minimization of the maximum peak to peak gain the general multiblock problem a comprehensive study of the general l sub 1 optimal multiblock problem and a new linear programming algorithm for computing suboptimal controllers are presented by formulating the interpolation conditions in a concise and natural way the general theory is developed in simpler terms and with a minimum number of assumptions in addition further insight is gained into the structure of the optimal solution and different classes of multiblock problems are distinguished this leads to a conceptually attractive iterative method for finding approximate solutions

a new approach to the numerical solution of weakly singular volterra integral equations we consider linear weakly singular volterra integral equations of the second kind with kernels of the form k x v x v k x v 0 1 or k x v log x v k x v k x v being a smooth function the solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration by a transformation of the unknown function we obtain an equation which is still weakly singular but whose solution is as smooth as we like this resulting equation is then solved by standard product integration methods

piecewise homotopy perturbation method for solving linear and nonlinear weakly singular vie of second kind the purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular volterra integral equation of the second kind especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem for this purpose we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like through modifying homotopy perturbation method an algorithm is successfully established to solve the linear and nonlinear weakly singular volterra integral equation of the second kind and the convergence of the algorithm is proved strictly comparisons are made between our method and other methods and the results reveal that the modified homotopy perturbation is effective

recovery of high order accuracy in jacobi spectral collocation methods for fractional terminal value problems with non smooth solutions abstract an open problem in the numerical analysis of spectral methods for fractional differential equations is how to maintain the high order accuracy for non smooth solutions the limited regularity of the exact solution to these equations causes a deterioration in the orders of convergence of standard schemes in this paper we derive and analyze an exponentially accurate jacobi spectral collocation method for the numerical solution of nonlinear terminal value problems involving the caputo fractional derivative of rational order 0 1 the main ingredient of the proposed approach is to regularize the solution by a suitable smoothing transformation which allows us to adjust a parameter in the solution according to different given data to maximize the convergence rate we systematically describe the necessary steps in the implementation process additionally a comprehensive numerical analysis including error estimates under the l and weighted l 2 norms is derived the extensive numerical examples that accompany our analysis confirm our theoretical estimates as well as give additional insights into the convergence behavior of our method for problems with smooth and non smooth solutions

random walks with self similar clusters abstract we construct a random walk on a lattice having a hierarchy of self similar clusters built into the distribution function of allowed jumps the random walk is a discrete analog of a levy flight and coincides with the levy flight in the continum limit the fourier transform of the jump distribution function is the continuous nondifferentiable function of weierstrass we show that for cluster formation it is necessary that the mean squared displacement per jump be infinite and that the random walk be transient we interpret our random walk as having an effective dimension higher than the spatial dimension available to the walker the difference in dimensions is related to the fractal hausdorff besicovitch dimension of the self similar clusters

fractal and lacunary stochastic processes discrete time random walks simulate diffusion if the single step probability density function jump distribution generating the walk is sufficiently shortranged in contrast walks with long ranged jump distributions considered in this paper simulate levy or stable processes a one dimensional walk with a selfsimilar jump distribution the weierstrass random walk and its higherdimensional generalizations generate fractal trajectories if certain transience criteria are met and lead to simple analogs of deep results on the hausdorff besicovitch dimension of stable processes the weierstrass random walk is lacunary has gaps in the set of allowed steps and its characteristic function is weierstrass non differentiable function other lacunary random walks with characteristic functions related to riemann s zeta function and certain numbertheoretic functions have very interesting analytic structure

the four block model matching problem in l 1 and infinite dimensional linear programming the purpose of this work is fourfold first the reader is introduced to the present state of the theory behind the solution of the general model matching problem in l 1 this solution is based on the fact that the model matching problem can be recast as an infinite dimensional linear programming problem however the transformation into linear programming form is highly nonunique which leads to the second question of discussion namely how to find a good linear programming formulation it is shown that the presently available formulations contain some redundancies that limit the applicability of the theory and lead to linear programming systems containing unnecessary degeneracies the third object of this work is to study problems related to the computational complexity of two different approximation methods for the solution of the infinite dimensional linear programming systems both of these methods are needed in order to get two sided error bounds on the cutoff error one complication is that ev

a polynomial approach to the l sub 1 mixed sensitivity optimal control problem it is shown how discrete time mixed sensitivity l sub 1 optimal control problems can be converted via polynomial techniques to linear least absolute data fitting problems and solved via efficient and stable numerical methods in particular two new sub superoptimization schemes are introduced by expressing the closed loop sensitivity and complementary sensitivity maps in terms of the free parameter of a stabilizing deadbeat controller parameterization and exploiting the underlying algebraic structure this approach induces the application of a consistent truncation strategy that leads to a redundance free constraint formulation and as a consequence to linear programming problems less affected by degeneracy further more insight on the algebraic structure of the problem and on the achievement of exact rational solutions is provided allowing the development of a simple and conceptually attractive theory

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

convergence analysis of the jacobi collocation method for nonlinear weakly singular volterra integral equations in this work we present an efficient spectral collocation method for numerical solution of a class of nonlinear weakly singular volterra integral equations this type of equations typically has a singular behavior at the left endpoint of the interval of integration for overcoming this non smooth behavior we apply the jacobi collocation method the convergence analysis of the proposed method is investigated in the l and the weighted l2 norms and the results of several numerical experiments are presented which support the theoretical results the computed results are compared wherever possible with those already available in the literature

z 2 gauge model on fractal lattices hamiltonians with z 2 gauge symmetry are studied on a class of fractal lattices the lattices considered have fourier and hausdorf besicovitch dimensions strictly greater than 2 but they have a low degree of connectivity characterised by their connectivity index 2 it is proved that there is no phase transition at non zero temperatures

the numerical solution of weakly singular volterra integral equations by collocation on graded meshes since the solution of a second kind volterra integral equation with weakly singular kernel has in general unbounded derivatives at the left endpoint of the interval of integration its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates in this paper we investigate the convergence rates with respect to graded meshes and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation

mimo l 1 optimal control problems via the polynomial equations approach the general mimo multi block l 1 optimal control problem is considered a novel solution is derived by resorting to polynomial techniques and simple algebraic conditions on the free parameter of the yjbk parameterization as a result we arrive at unconstrained lp formulations less affected by redundancy and solvable by standard lp solvers as usual because the optimal controller is in general infinite dimensional a solution scheme based on solving sequences of increasing larger finite dimensional sub super optimal optimization problem is proposed viz sequences of finite dimensional optimization problems whose solutions provide lower and respectively upper bounds to the optimum monotonically converging to it finally an example is presented in order to exemplify the theory and show the effectiveness of the method

fractal random walks we consider a class of random walks on lattices and in continuous spaces having infinite mean squared displacement per step the probability distribution functions considered generate fractal self similar trajectories the characteristic functions structure functions of the walks are nonanalytic functions and satisfy scaling equations

minimum distance to the range of a banded lower triangular toeplitz operator in l 1 and application in l 1 optimal control the subject of the paper is the best approximation in ell 1 of a given infinite sequence by sequences in the range of a given banded lower triangular toeplitz operator necessary and sufficient conditions for the existence of a minimizing solution are established and a numerical algorithm for finding such a solution is designed and theoretically founded it is also shown that an optimal error sequence is only finitely nonzero the relevancy of the problem in systems theory is outlined and numerical examples are presented

superconvergence of numerical solutions to volterra integral equations with singularities in this paper we discuss the beta polynomial discrete collocation method based on practical meshes for volterra integral equations with weakly singular kernels it will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and by a simple cheap postprocessing of the original discrete collocation solution

existence of solutions for a class of nonlinear volterra singular integral equations in this paper we prove some results concerning the existence of solutions for a large class of nonlinear volterra singular integral equations in the space c 0 1 consisting of real functions defined and continuous on the interval 0 1 the main tool used in the proof is the concept of a measure of noncompactness we also present some examples of nonlinear singular integral equations of volterra type to show the efficiency of our results moreover we compare our theory with the approach depending on the use of the theory of volterra stieltjes integral equations we also show that the results of the paper are applicable in the study of the so called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems

i 1 optimal control of multivariable systems with output norm constraints in this paper we consider the l1 optimal control problem for general rational plants it is shown that for plants with no poles or zeros on the unit circle an optimal compensator exists and that the resulting closed loop transfer function is polynomial whenever there are at least as many controls as regulated outputs and at least as many measurements as exogeneous inputs exactly or approximately optimal rational compensators can be obtained by solving a sequence of finite linear programs for the coefficients of a polynomial closed loop transfer function no assumptions on plant poles or zeros are required to obtain at least approximately optimal compensators it is shown that constrained problems in which a set of outputs is regulated subject to l norm constraints on another set of outputs can be solved using a slight modification of the same algorithm

extrapolation method for solving weakly singular nonlinear volterra integral equations of the second kind abstract based on a new generalization of discrete gronwall inequality in l tao h yong a generalization of discrete gronwall inequality and its application to weakly singular volterra integral equality of the second kind j math anal appl 282 2003 56 62 navot s quadrature rule for computing integrals with the end point singularity in i navot a further extension of euler maclaurin summation formula j math phys 41 1962 155 184 and a transformation in p baratella a palamara orsi a new approach to the numerical solution of weakly singular volterra integral equations j comput appl math 163 2004 401 418 a new quadrature method for solving nonlinear weakly singular volterra integral equations of the second kind is presented the convergence of the approximation solution and the asymptotic expansion of the error are proved so by means of the extrapolation technique we not only obtain a higher accuracy order of the approximation but also get a posteriori estimate of the error

further results on the optimal rejection of persistent bounded disturbances the problem of designing controllers that optimally reject persistent disturbances is studied the focus is on the case where the plant to be controlled has either zeros or poles on the stability boundary i e the unit circle in the discrete time case and the extended j omega axis in the continuous time case for the discrete time case the problem of minimizing a cost functional of the form f rg sub 1 where the transform g of g has some unit circle zeros is studied a previously published dual problem formulation is extended and it is shown that an optimal controller need not exist the construction of a sequence of suboptimal controllers whose performance approaches the unattainable infimum of the cost function is studied it is shown that two results which hold in the case of h sub infinity optimization do not hold in the presented situation specifically the introduction of unit circle zeros can increase the value of the infimum even when every unit circle zero of g is also a zero of f and a sequence of controllers constructed in an obvious fashion fails to be an optimizing sequence similar results are obtained for the continuous time case

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

a generalization of discrete gronwall inequality and its application to weakly singular volterra integral equation of the second kind this paper presents a new discrete gronwall inequality using the inequality we prove convergence and error estimate of the numerical solutions of the second weakly singular volterra integral equation where discrete equation is derived by novot s quadrature formula

optimal rejection of persistent disturbances robust stability and mixed sensitivity minimization the problem of optimal disturbance rejection of bounded persistent disturbances is solved in the general nonsquare case the minimum value of the objective function can be obtained by solving a semi infinite linear programming problem and an iterative procedure for obtaining approximate solutions is introduced application of the l sup 1 optimal problem to robustness is discussed a mixed sensitivity problem is formulated and shown to guarantee good disturbance rejection in the presence of plant perturbations

lattice dynamics random walks and nonintegral effective dimensionality definitions of the nonintegral effective dimensionality of recursively defined lattices fractal lattices may be based on scaling properties of the lattices or on the qualitative behavior of cooperative phenomena supported by the lattices we examine analogs of these definitions for regular i e periodic lattices supporting long range interactions in particular we show how to calculate a harmonic oscillator effective dimension a scaling dimension and a random walk effective dimension for simple cubic lattices with a class of long range interactions we examine the relationship between these three dimensions for regular lattices and conjecture a constraint on the analogs of these dimensions for fractal lattices

williams watts dielectric relaxation a fractal time stochastic process dielectric relaxation in amorphous materials is treated in a defect diffusion model where relaxation occurs when a mobile defect such as a vacancy reaches a frozen in dipole the random motion of the defect is assumed to be governed by a fractal time stochastic process where the mean duration between defect movements is infinite when there are many more defects than dipoles the williams watts decaying fractional exponential relaxation law is derived the argument of the exponential is related to the number of distinct sites visited by the random walk of the defect for the same reaction dynamics but with more traps than walkers an algebraically decaying relaxation is found

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method