preserving monotony of combined edge finite volume finite element scheme for a bone healing model on general mesh in this article we propose and analyze a combined finite volume finite element scheme for a bone healing model this choice of discretization allows to take into account anisotropic diffusions without imposing any restrictions on the mesh moreover following the work of cances et al 2013 we define a nonlinear correction of the diffusive terms to obtain a monotone scheme we provide under a numerical assumption a complete convergence analysis of this corrected scheme and present some numerical experiments which show its good behavior

brief paper root mean square gains of switched linear systems a variational approach we consider the problem of computing the root mean square rms gain of switched linear systems we develop a new approach which is based on an attempt to characterize the worst case switching law wcsl that is the switching law that yields the maximal possible gain our main result provides a sufficient condition guaranteeing that the wcsl can be characterized explicitly using the differential riccati equations dres corresponding to the linear subsystems this condition automatically holds for first order siso systems so we obtain a complete solution to the rms gain problem in this case

a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes abstract we construct a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes where the strict convexity restriction on the meshes is removed the main contributions of this paper include three aspects firstly the classical finite volume element c fve method is extended to severely distorted quadrilateral meshes even with concave cells by virtue of a new overlapping dual partition and a new gradient approximation in fact the choice of this dual partition and gradient approximation is also conducive to the construction of a monotone scheme secondly a new monotonicity correction is suggested based on which we obtain a monotone finite volume element m fve method the resulting m fve method still keeps the local conservation and is easy for implementation finally we analyze theoretically the truncation error and the monotonicity for this scheme besides the existence of a solution to this nonlinear scheme is proved by applying the brouwer s fixed point theorem numerical results demonstrate that the m fve method has the approximate second order accuracy and preserves well the positivity of the solution for both isotropic and anisotropic diffusion problems on severely distorted quadrilateral meshes

numerically satisfactory solutions of hypergeometric recursions each family of gauss hypergeometric functions fn 2f1 a 1n b 2n c 3n z n z for fixed j 0 1 not all j equal to zero satisfies a second order linear difference equation of the form anfn 1 bnfn cnfn 1 0 because of symmetry relations and functional relations for the gauss functions many of the 26 cases for different j values can be transformed into each other in this way only with four basic difference equations all other cases can be obtained for each of these recurrences we give pairs of numerically satisfactory solutions in the regions in the complex plane where t1 6 t2 t1 and t2 being the roots of the characteristic equation

stability analysis of switched systems using variational principles an introduction many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode switched systems provide a suitable mathematical model for such processes and their stability analysis is important for both theoretical and practical reasons we review a specific approach for stability analysis based on using variational principles to characterize the most unstable solution of the switched system we also discuss a link between the variational approach and the stability analysis of switched systems using lie algebraic considerations both approaches require the use of sophisticated tools from many different fields of applied mathematics the purpose of this paper is to provide an accessible and self contained review of these topics emphasizing the intuitive and geometric underlying ideas

convergent upper bounds of peak response of lti and polytopic ltv systems through lmis abstract this paper addresses the problem of determining the response peak of dynamical systems a classical problem in control systems the paper starts by considering the impulse response of linear time invariant lti systems a linear matrix inequality lmi condition for establishing upper bounds of the sought peak is proposed by embedding the trajectory onto the level set of a polynomial and by introducing a projection technique for evaluating the extension of this set this condition is sufficient for any degree of the polynomial which has to be chosen a priori and is also necessary whenever this degree is large enough hence the proposed methodology is extended to address the impulse response of polytopic linear time varying ltv systems in particular linear systems affected linearly by structured time varying uncertainty constrained into a polytope lastly generalizations to various responses and specializations to some structures are also presented as shown by several examples which include randomly generated systems and physical systems the proposed conditions may provide significantly less conservative results than the existing lmi methods

the finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes we construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes

on the computation and inversion of the cumulative noncentral beta distribution function the computation and inversion of the noncentral beta distribution bp q x y or the noncentral f distribution a particular case of bp q x y play an important role in different applications in this paper we study the stability of recursions satisfied by bp q x y and its complementary function and describe asymptotic expansions useful for computing the function when the parameters are large we also consider the inversion problem of finding x or y when a value of bp q x y is given we provide approximations to x and y which can be used as starting values of methods for solving nonlinear equations such as newton if higher accuracy is needed

root mean square gains of switched linear systems in this note we compute the root mean square rms gain of a switched linear system when the interval between consecutive switchings is large the algorithm proposed is based on the fact that a given constant spl gamma provides an upper bound on the rms gain whenever there is a separation between the stabilizing and the antistabilizing solutions to a set of spl gamma dependent algebraic riccati equations the motivation for this problem is the application of robust stability tools to the analysis of hybrid systems

convergence of a positive nonlinear control volume finite element scheme for solving an anisotropic degenerate breast cancer development model abstract in this paper a nonlinear control volume finite element cvfe scheme for solving an anisotropic degenerate breast cancer development model is introduced and analyzed this model includes both ordinary differential equations and convection diffusion reaction equations modeling the stepwise mutations from a normal breast stem cell to a tumor cell the diffusion term which generally involves an anisotropic and heterogeneous diffusion tensor is discretized on a dual mesh by means of the piecewise linear conforming finite element method and using the godunov scheme to approximate the diffusion fluxes provided by the conforming finite element reconstruction the other terms are discretized using a nonclassical upwind finite volume scheme on the dual mesh where the dual volumes are constructed around the vertices of the original mesh this technique ensures the positivity and boundedness of discrete solutions without any restriction on the diffusion tensor nor the transmissibility coefficients the convergence of the scheme is proved only supposing the shape regularity condition for the original mesh and using a priori estimates as well as the kolmogorov relative compactness theorem the proposed scheme is robust locally conservative efficient and stable which is confirmed by numerical experiments over a general mesh

randomized algorithms for synthesis of switching rules for multimodal systems in this paper we consider the design of globally asymptotically stabilizing state dependent switching rules for multimodal systems first restricting attention to linear time invariant lti systems with only two states for the switch and then generalizing the results to multimodal lti systems and to nonlinear systems in all cases the systems considered do not allow the construction of a single quadratic lyapunov function and hence fall in the class of problems that require multiple lyapunov functions and thus are nonconvex to address the challenge of nonconvexity we introduce probabilistic algorithms and prove their probability one convergence under a new notion of convergence then to reduce complexity we develop modified versions of the algorithm we also present a class of more general nonconvex problems to which this approach can be applied the results are illustrated using two and three dimensional systems with multiple switch states

fast reliable and unrestricted iterative computation of gauss hermite and gauss laguerre quadratures methods for the computation of classical gaussian quadrature rules are described which are effective both for small and large degree these methods are reliable because the iterative computation of the nodes has guaranteed convergence and they are fast due to their fourth order convergence and its asymptotic exactness for an appropriate selection of the variables for gauss hermite and gauss laguerre quadratures local taylor series can be used for computing efficiently the orthogonal polynomials involved with exact initial values for the hermite case and first values computed with a continued fraction for the laguerre case the resulting algorithms have almost unrestricted validity with respect to the parameters full relative precision is reached for the hermite nodes without any accuracy loss and for any degree and a mild accuracy loss occurs for the hermite and laguerre weights as well as for the laguerre nodes these fast methods are exclusively based on convergent processes which together with the high order of convergence of the underlying iterative method makes them particularly useful for high accuracy computations we show examples of very high accuracy computations of up to 1000 digits of accuracy

monotone corrections for generic cell centered finite volume approximations of anisotropic diffusion equations we present a nonlinear technique to correct a general finite volume scheme for anisotropic diffusion problems which provides a discrete maximum principle we point out general properties satisfied by many finite volume schemes and prove the proposed corrections also preserve these properties we then study two specific corrections proving under numerical assumptions that the corresponding approximate solutions converge to the continuous one as the size of the mesh tends to zero finally we present numerical results showing that these corrections suppress local minima produced by the original finite volume scheme

monotone finite volume schemes for diffusion equations on polygonal meshes we construct a nonlinear finite volume fv scheme for diffusion equation on star shaped polygonal meshes and prove that the scheme is monotone i e it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients our scheme has only cell centered unknowns and it treats material discontinuities rigorously and offers an explicit expression for the normal flux numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non smooth highly anisotropic solutions and numerical results show that our scheme appears to be approximate second order accuracy for the solution and first order accuracy for the flux

numerically satisfactory solutions of kummer recurrence relations pairs of numerically satisfactory solutions as n rightarrow infty for the three term recurrence relations satisfied by the families of functions 1 rm f 1 a epsilon 1 n b epsilon 2 n z epsilon i in mathbb z are given it is proved that minimal solutions always exist except when epsilon 2 0 and z is in the positive or negative real axis and that 1 rm f 1 a epsilon 1 n b epsilon 2 n z is minimal as n rightarrow infty whenever epsilon 2 0 the minimal solution is identified for any recurrence direction that is for any integer values of epsilon 1 and epsilon 2 when epsilon 2 neq 0 the confluent limit lim b rightarrow infty 1 rm f 1 gamma b b z e gamma z with gamma in mathbb c fixed is the main tool for identifying minimal solutions together with a connection formula for epsilon 2 0 lim a rightarrow infty 1 rm f 1 a b z 0 rm f 1 b az e z 2 is the main tool to be considered

interpolation based second order monotone finite volume schemes for anisotropic diffusion equations on general grids we propose two interpolation based monotone schemes for the anisotropic diffusion problems on unstructured polygonal meshes through the linearity preserving approach the new schemes are characterized by their nonlinear two point flux approximation which is different from the existing ones and has no constraint on the associated interpolation algorithm for auxiliary unknowns thanks to the new nonlinear two point flux formulation it is no longer required that the interpolation algorithm should be a positivity preserving one the first scheme employs vertex unknowns as the auxiliary ones and a second order but not positivity preserving interpolation algorithm is utilized the second scheme uses the so called harmonic averaging points located on cell edges to define the auxiliary unknowns and a second order positivity preserving interpolation method is employed both schemes have nearly the same convergence rates as compared with their related second order linear schemes numerical results demonstrate that the new schemes are monotone and have the second order accuracy for the solution and first order for its gradient on severely distorted meshes

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

a converse lyapunov theorem for linear parameter varying and linear switching systems we study families of linear time varying systems where time variations have to satisfy restrictions on the dwell time that is on the minimum distance between discontinuities as well as on the derivative in between discontinuities such classes of systems may be formulated as linear flows on vector bundles the main objective of this paper is to construct parameter dependent lyapunov functions which characterize the exponential growth rate this is possible in the generic irreducible case as an application the gelfand formula is generalized to the class of systems studied here in other words the maximal exponential growth rate may be approximated by only considering the periodic systems in the family of time varying systems a perspective on the question of continuous dependence of the exponential growth rate on the data is given

a decoupled and positivity preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes abstract we suggest a new positivity preserving discrete duality finite volume ddfv scheme for anisotropic diffusion problems on general polygonal meshes like existing ddfv schemes this scheme is built on the primary and dual meshes and has finite volume fv equations for both vertex and cell centered unknowns the transpose of the coefficient matrix for the fv equations of the cell centered unknowns is an m matrix while that for the vertex unknowns is not an m matrix but a symmetric and positive definite matrix by employing a certain truncation technique for the vertex unknowns the positivity preserving property for both categories of unknowns is guaranteed local conservation is strictly maintained for the cell centered unknowns and conditionally maintained for the vertex unknowns since the fv equations of the vertex unknowns can be solved independently the two sets of fv equations are decoupled in contrast to existing nonlinear positivity preserving schemes the new scheme requires no nonlinear iterations for linear problems for nonlinear problems the positivity preserving mechanism of the new scheme is decoupled from its nonlinear iteration so that any nonlinear solver can be adopted moreover the positivity of the discrete solution is proved and the well posedness is analyzed rigorously for linear problems in numerical experiments the new scheme is examined extensively and compared with two positivity preserving cell centered schemes and a nonlinear ddfv scheme numerical results show that the present ddfv scheme achieves second order accuracy and preserves the positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids the high efficiency of the scheme is also demonstrated by the comparison of computation time and number of nonlinear iterations

asymptotic inversion of the erlang b formula the erlang b formula represents the steady state blocking probability in the erlang loss model or m m s s queue we derive asymptotic expansions for the offered load that matches for a given number of servers a certain blocking probability in addressing this inversion problem we make use of various asymptotic expansions for the incomplete gamma function a similar inversion problem is investigated for the erlang c formula

monotonicity of control volume methods robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications here we discuss monotonicity for a simplified problem of single phase flow but where the simulation grids and media are allowed to be general posing challenges to control volume methods we discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine point control volume discretizations for conforming quadrilateral grids in 2d these criteria are less restrictive than the m matrix property it is shown that it is impossible to construct nine point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields numerical examples are presented which show the validity of the criteria for monotonicity further the impact of nonmonotonicity is studied different behavior for different discretization methods is illuminated and simple ideas are presented for improvement in terms of monotonicity

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

cal h _ infty and dwell time specifications of continuous time switched linear systems this technical note deals with continuous time switched linear systems and addresses the problem of characterizing open loop switching strategies based on dwell time specifications assuring a pre specified root mean square gain rms as a natural consequence of treating general systems of this particular class in terms of order and number of subsystems only sufficient conditions are worked out however as positive features they are expressed through linear matrix inequalities lmi being thus numerically solvable in polynomial time and allow the treatment of stable switched linear systems which do not admit a common lyapunov function the conservativeness of the proposed design method is evaluated by comparison with other results available in the literature by means of two simple illustrative examples

dynamic output feedback control of switched linear systems this paper is devoted to stability analysis and control design of switched linear systems in both continuous and discrete time domains a particular class of matrix inequalities the so called lyapunov metzler inequalities provides conditions for open loop stability analysis and closed loop switching control using state and output feedback switched linear systems are analyzed in a general framework by introducing a quadratic in the state cost determined from a series of impulse perturbations lower bounds on the cost associated with the optimal switching control strategy are derived from the determination of a feasible solution to the hamilton jacobi bellman inequality an upper bound on the optimal cost associated with a closed loop stabilizing switching strategy is provided as well the solution of the output feedback problem is based on the construction of a full order linear switched filter whose state variable is used by the mechanism for the determination of the switching rule throughout the theoretical results are illustrated by means of academic examples a realistic practical application related to the optimal control of semiactive suspensions in road vehicles is reported

technical communique lmi conditions for time varying uncertain systems can be non conservative establishing robust asymptotic stability of uncertain systems affected by time varying uncertainty is a key problem lmi sufficient conditions have been proposed in the literature for addressing this problem based on homogeneous polynomial lyapunov functions unfortunately till now it has been unclear whether these conditions are also necessary this paper proposes a proof in order to show that one of these conditions is not only sufficient but also necessary for a sufficiently large degree of the lyapunov function

computing the kummer function u a b z for small values of the arguments we describe methods for computing the kummer function u a b z for small values of z with special attention to small values of b for these values of b the connection formula that represents u a b z as a linear combination of two 1f1 functions needs a limiting procedure we use the power series of the 1f1 functions and consider the terms for which this limiting procedure is needed we give recursion relations for higher terms in the expansion and we consider the derivative u a b z as well we also discuss the performance for small z of an asymptotic approximation of the kummer function in terms of modified bessel functions

stability and l_2 synthesis of a class of periodic piecewise time varying systems in this paper the stability stabilization and l 2 gain problems are investigated for periodic piecewise systems with time varying subsystems continuous lyapunov function with time varying lyapunov matrix is adopted a condition guaranteeing the negative definiteness of a matrix polynomial deriving from the lyapunov derivative is first obtained based on such a condition an exponential stability condition is provided moreover a state feedback controller with time varying gain is developed to stabilize the unstable periodic piecewise time varying system the l 2 gain criterion for periodic piecewise time varying system is also studied numerical examples are given to show the validity of the proposed techniques

an admissibility and asymptotic preserving scheme for systems of conservation laws with source term on 2d unstructured meshes the objective of this work is to design explicit finite volumes schemes for specific systems of conservations laws with stiff source terms which degenerate into diffusion equations we propose a general framework to design an asymptotic preserving scheme that is stable and consistent under a classical hyperbolic cfl condition in both hyperbolic and diffusive regime for any two dimensional unstructured mesh moreover the scheme developed also preserves the set of admissible states which is mandatory to keep physical solutions in stiff configurations this construction is achieved by using a non linear scheme as a target scheme for the diffusive equation which gives the form of the global scheme for the complete system of conservation laws numerical results are provided to validate the scheme in both regimes

asymptotic inversion of incomplete gamma functions the normalized incomplete gamma functions p a x and q a x are inverted for large values of the parameter a that is x solutions of the equations p a x p q a x q p e o 1 q i p are considered especially for large values of a the approximations are obtained by using uniform asymptotic expansions of the incomplete gamma functions in which an error function is the dominant term the inversion problem is started by inverting this error function term numerical results indicate that for obtaining an accuracy of four correct digits the method can already be used for a 2 although a is a large parameter it is indicated that the method can be applied to other cumulative distribution functions

convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes abstract in the present work we deal with the convergence of cell centered nonlinear finite volume schemes for anisotropic and heterogeneous diffusion operators a general framework for the convergence study of finite volume methods is provided and used to establish the convergence of the new methods thorough assessment on a set of anisotropic heterogeneous problems as well as a comparison with linear finite volume schemes is provided

homogeneous rational lyapunov functions for performance analysis of switched systems with arbitrary switching and dwell time constraints this paper addresses the problems of determining the mathcal h 2 norm and the root mean square rms gain of continuous time switched linear systems a novel class of lyapunov functions is proposed for reaching this goal called homogeneous rational lyapunov functions hrlfs it is shown that sufficient conditions for establishing upper bounds of the sought performance indexes in the case of arbitrary switching can be given in terms of linear matrix inequality lmi feasibility tests by searching for an hrlf of chosen degree moreover it is shown that these conditions are also necessary by searching for an hrlf of degree sufficiently large it is worth mentioning that necessary and sufficient lmi conditions have not been proposed yet in the literature for the considered problems hence the paper continues by considering the case of switching with dwell time constraints showing that analogous lmi conditions can be obtained for this case by searching for a family of hrlfs mutually constrained by the dwell time specification some numerical examples illustrate the proposed methodology and highlight the advantages with respect to the existing works

lmi techniques for optimization over polynomials in control a survey numerous tasks in control systems involve optimization problems over polynomials and unfortunately these problems are in general nonconvex in order to cope with this difficulty linear matrix inequality lmi techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems this survey aims to provide the reader with a significant overview of the lmi techniques that are used in control systems for tackling optimization problems over polynomials describing approaches such as decomposition in sum of squares positivstellensatz theory of moments polya s theorem and matrix dilation moreover it aims to provide a collection of the essential problems in control systems where these lmi techniques are used such as stability and performance investigations in nonlinear systems uncertain systems time delay systems and genetic regulatory networks it is expected that this survey may be a concise useful reference for all readers

brief homogeneous lyapunov functions for systems with structured uncertainties the problem addressed in this paper is the construction of homogeneous polynomial lyapunov functions hplfs for linear systems with time varying structured uncertainties a sufficient condition for the existence of an hplf of given degree is formulated in terms of a linear matrix inequalities lmi feasibility problem this condition turns out to be also necessary in some cases depending on the dimension of the system and the degree of the lyapunov function the maximum norm of the parametric uncertainty for which there exists a homogeneous polynomial lyapunov function is computed by solving a generalized eigenvalue problem the construction of such lyapunov functions is efficiently performed by means of popular convex optimization tools for the solution of problems in lmi form comparisons with other classes of lyapunov functions through numerical examples taken from the literature show that hplfs are a powerful tool for robustness analysis

a new nonlinear finite volume scheme preserving positivity for diffusion equations in this paper we present a new nonlinear finite volume scheme preserving positivity for diffusion equations the main feature of the scheme is the assumption that the values of auxiliary unknowns are nonnegative is avoided two nonnegative parameters are introduced to define a new nonlinear two point flux in which one point is the cell center and the other is the midpoint of cell edge the final flux on the edge is obtained by the continuity of normal flux numerical results show that the accuracy of both solution and flux for our new scheme is superior to that of some existing monotone schemes

a nonconservative lmi condition for stability of switched systems with guaranteed dwell time ensuring stability of switched linear systems with a guaranteed dwell time is an important problem in control systems several methods have been proposed in the literature to address this problem but unfortunately they provide sufficient conditions only this technical note proposes the use of homogeneous polynomial lyapunov functions in the non restrictive case where all the subsystems are hurwitz showing that a sufficient condition can be provided in terms of an lmi feasibility test by exploiting a key representation of polynomials several properties are proved for this condition in particular that it is also necessary for a sufficiently large degree of these functions as a result the proposed condition provides a sequence of upper bounds of the minimum dwell time that approximate it arbitrarily well some examples illustrate the proposed approach

transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions three term recurrence relations yn 1 bnyn anyn 1 0 can be used for computing recursively a great number of special functions depending on the asymptotic nature of the function to be computed different recursion directions need to be considered backward for minimal solutions and forward for dominant solutions however some solutions interchange their role for finite values of n with respect to their asymptotic behaviour and certain dominant solutions may transitorily behave as minimal this phenomenon related to gautschi s anomalous convergence of the continued fraction for ratios of confluent hypergeometric functions is shown to be a general situation which takes place for recurrences with an negative and bn changing sign once we analyze the anomalous convergence of the associated continued fractions for a number of different recurrence relations modified bessel functions confluent and gauss hypergeometric functions and discuss the implication of such transitory behaviour on the numerical stability of recursion

identifying minimal and dominant solutions for kummer recursions we identify minimal and dominant solutions of three term recurrence relations for the confluent hypergeometric functions 1f1 a 1n c 2n z and u a 1n c 2n z where i 0 1 not both equal to 0 the results are obtained by applying perron s theorem together with uniform asymptotic estimates derived by t m dunster for whittaker functions with large parameter values the approximations are valid for complex values of a c and z with ph z

robust stability of time varying polytopic systems abstract this paper provides global asymptotic stability conditions for time varying continuous time polytopic systems using a parameter dependent lyapunov function the time varying parameter uncertainty as well as its time derivative are modelled as belonging to polytopic convex sets and their dependence is made explicit to get less conservative results a particular case characterized by the parameter uncertainty satisfying a linear differential equation is analyzed and a simpler version of the aforementioned stability conditions is presented the results are expressed in terms of linear matrix inequalities being thus numerically solvable with no big difficulty the theory is illustrated by the determination of the asymptotic stability region of a mathieu s type equation with an uncertain time varying parameter