Last decade witnessed the explosive growth in mobile devices and their traffic demand, and hence the significant increase in the energy cost of the cellular service providers. One major component of energy expenditure comes from the operation of base stations. How to reduce energy cost of base stations while satisfying users’ soaring demands has become an imperative yet challenging problem. In this dissertation, we investigate the minimization of the long-term time-averaged expected energy cost while guaranteeing network strong stability. Specifically, considering flow routing, link scheduling, and energy constraints, we formulate a time-coupling stochastic Mixed-Integer Non-Linear Programming (MINLP) problem, which is prohibitively expensive to solve. We reformulate the problem by employing Lyapunov optimization theory and develop a decomposition based algorithm which ensures network strong stability. We obtain the bounds on the optimal result of the original problem and demonstrate the tightness of the bounds and the efficacy of the proposed scheme.

The main goal of the thesis is to construct explicit, s-stage, strong-stability-preserving (SSP) Hermite–Birkhoff (HB) time discretization methods of order p with nonnegative coefficients for the integration of hyperbolic conservation laws. The Shu–Osher form and the canonical Shu–Osher form by means of the vector formulation for SSP Runge–Kutta (RK) methods are extended to SSP HB methods. The SSP coefficients of k-step, s-stage methods of order p, HB(k,s,p), as combinations of k-step methods of order (p − 3) with s-stage explicit RK methods of order 4, and k-step methods of order (p-4) with s-stage explicit RK methods of order 5, respectively, for s = 4, 5,..., 10 and p = 4, 5,..., 12, are constructed and compared with other methods. The good efficiency gains of the new, optimal, SSP HB methods over other SSP methods, such as Huang’s hybrid methods and RK methods, are numerically shown by means of their effective SSP coefficients and largest effective CFL numbers. The formulae of these new, optimal methods are presented in their Shu–Osher form.

The main goal of the thesis is to construct explicit, s-stage, strong-stability-preserving (SSP) Hermite–Birkhoff (HB) time discretization methods of order p with nonnegative coefficients for the integration of hyperbolic conservation laws. The Shu–Osher form and the canonical Shu–Osher form by means of the vector formulation for SSP Runge–Kutta (RK) methods are extended to SSP HB methods. The SSP coefficients of k-step, s-stage methods of order p, HB(k,s,p), as combinations of k-step methods of order (p − 3) with s-stage explicit RK methods of order 4, and k-step methods of order (p-4) with s-stage explicit RK methods of order 5, respectively, for s = 4, 5,..., 10 and p = 4, 5,..., 12, are constructed and compared with other methods. The good efficiency gains of the new, optimal, SSP HB methods over other SSP methods, such as Huang’s hybrid methods and RK methods, are numerically shown by means of their effective SSP coefficients and largest effective CFL numbers. The formulae of these new, optimal methods are presented in their Shu–Osher form.

The main goal of the thesis is to construct explicit, s-stage, strong-stability-preserving (SSP) Hermite–Birkhoff (HB) time discretization methods of order p with nonnegative coefficients for the integration of hyperbolic conservation laws. The Shu–Osher form and the canonical Shu–Osher form by means of the vector formulation for SSP Runge–Kutta (RK) methods are extended to SSP HB methods. The SSP coefficients of k-step, s-stage methods of order p, HB(k,s,p), as combinations of k-step methods of order (p − 3) with s-stage explicit RK methods of order 4, and k-step methods of order (p-4) with s-stage explicit RK methods of order 5, respectively, for s = 4, 5,..., 10 and p = 4, 5,..., 12, are constructed and compared with other methods. The good efficiency gains of the new, optimal, SSP HB methods over other SSP methods, such as Huang’s hybrid methods and RK methods, are numerically shown by means of their effective SSP coefficients and largest effective CFL numbers. The formulae of these new, optimal methods are presented in their Shu–Osher form.

A plethora of physical phenomena are modelled by hyperbolic partial differential equations, for which the exact solution is usually not known. Numerical methods are employed to approximate the solution to hyperbolic problems; however, in many cases it is difficult to satisfy certain physical properties while maintaining high order of accuracy. In this thesis, we develop high-order time-stepping methods that are capable of maintaining stability constraints of the solution, when coupled with suitable spatial discretizations. Such methods are called strong stability preserving (SSP) time integrators, and we mainly focus on perturbed methods that use both upwind- and downwind-biased spatial discretizations. Firstly, we introduce a new family of third-order implicit Runge–Kuttas methods with arbitrarily large SSP coefficient. We investigate the stability and accuracy of these methods and we show that they perform well on hyperbolic problems with large CFL numbers. Moreover, we extend the analysis of SSP linear multistep methods to semi-discretized problems for which different terms on the right-hand side of the initial value problem satisfy different forward Euler (or circle) conditions. Optimal perturbed and additive monotonicity-preserving linear multistep methods are studied in the context of such problems. Optimal perturbed methods attain augmented monotonicity-preserving step sizes when the different forward Euler conditions are taken into account. On the other hand, we show that optimal SSP additive methods achieve a monotonicity-preserving step-size restriction no better than that of the corresponding non-additive SSP linear multistep methods. Furthermore, we develop the first SSP linear multistep methods of order two and three with variable step size, and study their optimality. We describe an optimal step-size strategy and demonstrate the effectiveness of these methods on various one- and multi-dimensional problems. Finally, we establish necessary conditions to preserve the total variation of the solution obtained when perturbed methods are applied to boundary value problems. We implement a stable treatment of nonreflecting boundary conditions for hyperbolic problems that allows high order of accuracy and controls spurious wave reflections. Numerical examples with high-order perturbed Runge–Kutta methods reveal that this technique provides a significant improvement in accuracy compared with zero-order extrapolation.

La thèse est portée essentiellement sur la stabilisation indirecte d’un système de deux équations des ondes couplées et sur la stabilisation frontière de poutre de Rayleigh.Dans le cas de la stabilisation d’un système d’équations d’onde couplées, le contrôle est introduit dans le système directement sur le bord du domaine d’une seule équation dans le cas d’un domaine borne ou à l’intérieur d’une seule équation mais dans le cas d’un domaine non borné. La nature du système ainsi couplé dépend du couplage des équations et de la nature arithmétique des vitesses de propagations, et ceci donne divers résultats pour la stabilisation polynomiale ainsi la non stabilité.Dans le cas de la stabilisation de poutre de Rayleigh, l’équation est considérée avec un seul contrôle force agissant sur bord du domaine. D’abord, moyennant le développement asymptotique des valeurs propres et des vecteurs propres du système non contrôlé, un résultat d’observabilité ainsi qu’un résultat de bornétude de la fonction de transfert correspondant sont obtenus. Alors, un taux de décroissance polynomial de l’énergie du système est établi. Ensuite, moyennant une étude spectrale combinée avec une méthode fréquentielle, l’optimalité du taux obtenu est assurée. / The thesis is driven mainly on indirect stabilization system of two coupled wave equations and the boundary stabilization of Rayleigh beam equation. In the case of stabilization of a coupled wave equations, the Control is introduced into the system directly on the edge of the field of a single equation in the case of a bounded domain or inside a single equation but in the case of an unbounded domain. The nature of thus coupled system depends on the coupling equations and arithmetic Nature of speeds of propagation, and this gives different results for the polynomial stability and the instability. In the case of stabilization of Rayleigh beam equation, we consider an equation with one control force acting on the edge of the area. First, using the asymptotic expansion of the eigenvalues and vectors of the uncontrolled system an observability result and a result of boundedness of the transfer function are obtained. Then a polynomial decay rate of the energy of the system is established. Then through a spectral study combined with a frequency method, optimality of the rate obtained is assured.

Cette thèse est consacrée à l’étude de la stabilisation directe et indirecte de différents systèmes d’équations d’ondes avec un contrôle frontière de type fractionnaire ou un contrôle local viscoélastique de type Kelvin-Voight. Nous considérons, d’abord, la stabilisation de l’équation d’ondes multidimensionnel avec un contrôle frontière fractionnaire au sens de Caputo. Sous des conditions géométriques optimales, nous établissons un taux de décroissance polynomial de l’énergie de système. Ensuite, nous nous intéressons à l’étude de la stabilisation d’un système de deux équations d’ondes couplées via les termes de vitesses, dont une seulement est amortie avec contrôle frontière de type fractionnaire au sens de Caputo. Nous montrons différents résultats de stabilités dans le cas 1-d et N-d. Finalement, nous étudions la stabilité d’un système de deux équations d’ondes couplées avec un seul amortissement viscoélastique localement distribué de type Kelvin-Voight. / This thesis is devoted to study the stabilization of the system of waves equations with one boundary fractional damping acting on apart of the boundary of the domain and the stabilization of a system of waves equations with locally viscoelastic damping of Kelvin-Voight type. First, we study the stability of the multidimensional wave equation with boundary fractional damping acting on a part of the boundary of the domain. Second, we study the stability of the system of coupled onedimensional wave equation with one fractional damping acting on a part of the boundary of the domain. Next, we study the stability of the system of coupled multi-dimensional wave equation with one fractional damping acting on a part of the boundary of the domain. Finally, we study the stability of the multidimensional waves equations with locally viscoelastic damping of Kelvin-Voight is applied for one equation around the boundary of the domain.