affine parameter dependent lyapunov functions and real parametric uncertainty this paper presents new tests to analyze the robust stability and or performance of linear systems with uncertain real parameters these tests are extensions of the notions of quadratic stability and performance where the fixed quadratic lyapunov function is replaced by a lyapunov function with affine dependence on the uncertain parameters admittedly with some conservatism the construction of such parameter dependent lyapunov functions can be reduced to a linear matrix inequality lmi problem and hence is numerically tractable these lmi based tests are applicable to constant or time varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations they also avoid the frequency sweep needed in real spl mu analysis and numerical experiments indicate that they often compare favorably with spl mu analysis for time invariant parameter uncertainty

stabilization of uncertain systems via linear control this note considers the problem of stabilizing a linear dynamical system whose state equation includes a time varying uncertain parameter vector q cdot given the dynamics dot x t a q t x t b q t u t and a bounding set q for the values q t the objective is to choose a control law u t p x t guaranteeing uniform asymptotic stability for all admissible variations of q cdot our results differ from previous work in one fundamental way that is we show that when working with linear controllers it is possible to dispense with all assumptions on b cdot which have been made by previous authors e g see 1 9 this elimination of hypotheses on b cdot is accomplished roughly as follows the system sigma underline underline delta a q b q is shown to be equivalent to another system sigma underline underline delta a q b as far as stabilization is concerned since b is a constant matrix independent of q the desired result is readily obtained

outer factorizations in one and several variables a multivariate version of rosenblum s fejer riesz theorem on outer factorization of trigonometric polynomials with operator coefficients is considered due to a simplification of the proof of the single variable case new necessary and sufficient conditions for the multivariable outer factorization problem are formulated and proved

brief computational complexity of randomized algorithms for solving parameter dependent linear matrix inequalities randomized algorithms are proposed for solving parameter dependent linear matrix inequalities and their computational complexity is analyzed the first proposed algorithm is an adaptation of the algorithms of polyak and tempo syst control lett 43 5 2001 343 and calafiore and polyak ieee trans autom control 46 11 2001 1755 for the present problem it is possible however to show that the expected number of iterations necessary to have a deterministic solution is infinite in order to make this number finite the improved algorithm is proposed the number of iterations necessary to have a probabilistic solution is also considered and is shown to be independent of the parameter dimension a numerical example is provided

on intertwining dilations on considere la description de l ensemble l x de tous les relevements de nagy foias de x en combinant la methode des noyaux de toeplitz generalises avec certaines idees de arov et grossman sur les fonctions analytiques contractibles

robust adaptive l2 gain control of polytopic mimo lti systems lmi results passification based direct adaptive control is considered for polytopic uncertain linear time invariant multi input multi output systems linear matrix inequality based results are provided to guarantee that the adaptive algorithm passifies the system whatever the uncertain parameters in some given set the contributions are based on the introduction of a parallel feed forward shunt that liberates from the strong equality constraint pb c t g t often used for strict passification the shunt combined with the introduction of slack variables allows to conclude with easy to test conditions without restrictions on which data is uncertain in the process model the resulting adaptive control is such that the control gain is bounded in a chosen set whatever bounded disturbances moreover it proves to be not worse than computable parameter dependent static output feedback controls with respect to l2 gain attenuation a simple academic example illustrates the results keywords passivity robustness adaptive control lmi output feedback polytopic systems l2 gain

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

multivariable mrac design without gain symmetry conditions using a stabilizing multiplier a new multivariable mrac design for plants of arbitrary relative degree which does not require a stringent symmetry assumption related with the plant high frequency gain matrix is presented in contrast to previous results the new solution does not involve additional parametrization and filtering being thus close in structure and complexity to conventional solutions instead of the fragile symmetry assumption a less restrictive and robust condition is required which can be achieved using a multiplier

extensions of positive definite functions on free groups abstract an analogue of krein s extension theorem is proved for operator valued positive definite functions on free groups the proof gives also the parametrization of all extensions by means of a generalized type of szego parameters one singles out a distinguished completion called central which is related to quasi multiplicative positive definite functions an application is given to factorization of noncommutative polynomials

shift invariant subspaces factorization and interpolation for matrices i the canonical case abstract this article begins a program of classifying subspaces of block matrices that are invariant under multiplication on the right by the algebra of block upper or lower triangular matrices this gives a unifying point of view to several general kinds of factorization problems including lu qr and cholesky factorization a second application is to some matrix completion problems such as that of filling in the upper triangular entries of a partially defined matrix to make the result of norm less than or equal to 1 included is a linear fractional parametrization of the set of all solutions of this problem

design of strictly positive real systems using constant output feedback the authors present a linear matrix inequality lmi approach to the strictly positive real spr synthesis problem find an output feedback k such that the closed loop system t s is spr the authors establish that if no such constant output feedback k exists then no dynamic output feedback with a proper transfer matrix exists to make the closed loop system spr the existence of k to guarantee the spr property of the closed loop system is used to develop an adaptive control scheme that can stabilize any system of arbitrary unknown order and unknown parameters

a new bounded real lemma representation for the continuous time case a differential linear matrix inequality dlmi approach is introduced for the solution of various linear continuous time control problems the proposed method permits the application of linear matrix inequalities lmis to the solution of control design problems under uncertainty these problems are solved for finite horizon linear systems while considerably reducing the overdesign inherent in previous methods the new approach also allows for the solution of the output feedback control problem for systems belonging to a finite set of uncertain plants with hardly any overdesign four examples are given to demonstrate the applicability of the new method

hamiltonian odes in the wasserstein space of probability measures in this paper we consider a hamiltonian h on p 2 r 2d the set of probability measures with finite quadratic moments on the phase space r 2d r d r d which is a metric space when endowed with the wasserstein distance w 2 we study the initial value problem d t dt j d v t t 0 where j d is the canonical symplectic matrix 0 is prescribed and v t is a tangent vector to p 2 r 2d at t belonging to h t the subdifferential of h at t two methods for constructing solutions of the evolutive system are provided the first one concerns only the case where 0 is absolutely continuous it ensures that fit remains absolutely continuous and v t h t is the element of minimal norm in h t the second method handles any initial measure 0 if we further assume that h is convex proper and lower semicontinuous on p 2 r 2d we prove that the hamiltonian is preserved along any solution of our evolutive system h t h 0

an improved lmi condition for robust d stability of uncertain polytopic systems a sufficient condition for robust d stability of linear systems with polytope type uncertainties is proposed the result is based on a linear parameter dependent lyapunov function obtained from the feasibility test of a set of linear matrix inequalities lmis defined at the vertices of the polytope this improved lmi condition encompasses previous results based on additional inequalities as well as results based on extra variables

brief paper comparisons among robust stability criteria for linear systems with affine parameter uncertainties this paper is concerned with the problem of comparisons among robust stability criteria for a class of uncertain linear systems where the system state matrices considered are affinely dependent on the uncertain parameters at first a robust stability criterion for the class of systems to be affinely quadratically stable aqs is derived based on the vertex separator approach where affine parameter dependent lyapunov functions are exploited to prove stability then comparison results between the robust stability criterion and the existing tests for aqs are given in terms of degree of conservatism a numerical example is given to illustrate the results

lmi based algorithm for strictly positive real systems with static output feedback abstract an algorithm based on linear matrix inequalities lmis is proposed to find a constant output feedback matrix k o and a constant output tandem matrix f such that the controlled system a b k o c b f c is strictly positive real spr the number of output variables of the plant a b c is greater than or equal to the number of its input variables considering that spr systems with static output feedback are related to spr systems with static state feedback as shown in this manuscript the first step of the algorithm is to find a matrix f such that all transmission zeros of the system a b f c have negative real parts after finding this matrix f an output feedback matrix k o such that the system a b k o c b f c is spr is found another algorithm is proposed to specify a decay rate the results are applied to the simulation of electrical stimulation for paraplegic patients to vary knee joint angle from 0 to 60

robust discrete time simple adaptive regulation robust simple adaptive controllers are proposed for regulation of linear discrete time systems with parameter uncertainties sufficient conditions for closed loop stability of the proposed control scheme are derived these lead to the requirement of almost strictly positive realness of the plant which is shown to be equivalent under certain conditions to a minimum phase requirement a numerical example is given that demonstrates the applicability of the proposed method and the simplicity of its implementation

a variational finite volume scheme for wasserstein gradient flows we propose a variational finite volume scheme to approximate the solutions to wasserstein gradient flows the time discretization is based on an implicit linearization of the wasserstein distance expressed thanks to benamou brenier formula whereas space discretization relies on upstream mobility two point flux approximation finite volumes our scheme is based on a first discretize then optimize approach in order to preserve the variational structure of the continuous model at the discrete level our scheme can be applied to a wide range of energies guarantees non negativity of the discrete solutions as well as decay of the energy we show that our scheme admits a unique solution whatever the convex energy involved in the continuous problem and we prove its convergence in the case of the linear fokker planck equation with positive initial density numerical illustrations show that it is first order accurate in both time and space and robust with respect to both the energy and the initial profile

a schur analysis approach to minimum distance problems abstract some minimum distance problems in hilbert space are to be transformed into associated matrix optimization problems which is their turn are studied using schur parametrization of nonnegative hermitian block matrices this enables a clear geometrical insight into the structure of the feasible region and a complete description of the set of all solutions

continuous time analysis eigenstructure assignment and h sub 2 synthesis with enhanced linear matrix inequalities lmi characterizations this note describes a new framework for the analysis and synthesis of control systems which constitutes a genuine continuous time extension of results that are only available in discrete time in contrast to earlier results the proposed methods involve a specific transformation on the lyapunov variables and a reciprocal variant of the projection lemma in addition to the classical linearizing transformations on the controller data for a wide range of problems including robust analysis and synthesis multichannel h sub 2 stateand output feedback syntheses the approach leads to potentially less conservative linear matrix inequality lmi characterizations this comes from the fact that the technical restriction of using a single lyapunov function is to some extent ruled out in this new approach moreover the approach offers new potentials for problems that cannot be handled using earlier techniques an important instance is the eigenstructure assignment problem blended with lyapunov type constraints which is given a simple and tractable formulation

global stability and performance of a simplified adaptive algorithm abstract most self tuning and adaptive control algorithms usually use reference models controllers or identifiers of about the same order as the controlled plant since the dimension of the plants in the real world may be very large or unknown implementation of adaptive control procedures may be difficult or sometimes impossible in this paper we prove global stability for a simple adaptive algorithm that can use low order model reference and controllers since no observers or identifiers are used in the adaptation process the algorithm is basically fitted for systems that are denominated as almost positive real it is shown that at the price of bounded rather than vanishing output tracking errors the simple algorithm can be applied in systems that can be stabilized via constant output feedback these procedures are believed to reduce considerably the effort required for implementation of adaptive control in practical applications especially in multivariable large scale systems

stabilizability conditions of multivariable uncertain systems via output feedback control a stabilizability condition is established for stabilizing multivariable uncertain dynamical systems with matched uncertainties by linear static output feedback control it is shown that under the assumptions of minimum phase and nonsingular high frequency gain for the nominal plant stabilization with linear static output feedback control is achievable a necessary and sufficient condition is also obtained for the existence of constant output feedback which makes the closed loop system strictly positive real

dilation of generalized toeplitz kernels on ordered groups abstract we introduce the concept of toeplitz kre n cotlar triplet on ordered groups and we prove a general dilation result and a general representation result for the positive definite case this general result includes and extends previous generalizations of the kre n extension theorem the sz nagy and foias commutant lifting theorem and the generalized herglotz bochner weil theorem

on the stabilization of uncertain linear systems via bound invariant lyapunov functions this paper considers the problem of stabilizing a class of systems with an arbitrarily large uncertainty bounding set in particular the stabilization of linear uncertain systems via a bound invariant lyapunov function is examined necessary and sufficient conditions for the existence of a bound invariant lyapunov function are derived for systems with block structured uncertainty and a class of systems with scalar linear uncertainty

a gradient flow approach to an evolution problem arising in superconductivity we study an evolution equation proposed by chapman rubinstein and schatzman as a mean field model for the evolution of the vortex density in a superconductor we treat the case of a bounded domain where vortices can exit or enter the domain we show that the equation can be derived rigorously as the gradient flow of some specific energy for the riemannian structure induced by the wasserstein distance on probability measures this leads us to some existence and uniqueness results and energy dissipation identities we also exhibit some entropies that decrease through the flow and allow us to get regularity results solutions starting in lp p 1 remain in lp 2007 wiley periodicals inc

congested aggregation via newtonian interaction we consider a congested aggregation model that describes the evolution of a density through the competing effects of nonlocal newtonian attraction and a hard height constraint this provides a counterpoint to existing literature on repulsive attractive nonlocal interaction models where the repulsive effects instead arise from an interaction kernel or the addition of diffusion we formulate our model as the wasserstein gradient flow of an interaction energy with a penalization to enforce the constraint on the height of the density from this perspective the problem can be seen as a singular limit of the keller segel equation with degenerate diffusion two key properties distinguish our problem from previous work on height constrained equations nonconvexity of the interaction kernel which places the model outside the scope of classical gradient flow theory and nonlocal dependence of the velocity field on the density which causes the problem to lack a comparison principle to overcome these obstacles we combine recent results on gradient flows of nonconvex energies with viscosity solution theory we characterize the dynamics of patch solutions in terms of a hele shaw type free boundary problem and using this characterization show that in two dimensions patch solutions converge to a characteristic function of a disk in the long time limit with an explicit rate on the decay of the energy we believe that a key contribution of the present work is our blended approach combining energy methods with viscosity solution theory

improved lmi representations for the analysis and the design of continuous time systems with polytopic type uncertainty simple modifications to the representation of the bounded real lemma brl in the linear continuous time invariant case are introduced these modifications reduce the overdesign that occurs in the analysis and the design of systems with polytopic type uncertainties a different lyapunov function accompanies each of the vertices of the uncertainty polytope thus eliminating the need for quadratic stability or stabilizability the advantages of these new representations are demonstrated by way of two examples

output feedback stabilization of uncertain dynamical systems given an uncertain dynamical system with a linear nominal part matched uncertainties and a measured output a sufficient condition is given for the existence of an output feedback stabilizing controller for the general multiinput multioutput case checking the sufficient condition involves the selection of a matrix f for the special class of single input single output systems f is a positive or negative scalar and it is shown that if the nominal system transfer function is strictly positive real then an output feedback stabilizing controller does indeed exist

brief paper performance improvement of robust controllers for polynomially uncertain systems this paper is concerned with the high performance robust control of discrete time linear time invariant lti systems with semi algebraic uncertainty regions it is assumed that a robustly stabilizing static controller is given whose gain depends polynomially on the uncertain variables the problem of tuning this parameter dependent gain with respect to a prescribed quadratic cost function is formulated as a sum of squares sos optimization this method leads to a near optimal controller whose performance is better than that of the initial controller it is shown that the results derived in the present work encompass the ones obtained in a recent paper the efficacy of the results is elucidated by an example

simplified adaptive control via improved robust positive real conditions a new method is derived for embedding plants in a robust state feedback scheme to achieve strictly positive realness of the resulting augmented plants a state feedback gain is derived that guarantees the strictly positive realness of the closed loop in presence of polytopic type possibly time varying parameter uncertainties in the model that describes the plant this is achieved by assigning different lyapunov functions to each of the vertices of the uncertainty polytope the obtained feedback gain is used to apply existing methods for robust simplified adaptive control on systems with possibly time varying polytopic uncertainties

evolution of microstructure in unstable porous media flow a relaxational approach we study the flow of two immiscible fluids of different density and mobility in a porous medium if the heavier phase lies above the lighter one the interface is observed to be unstable the two phases start to mix on a mesoscopic scale and the mixing zone grows in time an example of evolution of microstructure a simple set of assumptions on the physics of this two phase flow in a porous medium leads to a mathematically ill posed problem when used to establish a continuum free boundary problem we propose and motivate a relaxation of this nonconvex constraint of a phase distribution with a sharp interface on a macroscopic scale we prove that this approach leads to a mathematically well posed problem that predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient 1999 john wiley sons inc

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

guaranteed cost control of uncertain nonlinear systems via polynomial lyapunov functions in this paper we consider the problem of guaranteed cost control for a class of uncertain nonlinear systems we derive linear matrix inequality conditions for the regional robust stability and performance problems based on lyapunov functions which are polynomial functions of the state and uncertain parameters the performance index is calculated over a set of initial conditions also we discuss the synthesis problem for a class of affine control systems numerical examples illustrate our method

the maximum distance problem and band sequences abstract we solve the following problem for 1 j k n and j k m let a jk be a given complex number with a kj jk we wish to find linearly independent vectors x 1 x n such that x k x j a jk for j k m and such that the distance from x k to the linear span of x 1 x k 1 is maximal for 2 k n we construct and characterize all such sequences of vectors our solution leads naturally to the class of m band sequences of vectors in an inner product space we study these sequences and characterize their equivalence classes under unitary transformations

a gradient flow approach to the porous medium equation with fractional pressure we consider a family of porous media equations with fractional pressure recently studied by caffarelli and vazquez we show the construction of a weak solution as the wasserstein gradient flow of a square fractional sobolev norm the energy dissipation inequality regularizing effect and decay estimates for the l p norms are established moreover we show that a classical porous medium equation can be obtained as a limit case

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

frequency domain conditions for strictly positive real functions frequency domain conditions for strictly positive real spr functions which appear in literature are often only necessary or only sufficient this point is raised in 1 2 where necessary and sufficient conditions in the s domain are given for a transfer function to be spr in this note the points raised in 1 i2 are clarified further by giving necessary and sufficient conditions in the frequency domain for transfer functions to be spr these frequency domain conditions are easier to test than those given in the s domain or time domain 1 2

partial matrix contractions and intersections of matrix balls abstract the balls of maximal volume in the intersection of two concentric matrix balls are computed the results are applied to the 2 2 contractive completion problem where the diagonal elements are given

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

a new robust d stability condition for real convex polytopic uncertainty abstract the problem of robust d stability analysis with respect to real convex polytopic uncertainties is tackled a new lmi based sufficient condition for the existence of parameter dependent lyapunov functions is proposed this condition generalises previously published conditions numerical comparisons with quadratic stability results as well as previous results based on parameter dependent lyapunov functions illustrate the relevance of this new condition finally this result appears to be promising for robust multi objective performance analysis and control synthesis purposes

schur analysis of some completion problems abstract the complete schur analysis for contractive block matrices is given together with its connections with positiveness these are used for reobtaining some results on completion problems sometimes in a more general setting these completion problems include positive completions contractive and isometric completions and the realizability of upper triangular contractions

time domain and frequency domain conditions for strict positive realness the author states various time domain and frequency domain conditions pertaining to multivariate strict positive real systems and establishes the relationship between these conditions their relationship is clarified by means of a three tier diagram each tier represents a set of equivalent conditions successively stronger than the lower tier the main theorem in the present study can be considered as the positive realness lemma for strictly positive real systems

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

comments on design of strictly positive real systems using constant output feedback a recent publication states and proves the conditions under which a linear time invariant system with state space realization a b c can be made strictly positive real via constant output feedback this note is intended to briefly present the development of the proof and to give due credit to the first proofs of this statement

a new approach to eigenstructure assignment by output feedback the eigenstructure assignment problem for a linear time invariant multi input multi output system using output feedback is considered a new approach is developed which identifies the eigenspaces for the desired set of all the closed loop eigenvalues for the assignment of this desired set necessary and sufficient conditions are established these conditions contain two coupled sylvester matrix equations one of which is proven to be a reduced order square sylvester matrix equation this results in an efficient analytical procedure numerically superior to known techniques for the determination of the output feedback gain matrix

exponential feedback passivity and stabilizability of nonlinear systems abstract motivated by n krasovskii s characterisation of exponential stability the concept of exponential passivity is introduced it is shown that to make a nonlinear system with factorisable high frequency gain matrix exponentially passive via either state or output feedback exponential minimum phaseness and invertibility conditions are necessary and sufficient these conditions also guarantee exponential output feedback stabilisability this result extends previous results concerning linear systems

the intertwining lifting theorem for ordered groups abstract sz nagy s dilation theorem for a contraction on a hilbert space has been extended by mlak to the case of a contraction semigroup whose indices are the positive elements of a totally ordered discrete group we generalize to this case the intertwining lifting theorem of sz nagy and foias some previous interpolation results on ordered groups are obtained as consequences

global existence for a system of non linear and non local transport equations describing the dynamics of dislocation densities in this paper we study the global in time existence problem for the groma balogh model describing the dynamics of dislocation densities this model is a two dimensional model where the dislocation densities satisfy a system of transport equations such that the velocity vector field is the shear stress in the material solving the equations of elasticity this shear stress can be expressed as some riesz transform of the dislocation densities the main tool in the proof of this result is the existence of an entropy for this system

convergence and non convergence of many particle evolutions with multiple signs we address the question of the convergence of evolving interacting particle systems as the number of particles tends to infinity we consider two types of particles called positive and negative same sign particles repel each other and opposite sign particles attract each other the interaction potential is the same for all particles up to the sign and has a logarithmic singularity at zero the central example of such systems is that of dislocations in crystals because of the singularity in the interaction potential the discrete evolution leads to blow up in finite time we remedy this situation by regularising the interaction potential at a length scale delta n 0 which converges to zero as the number of particles n tends to infinity we establish two main results the first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled pdes which describe the evolution of their continuum densities in the setting of dislocations these pdes are known as the groma balogh equations in the proof we rely on both the theory of lambda convex gradient flows to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a delta n regularised version of the groma balogh equations and a priori estimates for the groma balogh equations to pass to the small regularisation limit in a functional setting based on orlicz spaces in order for the quantitative bound not to degenerate too fast in the limit n rightarrow infty we require delta n to converge to zero sufficiently slowly the second result is a counterexample demonstrating that if delta n converges to zero sufficiently fast then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the groma balogh equations these results show how the validity of the groma balogh equations as the limit of many particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised

technical communique mitigation of symmetry condition in positive realness for adaptive control feasibility of nonlinear and adaptive control methodologies in multivariable linear time invariant systems with state space realization a b c is apparently limited by the standard strictly positive realness conditions that imply that the product cb must be positive definite symmetric this paper expands the applicability of the strictly positive realness conditions used for the proofs of stability of adaptive control or control with uncertainty by showing that the not necessarily symmetric cb is only required to have a diagonal jordan form and positive eigenvalues the paper also shows that under the new condition any minimum phase systems can be made strictly positive real via constant output feedback the paper illustrates the usefulness of these extended properties with an adaptive control example

analysis and synthesis of robust control systems via parameter dependent lyapunov functions in this paper the problem of robust stability of systems subject to parametric uncertainties is considered sufficient conditions for the existence of parameter dependent lyapunov functions are given in terms of a criterion which is reminiscent of but less conservative than popov s stability criterion an equivalent frequency domain criterion is demonstrated the relative sharpness of the proposed test and existing stability criteria is then discussed the use of parameter dependent lyapunov functions for robust controller synthesis is then considered it is shown that the search for robustly stabilizing controllers may be limited to controllers with the same order as the original plant a possible synthesis procedure and a numerical example are then discussed

nonlocal interactions by repulsive attractive potentials radial ins stability abstract we investigate nonlocal interaction equations with repulsive attractive radial potentials such equations describe the evolution of a continuum density of particles in which they repulse resp attract each other in the short resp long range we prove that under some conditions on the potential radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state otherwise we prove that it is not possible for a radially symmetric solution to converge weakly toward the spherical shell stationary state we also investigate under which condition it is possible for a non radially symmetric solution to converge toward a singular stationary state supported on a general hypersurface finally we provide a detailed analysis of the specific case of the repulsive attractive power law potential as well as numerical results

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar

longtime behavior and weak strong uniqueness for a nonlocal porous media equation abstract in this manuscript we consider a non local porous medium equation with non local diffusion effects given by a fractional heat operator t u div u p t p s p u 2 in three space dimensions for 3 4 s 1 and analyze the long time asymptotics the proof is based on energy methods and leads to algebraic decay towards the stationary solution u 0 and p 0 in the l 2 r 3 norm the decay rate depends on the exponent s we also show weak strong uniqueness of solutions and continuous dependence from the initial data as a side product of our analysis we also show that existence of weak solutions previously shown in 4 for 3 4 s 1 holds for 1 2 s 1 if we consider our problem in the torus