictive objectives arising from the legal protection of threatened species vs. the possible necessity of culling individuals to prevent severe damages on human activities. Since the wolf recovery in the French Alps in the early 90's, the population has been numerically and spatially increasing. In parallel, livestock depredations have been following the same trend. As an EU member state, France is bound to the European Habitat Directive, which provides full protection of wolf populations and their habitat. Nevertheless, derogatory killings are allowed for individuals causing problems on livestock and some lethal control is now incorporated into the national management plan, as long as the population growth and its distribution range are not being threatened. Illustrating with the case study of the wolf in France, my dissertation proposes a structured decision making framework for the management and the conservation of elusive species that are legally protected but, in a conflictive context, are subject to population control. The sequential modeling of our decision process occurred in the following steps: 1) define the multiple objectives and formulate them in terms of a utility function, 2) monitor the population through a non-invasive approach in order to define the population conservation status, 3) build a demographic model to predict the consequences of harvesting on population dynamics and social structure, 4) obtain optimal state-dependent decisions. The last step is done with stochastic dynamic programming (SDP), acknowledged to be one of the most useful optimization methods in decision making. We provide an optimal solution for wolf management that gives the highest chance of meeting objectives, defined on population growth rate. This demographic indicator was found to be robust to sampling uncertainty arising from partial and heterogeneous detection of individuals. We ran decision sensibility analyses and found a strong effect of the utility function on the optimal strategy, highlighting the importance of defining explicit objectives. We also found that the optimal strategy was sensitive to demographic parameters, which demonstrate the general need of using solid statistical approaches to estimate them properly. This structured decision making framework can further be extended to adaptive management, acknowledged as being a convenient framework for wildlife management.

Initialement, la théorie du risque supposait l’indépendance entre les différentes variables aléatoires et autres paramètres intervenant dans la modélisation actuarielle. De nos jours, cette hypothèse d’indépendance est souvent relâchée afin de tenir compte de possibles interactions entre les différents éléments des modèles. Dans cette thèse, nous proposons d’introduire des modèles de dépendance pour différents aspects de la théorie du risque. Dans un premier temps, nous suggérons l’emploi des copules comme structure de dépendance. Nous abordons tout d’abord un problème d’allocation de capital basée sur la Tail-Value-at-Risk pour lequel nous supposons un lien introduit par une copule entre les différents risques. Nous obtenons des formules explicites pour le capital à allouer à l’ensemble du portefeuille ainsi que la contribution de chacun des risques lorsque nous utilisons la copule Farlie-Gumbel-Morgenstern. Pour les autres copules, nous fournissons une méthode d’approximation. Au deuxième chapitre, nous considérons le processus aléatoire de la somme des valeurs présentes des sinistres pour lequel les variables aléatoires du montant d’un sinistre et de temps écoulé depuis le sinistre précédent sont liées par une copule Farlie-Gumbel-Morgenstern. Nous montrons comment obtenir des formes explicites pour les deux premiers moments puis le moment d’ordre m de ce processus. Le troisième chapitre suppose un autre type de dépendance causée par un environnement extérieur. Dans le contexte de l’étude de la probabilité de ruine d’une compagnie de réassurance, nous utilisons un environnement markovien pour modéliser les cycles de souscription. Nous supposons en premier lieu des temps de changement de phases de cycle déterministes puis nous les considérons ensuite influencés en retour par les montants des sinistres. Nous obtenons, à l’aide de la méthode d’erlangisation, une approximation de la probabilité de ruine en temps fini. / Initially, it was supposed in risk theory that the random variables and other parameters of actuarial models were independent. Nowadays, this hypothesis is often relaxed to take into account possible interactions. In this thesis, we propose to introduce some dependence models for different aspects of risk theory. In a first part, we use copulas as dependence structure. We first tackle a problem of capital allocation based on the Tail-Value-at-Risk where the risks are supposed to be dependent according to a copula. We obtain explicit formulas for the capital to be allocated to the overall portfolio but also for the contribution of each risk when we use a Farlie-Gumbel-Morenstern copula. For the other copulas, we give an approximation method. In the second chapter, we consider the stochastic process of the discounted aggregate claims where the random variables for the claim amount and the time since the last claim are linked by a Farlie-Gumbel-Morgenstern copula. We show how to obtain exact expressions for the first two moments and for the moment of order m of the process. The third chapter assumes another type of dependence that is caused by an external environment. In the context of the study of the ruin probability for a reinsurance company, we use a Markovian environment to model the underwriting cycles. We suppose first deterministic cycle phase changes and then that these changes can also be influenced by the claim amounts. We use the erlangization method to obtain an approximation for the finite time ruin probability.

Esta dissertação desenvolve filtro de informação, algoritmos array para estimador do erro médio mínimo quadrático para sistemas lineares sujeitos a saltos Markovianos e algoritmos array rápidos para filtragem de sistemas singulares convencionais. Exemplos numéricos serão apresentados para mostrarem as vantagens dos algoritmos array deduzidos. Parte dos resultados obtidos nesta pesquisa serão publicados no seguinte artigo: Terra et al. (2007). Terra, M. H., Ishihara, J. Y. and Jesus, G. Q. (2007). Information filtering and array algorithms for discrete-time Markovian jump linear systems. Proceedings of the American Control Conference ACC07. / This dissertation develops information filter and array algorithms for linear minimum mean square error estimator (LMMSE) of discrete-time Markovian jump linear systems (MJLSs) and fast array algorithms for filtering of standard singular systems. Numerical examples to show the advantage of the array algorithms are presented. Some results obtained in this research are published in the following paper: Terra et al. (2007). Terra, M. H., Ishihara, J. Y. and Jesus, G. Q. (2007). Information filtering and array algorithms for discrete-time Markovian jump linear systems. Proceedings of the American Control Conference ACC07.

Nesta tese de doutorado são apresentados sistemas globais de estimativa baseados em modelos Markovianos aplicados na área de reabilitação robótica. Os sistemas propostos foram desenvolvidos para estimar as posições angulares dos elos de exoesqueletos para membros inferiores, desenvolvidos para reabilitação motora em pacientes que sofreram Acidente Vascular Cerebral (AVC) ou lesão medular. Filtros baseados no filtro de Kalman, um nominal e outro considerando incertezas no modelo, foram utilizados em estratégias de fusão de dados de sensores provenientes de sensores inerciais, possibilitando estimativas de posicionamentos angulares. Algoritmos genéticos são utilizados na otimização dos filtros, ajustando as matrizes de peso destes. Em oposição as modelagens tradicionais, via estimativa local, utilizando somente uma unidade inercial para cada modelo, propõe-se um sistema global de estimativa, obtendo-se a melhor informação de cada sensor combinando-os em um modelo Markoviano. Resultados experimentais com um exoesqueleto foram utilizados para comparar a abordagem Markoviana às convencionais. / In this thesis are presented global estimation systems based on Markov models applied in robotic rehabilitation area. The proposed systems have been developed to estimate the angular positions of the exoskeletons for lower limbs, designed to provide motor rehabilitation of stroke and spinal cord injured people. Filters based on the Kalman filter, one nominal and other considering uncertainties in the model, were used in sensor data fusion strategies from inertial sensors, to estimate angular positions. Genetic algorithms are used to the optimization of filters, tuning the weighting matrices. In opposition to these modelling via local estimation, using only one inertial unit, we also chose a global modelling getting the best information from each sensor, combining them in a Markov model. Experimental results with an exoskeleton were used to compare the Markovian approach to conventional.

Ziel dieser Arbeit ist es, den Einfluss der gekoppelten Dynamik in nichtlinearen Polymerketten auf Prozesse wie den thermischen Zerfall und den kraftinduzierten Abriss zu untersuchen. Dafür betrachten wir zunächst die Gleichgewichtsrelaxationseigenschaften von Ketten mit nichtlinearen Wechselwirkungspotentialen. Es kann gezeigt werden, dass die Relaxationseigenschaften des End-zu-End-Abstandes wie auch der Hauptkomponenten jene der harmonischen Kette sind. Die Korrelationszeiten weichen jedoch ab. Sogenannte weiche nichtlineare Potentiale erhöhen die Korrelationszeit. Doppelmuldenpotentiale können zum Anwachsen typischer Korrelationszeiten um mehrere Größenordnungen führen. Die Hauptkomponenten, deren Richtungen mit denen der Normalmoden der harmonischen Kette übereinstimmen, zeigen dabei unter Umständen eine subdiffusive Dynamik. In Bezug auf das thermisch aktivierte Bindungsversagen gehen wir auf zwei experimentell wichtige Problemstellungen ein. Zuerst untersuchen wir den thermisch aktivierten Zerfallsprozess einer homopolymeren Kette. Wir berechnen mittlere Aktivierungszeiten einzelner Bindungen für freie und befestigte Ketten. Wir können zeigen, dass diese Zeiten entscheidend von der Gesamtlänge der Kette sowie der Position der Bindung abhängen. An den freien Kettenenden weisen sie ein Minimum auf. Die zweite experimentell relevante Situation entspricht jener in Einzelmolekül-Zugexperimenten. Zusätzlich zum Einfluss nicht Markovscher Fluktuationen im gekoppelten System hat die verzögerte Kraftpropagation entlang der Kette einen weiteren Einfluss auf die Abrissdynamik. In langen Ketten brechbarer Bindungen verursacht das komplexe Zusammenspiel zwischen Kraftpropagation und der dem Abriss zugrundeliegenden Extremwertstatistik eine nicht monotone Abhängigkeit der wahrscheinlichsten Abrisskraft von der Kettenlänge. Abschließend analysieren wir experimentelle Abrissdaten von kovalenten Bindungen in zweisträngigen DNS Ringen. / This work intends to show how the coupled dynamics of a nonlinear polymer chain alters processes such as thermal fragmentation and force-induced rupture. For that purpose we first examine the equilibrium relaxation properties of chains with nonlinear interaction potentials. We find that the relaxation properties of the end-to-end distance and the principal components are essentially those of the harmonic chain, though with shifted correlation times. Soft nonlinear potentials increase the correlation times. Double-well interaction potentials may lead to an increase of the relaxation times by orders of magnitude. The principal components, whose directions follow the normal modes of the harmonic chain, can exhibit vastly different subdiffusive kinetics. Concerning the thermally activated bond rupture in polymer chains we focus on two experimentally relevant situations. First, we consider the thermally activated fragmentation of a homopolymer chain. We calculate mean activation times of individual bonds for free and grafted chains. We show that these times crucially depend on the length of the chain and the location of the bond yielding a minimum at the free chain ends. Second, we study a set up corresponding to the one found in single molecule pulling experiments. In addition to the influence of non-Markovian fluctuations in the coupled system the delayed force propagation into the chain has a further impact on its overall rupture dynamics. In long chains of breakable bonds the complex interplay between the force propagation into the chain and the extreme value statistics underlying rupture causes a non-monotonic scaling of the most probable rupture force as a function of the chain length. Finally we analyze experimental data of the rupture of covalent bonds in ds-DNA loops.

A presente pesquisa se refere à aplicação dos métodos estocásticos para estudar fenômenos críticos em modelos de sistemas classificados como desordenados que apresentam transição de fase do tipo-ordem desordem. Essa pesquisa é definida tanto no quadro teórico da Mecânica Estatística dos fenômenos críticos e transição de fase de equilíbrio e fora de equilibrio, com os recursos associados à análise de escala de tamanho finito quanto no quadro dos recursos aos processos estocásticos markovianos, descritos pela equação-mestra e associados a técnicas essencialmente numéricas como o método estocástico computacional de Monte Carlo. Na primeira etapa desta pesquisa, os modelos estudados são da classe denominada de votante majoritário. Eles são indexados pelo número z de vizinhos mais próximos com spin central, tem dois estados e são construidos em redes quadradas. A evolução dinâmica é dada pela regra da maioria junto com regradfe desempate. Eles não satisfazem a propriedade do principio do balanceamento detalhado, portanto, são classificados como descrevendo fenômenos fora do equilíbrio. Contudo, eles satisfazem a propriedade de simetria de inversão de sinal, o que os coloca teoricamente na classe de universalidade do modelo de Ising. Desta forma, a evolução dinâmica desses modelos é estudada com os recursos da equação mestra ou equação de evolução. No entanto, essa abordagem teórica é feita apenas na aproximação de campo médio, a qual fornece, na solução estacionária, os valores clássicos para os parametros relevantes. Em contrapartida, os valores numéricos exatos para os valores do ponto crítico e dos expoentes críticos, que são não clássicos, é dada por meio do recurso ao método de simulação computacional e à análise de escala de tamanho finito. Esses valores confirmam o resultado teórico quanto à classe de universalidade para cada modelo específico. Na sequência, estudam-se as propriedades dos modelos resultantes da combinação convexa do votante majoritário. Os resultados são semelhantes aos anteriores. Um resultado extra permitido por essas combinações convexa éa construção de uma relação contínua entre o valor crítico indutor da transição de fase e o número de vizinhos. Neste contexto foi apresentada uma solução para o problema do modelo mais simples desta classe de modelos. Com o modelo mais simples ilustram-se as condições universais de transição de fase, em particular o papel da dimensão do sistema. Na segunda etapa da pesquisa, cvonstrí-se, então, outra classe de modelos do votante que, por analogia com o modelo de Ising, tem como estado fundamental a fasse antiferromagnética: a classe dos modelos do votante minoritário. Essa classe de modelos possue as mesmas propriedades da classe de modelos do votante majoritário e por isso obtem-se os mesmos resultados. A analogia com o modelo de Ising é levada um pouco mais longe com a construção de um análogo aos modelo +-J: a construção da combinação convexa do votante majoritário com o minoritário. Para esse novo modelo constrói-se tanto o diagrama com as três fases, ferromagnética, paramagnética e antiferromagnética quanto as concentrações críticas que as distinguem. Não se obtém uma possível fase de vidro de spin. Uma vez que os modelos do votante são originalmente tidos como sistemas desordenados, comparam-se, para um mesmo modelo, resultados obtidos pela aplicação de dois diferentes métodos de tratar os modelos de sistemas desordenados: o método temperado-\"quenched\" - e o método recozido - \"annealed\". Na terceira etapa desta pesquisa e na mesma linha dos modelos estocásticos irreversíveis tratados anteriormente, estuda-se ainda outro modelo, classificado como jogo espacial nos sítios de uma rede quadrada. Simulações mostram que além de dois estados absorventes ha\'também a presença de um estado ativo definido por uma densidade finita de cooperadores e não cooperadores e que esse modelo se encontra na classe de universalidade do modelo de percolação direcionada. Nesta mesma etapa, mas, agora, no contexto da Mecânica sStatística de Equilíbrio, aborda-se o modelo de Ising quântico unidimensional com campo transverso por meio de simulação de Monte Carlo.Com o uso do método estiocástico e por meio da curva do colapso calculam-se os valores do ponto crítico e dos expoentes críticos desse modelo. / This research refers to the applications of the stochastic methods to the study of the critical phenomena in models of systems classified as disordered that undergo phase transition of the order-disorder kind. This research is defined as in the theoretical framework of the Statistical mechanics of the equilibrium and non-equilibrium of the critical phenomena and phase transition with the resources associated to the analysis of finite-size scale, as in the frame of the resources of markovian stochastic process described by the master equation associated with essentially numerical techniques such as stochastic computational method of Monte Carlo.l In this first stage of this research, the studied models belong to the class of the majority voter. They are described by a lattice with spins in each site with two states. The dynamic of these models is described by the majority rule together with a rule for solving problems of indecision. These models do not obey the principle of microscopic reversibility therefore they are classified as describing phenomena of non-equilibrium. However, they satisfy the property of \"up-down\" symmetry which make theoretically belong to the universality class of the Ising model. The mean field approach to the master equation is done and the exact value is pursued by the use of the method of the computational simulation with theuse of the analysis of finite-size scale. The results obtained for the critical exponents support the hypothesis of universality class of these models. There are constructions of the convex combination of these models. A question is raised about the simplest model and a possible solution is presented. There is a search for another kind of majority voter, but with an antiferromagnetic ground state, which leads to the minority voter. It is also to be classified in the same universality class. A natural unfold of this research is making the convex combination of the minority and majority voter models by analogy with the Ising model +- J and ask for the phase diagram class.Some results are also obtained by comparing the quenched and annealed approach to a same majority voter model. Finally, there are two more applications of these methods for obtaining critical point and critical exponents. The first refers to a model with absorbing state which is classified in the universality class of direct percolation. The second refers to a quantum model with transverse field.

Esta tese aborda o problema de regulação para sistemas lineares sujeitos a saltos Markovianos de tempo discreto com matrizes de transição incertas. Considera-se que as incertezas são limitadas em norma e os estados da cadeia de Markov podem não ser completamente observados pelo controlador. No cenário com observação completa dos estados, a solução é deduzida com base em um funcional quadrático dado em termos das probabilidades de transição incertas. Enquanto que no cenário sem observação, a solução é obtida por meio da reformulação do sistema Markoviano como um sistema determinístico, independente da cadeia de Markov. Três modelos são propostos para essa reformulação: um modelo é baseado no primeiro momento do sistema Markoviano, o segundo é obtido a partir da medida de Dirac e resulta em um sistema aumentado, e o terceiro fornece um sistema aumentado singular. Os reguladores recursivos robustos são projetados a partir de critérios de custo quadrático, dados em termos de problemas de otimização restritos. A solução é derivada da técnica de mínimos quadrados regularizados robustos e apresentada em uma estrutura matricial. A recursividade é estabelecida por equações de Riccati, que se assemelham às soluções dos reguladores clássicos, para essa classe de sistemas, quando não estão sujeitos a incertezas. / This thesis deals with regulation problem for discrete-time Markovian jump linear systems with uncertain transition matrix. The uncertainties are assumed to be normbounded type. The states of the Markov chain can not be completely observed by the controller. In the scenario with complete observation of the states, the solution is deduced based on a quadratic functional given in terms of uncertain transition probabilities. While in the scenario without observation, the solution is obtained from reformulation of the Markovian system as a deterministic system, independent of the Markov chain. Three models are proposed for the reformulation process: a model is based on the first moment of the Markovian system, the second is obtained from Dirac measure which results in an augmented system, and the third provides a singular augmented system. Recursive robust regulators are designed from quadratic cost criteria given in terms of constrained optimization problems. The solution is derived from the robust regularized least-square approach, whose framework is given in terms of a matrix structure. The recursiveness is established by Riccati equations which resemble the solutions of standard regulators for this class of systems, when they are not subject to uncertainties.