on the long time behavior of type iii ricci flow solutions we show that three dimensional homogeneous ricci flow solutions that admit finite volume quotients have long time limits given by expanding solitons we show that the same is true for a large class of four dimensional homogeneous solutions we give an extension of hamilton s compactness theorem that does not assume a lower injectivity radius bound in terms of riemannian groupoids using this we show that the long time behavior of type iii ricci flow solutions is governed by the dynamics of an mathbb r action on a compact space

h control for discrete time linear systems with frobenius norm bounded uncertainties in this paper we consider the problems of robust stability and control for the class of uncertain discrete time linear systems with frobenius norm bounded parameter uncertainties in all matrices of the system and output equations necessary and sufficient conditions for the above problems are proposed a linear static state feedback control law is designed which is in terms of a riccati inequality the results obtained here show that the robust control problem of the uncertain system is equivalent to the control problem for a corresponding uncertainty free system

h sub infinity optimal control with state feedback a h sub infinity optimal control problem in which the measured outputs are the states of the plant is considered the main result shows that the infimum of the norm of the closed loop transfer function using linear static state feedback equals the infimum of the norm of the closed loop transfer function over all stabilizing dynamic even nonlinear time varying state feedback controllers

sliding mode control of linear systems with mismatched uncertainties abstract it is well known that sliding mode control is robust to matched uncertainties that lie in the range space of the input matrix however many systems are affected by mismatched uncertainties and yet do not enjoy the matching conditions it is the purpose of this paper to present a new dynamical approach of sliding variable formulation which when the system is in sliding mode can explicitly deal with mismatched uncertainties our first step is to treat certain states as inputs to a reduced order system and use adaptive techniques to design fictitious controllers for these inputs which can then tackle the mismatched uncertainties the second step is to use sliding control to realize the adaptive fictitious controllers the design is systematic modular and intuitively simple

a stabilization algorithm for a class of uncertain linear systems abstract this paper presents an algorithm for the stabilization of a class of uncertain linear systems the uncertain systems under consideration are described by state equations which depend on time varying unknown but bounded uncertain parameters the construction of the stabilizing controller involves solving a certain algebraic riccati equation furthermore the solution to this riccati equation defines a quadratic lyapunov function which is used to establish the stability of the closed loop system this leads to a notion of quadratic stabilizability it is shown that the stabilization procedure will succeed if and only if the given uncertain linear system is quadratically stabilizable the paper also deals with a notion of overbounding for uncertain linear systems this procedure enables the stabilization algorithm to be applied to a larger class of uncertain linear systems also included in the paper are results which indicate the degree of conservativeness introduced by this overbounding process

a linear matrix inequality approach for robust control of systems with delayed states abstract this paper addresses the robust stabilization problem for uncertain systems with delayed states the parameter uncertainties are unknown but norm bounded and the delay is time varying in this study a linear matrix inequality lmi approach for robust stability analysis for the nominal unforced system and a method for robust stabilization for a class of uncertain delay systems via linear memoryless state feedback control are presented the results depend on the size of the delay and are given in terms of linear matrix inequalities

h sub spl infin state feedback controller design for discrete time fuzzy systems using fuzzy weighting dependent lyapunov functions for discrete time takagi sugeno ts fuzzy systems we propose an h sub spl infin state feedback fuzzy controller associated with a fuzzy weighting dependent lyapunov function the controller which is designed via parameterized linear matrix inequalities plmis employs not only the current time but also the one step past information on the time varying fuzzy weighting functions appropriately selecting the structures of variables in the plmis allows us to find an lmi formulation as a special case

a direct characterization of l sub 2 gain controllers for lpv systems in this paper a class of linear parameter dependent output feedback controllers that satisfy quadratic stability and an induced l sub 2 norm bound for a given linear parameter varying lpv plant is considered by using a parameter independent common lyapunov function the solvability conditions are expressed in terms of finite dimensional linear matrix inequalities lmi evaluated at the extreme points of the admissible parameter set conditions under which strictly proper controllers can be used are obtained by restricting some of the controller matrices to be constant the input and output matrices can be parameter varying without destroying the problem s convexity cases where the controller matrices can be obtained without interpolation are also discussed thereby simplifying the implementation of the controller a numerical example is included which demonstrates the application of the results

quadratic stability with real and complex perturbations it is shown that the equivalence between real and complex perturbations in the context of quadratic stability to linear fractional unstructured perturbations does not hold when the perturbations are block structured for a limited class of problems quadratic stability in the face of structured complex perturbations is equivalent to a particular class of scaled norms and hence appropriate synthesis techniques coupled with diagonal constant scalings can be used to design quadratically stable systems

a new result for the porous medium equation derived from the ricci flow given r 2 with a good complete metric we show that the unique solution of the ricci flow approaches a soliton at time infinity solitons are solutions of the ricci flow which move only by diffeomorphism the ricci flow on r 2 is the limiting case of the porous medium equation when m is zero the results in the ricci flow may therefore be interpreted as sufficient conditions on the initial data which guarantee that the corresponding unique solution for the porous medium equation on the entire plane asymptotically behaves like a soliton solution

structured and simultaneous lyapunov functions for system stability problems abstract it is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic lyapunov function of a certain structure which establishes stability of x ax for some appropriate a the existence of such a lyapunov function can be determined by solving a convex program we present several numerical methods for these optimization problems a simple numerical example is given

brief rbust control of linear systems with real parametric uncertainty we discuss the problem of robust controller synthesis to provide quadratic stability and a desirable disturbance attenuation level through an appropriately small l 2 gain for systems with time varying real parametric uncertainty through the use of skewsymmetric matrices the conservatism of standard scaled h approach is reduced while the full state feedback problem is convex the output feedback problem is not a set of conditions under which the design of output feedback controllers can be broken into two sequential convex problems is presented the results are compared with recent results concerning the design of parameter varying controllers and a simple result regarding the mixed problem where some of the unknown parameters can be measured on line is discussed

on uniform stabilization of discrete time linear parameter varying control systems for discrete time polytopic linear parameter varying systems the uniform exponential stability which is equivalent to the asymptotic stability and includes the quadratic stability as a special case is characterized by the union of an increasing family of linear matrix inequality conditions on the other hand in certain cases nonconservative synthesis of uniformly stabilizing controllers is achieved via a system of finite number of linear matrix inequalities if the controller is allowed to have finite memory of past parameters two such cases are robust state feedback in a relaxed sense against polytopic parameter variations and multiple output injections under polytopic fusion rules

comments on sliding mode control of linear systems with mismatched uncertainties abstract in the above paper kwan 1995 sliding mode control of linear systems with mismatched uncertainties automatica 31 303 307 a sliding mode control for handling mismatched uncertainties explicitly was proposed in this correspondence it is pointed out that the proof of stability the adaptation law and the control given by kwan are incorrect previously published results for handling mismatched uncertainties explicitly are cited an alternative treatment of the problem is suggested

linear and dynamical stability of ricci flat metrics we can talk about two kinds of stability of the ricci flow at ricci flat metrics one of them is a linear stability defined with respect to perelman s functional f see 1 page 5 the other one is a dynamical stability and it refers to a convergence of a ricci flow starting at any metric in a neighborhood of a considered ricci flat metric we show that dynamical stability implies linear stability we also show that a linear stability together with the integrability assumption implies dynamical stability as a corollary we get a stability result for k3 surfaces part of which has been done in 11 corollary 4 15 theorem 4 16 our stability result applies to calabi yau manifolds as well

control of lpv systems with partly measured parameters the control of linear parameter varying lpv systems with time varying real uncertain parameters where only some of the parameters are measured and available for feedback is considered the control objectives are internal stability and disturbance attenuation in the sense of a bounded induced l sub 2 norm from the disturbance to the controlled output using parametric lyapunov functions the solvability conditions for dynamic output feedback controllers that depend on the measured parameters are expressed in terms of a set of linear matrix inequalities lmis and an additional coupling constraint which destroys the convexity of the overall problem by transforming the coupling constraint into a rank condition the problem is recast into a rank minimization problem with lmi constraints an example is included that demonstrates the application of the results

a unified approach to lmi based reduced order self scheduling control synthesis abstract we consider a reduced order controller synthesis for a general class of control specifications for linear parameter varying lpv systems when some of state variables are exactly available the class is defined in an abstract manner so that it uniformly deals with many significant specifications a necessary and sufficient condition for the existence of a reduced order controller is given in terms of linear matrix inequalities lmis we also show that the order of the controller can be reduced by the number of the state variables exactly available in the measurements moreover in the case of linear time invariant lti systems a parameterization of all desirable reduced order lti controllers is given by means of solutions of lmis the results in this paper generalize the class of control specifications in which a reduced order controller exists making it possible to synthesize a reduced order controller based on lmis for multi objective control specifications furthermore these results uniformly describe and generalize the existing results on synthesis of a constant state and a full order output feedback controller for lti and lpv systems such that the specification is given by the existence of a constant positive definite matrix

ricci soliton homogeneous nilmanifolds we study a notion weakening the einstein condition on a left invariant riemannian metric g on a nilpotent lie groupn we consider those metrics satisfying ric g ci d for some c in mathbb r and some derivationd of the lie algebra mathfrak n ofn where ric g denotes the ricci operator of n g this condition is equivalent to the metric g to be a ricci soliton we prove that a ricci soliton left invariant metric on n is unique up to isometry and scaling the following characterization is also given n g is a ricci soliton if and only if n g admits a metric standard solvable extension whose corresponding standard solvmanifold s tilde g is einstein this gives several families of new examples of ricci solitons by a variational approach we furthermore show that the ricci soliton homogeneous nilmanifolds n g are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set

on designing of sliding mode control for stochastic jump systems in this note we consider the problems of stochastic stability and sliding mode control for a class of linear continuous time systems with stochastic jumps in which the jumping parameters are modeled as a continuous time discrete state homogeneous markov process with right continuous trajectories taking values in a finite set by using linear matrix inequalities lmis approach sufficient conditions are proposed to guarantee the stochastic stability of the underlying system then a reaching motion controller is designed such that the resulting closed loop system can be driven onto the desired sliding surface in a limited time it has been shown that the sliding mode control problem for the markovian jump systems is solvable if a set of coupled lmis have solutions a numerical example is given to show the potential of the proposed techniques

ray congruences that generate conformal foliations given a riemannian 3 manifold m3 g0 endowed with a unit vector field u0 that is tangent to a conformal foliation we require that the pair extend to a space time m g endowed with a space like unit vector field ut in such a way that ut simultaneously generates null geodesics and is tangent to a conformal foliation on space like slices t constant the property that the foliation be conformal is intimately related to cr geometry

nonradial solvability structure of super diffusive nonlinear parabolic equations we study the solvability of the cauchy problem for the nonlinear parabolic equation u div u m 1 u at when m 0 then there is no local solution of the initial value problem on the other hand if such an interval does not exist then the initial value problem is locally solvable and the time of existence can be estimated explicitly

an lmi based switching surface design method for a class of mismatched uncertain systems in this note we consider the problem of designing linear sliding surfaces for multivariable uncertain systems with mismatched uncertainties in the state matrix in terms of linear matrix inequalities we give a new invariance condition guaranteeing the existence of a linear switching surface such that the reduced order equivalent sliding mode dynamics restricted to the switching surface is not only stable but completely invariant to mismatched uncertainties we also give an explicit formula of such linear switching surfaces together with a design example

h sub 2 norm checking for systems with parameter uncertainties this paper studies the problem of h sub 2 norm checking for a mimo system with parameter uncertainties under the assumptions that the coefficients of the numerator polynomial of the system s transfer function are multi affine functions of the uncertain parameter that varies in a convex polytope and that the coefficients of the denominator polynomial of the system s transfer function are affine functions of the uncertain parameter which varies in a convex polytope a vertex checking result is obtained

robust hyperplane design in multivariable variable structure control systems the problem of designing a stable sliding mode giving robust performance of a variable structure control system is studied a set of specified eigenvalues is assigned to the closed loop feedback system during the sliding mode the design philosophy seeks to build on the known robustness and invariance properties associated with the use of discontinuous or continuous non linear controls by assigning the eigenvectors associated with the sliding mode eigenvalues in a manner that leads to a robust control scheme a previously developed canonical form for the hyperplane design problem is again employed the design technique centres on a recently published eigenvalue assignment algorithm and is illustrated by the inclusion of an example

optimal sliding mode control for linear time delay systems with sinusoidal disturbances this paper develops a successive approximation approach saa of optimal sliding mode control smc for linear time delay systems with sinusoidal disturbances a sequence of two point boundary value tpbv problems with both time delay and time advance terms is derived from the optimal sliding mode design according to the saa the original tpbv problems are transformed into a sequence of linear tpbv problems without delay and advance terms the obtained smc ensures that the state trajectories reach the sliding surface in finite time and remain on it thereafter the stability of the sliding mode is proved a numerical simulation is employed to verify the effectiveness of the proposed approach

optimal constant i o similarity scaling for full information and state feedback control problems abstract in robust control problems capturing all robustness and performance objectives in a single h norm cost function is impossible an alternative approach still untilizing the h norm involves diagonal similarity scaling of certain closed loop transfer functions the set of allowable diagonal scalings is problem dependent and reflects assumptions about the uncertainty and desired performance objectives the scaling set considered here is a prescribed convex set of positive definite matrices we consider the optimal constant scaling problem for the full information h control problem the solution is obtained by transforming the original problem into a convex feasibility problem specifically a structured linear matrix inequality in special cases solvability of the full information problem is equivalent to solvability of the state feedback problem

constructing discontinuity surfaces for variable structure systems a lyapunov approach the problem of constructing discontinuity surfaces in variable structure systems is studied from a lyapunov point of view a switching surface determined by the control coefficient matrix and the associated lyapunov function is able to ensure asymptotic stability for the system in sliding mode the proposed method may also be used for systems with nonlinear dynamics and for linear systems with delays

sliding mode design via quadratic performance optimization with pole clustering constraint we consider a novel method for designing the sliding mode that minimizes the quadratic performance while keeping a pole clustering constraint our approach is based on the manipulations of linear matrix inequalities lmis imposed by the design objectives for this purpose we newly propose lmi conditions for the quadratic performance optimization and the pole clustering problem respectively in a full order state then they are combined in the lmi framework that is typically devised for the sliding mode design in the convex form an effort is made to reduce the generic conservatism by allowing different lyapunov matrices in addition a class of polytope uncertain systems is addressed to illustrate the advantages of the proposed method

robust performance of systems with structured uncertainties in state space abstract this paper considers robust performance analysis and state feedback design for systems with time varying parameter uncertainties the notion of a strongly robust h performance criterion is introduced and its applications in robust performance analysis and synthesis for nominally linear systems with time varying uncertainties are discussed and compared with the constant scaled small gain criterion it is shown that most robust performance analysis and synthesis problems under this strongly robust performance criterion can be transformed into linear matrix inequality problems and can be solved through finite dimensional convex programming the results are in general less conservative than those using small gain type criteria

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

a new variable structure controller for affine nonlinear systems with non matching uncertainties the problem of output tracking in siso uncertain non autonomous nonlinear systems is treated in this study it is shown that a nonlinear system experiencing non matching disturbances and uncertainties can be transformed to a form similar to the normal form then a novel variable structure controller with disturbance estimation and parameter adaptation is presented this controller while preserving the robustness properties of sliding control avoids a sliding mode and generates a continuous control signal which results in close output tracking

lmi based sliding surface design for integral sliding mode control of mismatched uncertain systems we propose a linear matrix inequality lmi based sliding surface design method for integral sliding mode control of mismatched uncertain systems the uncertain system under consideration may have mismatched norm bounded uncertainties in the state matrix as well as the input matrix we give a sufficient condition for the existence of a sliding surface guaranteeing asymptotic stability of the full order sliding mode dynamics we also give an lmi characterization of the sliding surface together with an integral sliding mode control law guaranteeing the existence of a sliding mode from the initial time additionally we give an lmi condition of sliding surfaces guaranteeing the alpha stability constraint finally we give a simulation result to show the effectiveness of our method

on singular diffusion equations with applications to self organized criticality we consider solutions of the singular diffusion equation t um 1ux x m 0 associated with the flux boundary condition limx um 1ux x 0 the evolutions defined by this problem depend on both m and we prove existence and stability of traveling wave solutions parameterized by each traveling wave is stable in its appropriate evolution these traveling waves are in l1 for 1 m 0 but have non integrable tails for m 1 we also show that these traveling waves are the same as those for the scalar conservation law ut f u x uxx for a particular nonlinear convection term f u f u m 1993 john wiley sons inc

robust h sub infinity control for linear systems with norm bounded time varying uncertainty robust h sub infinity control design for linear systems with uncertainty in both the state and input matrices is treated a state feedback control design which stabilizes the plant and guarantees an h sub infinity norm bound constraint on disturbance attenuation for all admissible uncertainties is presented the robust h sub infinity control problem is solved via the notion of quadratic stabilization with an h sub infinity norm bound necessary and sufficient conditions for quadratic stabilization with an h sub infinity norm bound are derived the results can be regarded as extensions of existing results on h sub infinity control and robust stabilization of uncertain linear systems

brief designing robust sliding hyperplanes for parametric uncertain systems a riccati approach in this paper we propose a method to design robust sliding hyperplanes in the presence of mismatched parametric uncertainty based on quadratic stability the robust sliding hyperplane is constructed from a riccati inequality associated with quadratic stabilizability the proposed method enables us to deal with structured uncertainty and optimize the sliding motion by applying the guaranteed cost control idea

adaptive sliding mode control of multi input nonlinear systems with perturbations to achieve asymptotical stability a design of adaptive sliding surfaces for a class of multi input mi nonlinear systems with matched and mismatched perturbations is proposed in this note to solve regulation problems based on the lyapunov stability theorem and backstepping technique by using some adaptive gains designed in the sliding surface function the virtual input and the controllers not only the mismatched perturbations are automatically overcome during the sliding mode but also the property of asymptotical stability of controlled systems is achieved at the same time furthermore the knowledge of the upper bound of partial perturbations is not required controlling a two link robot manipulator is given to demonstrate the feasibility of the proposed control scheme

harmonic morphisms with one dimensional fibres on einstein manifolds we prove that from an einstein manifold of dimension greater than or equal to five there are just two types of harmonic morphism with one dimensional fibres this generalizes a result of r l bryant who obtained the same conclusion under the assumption that the domain has constant curvature

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

robust h control for linear time invariant systems with norm bounded uncertainty in the input matrix abstract this paper focuses on the problem of robust h control design for a class of linear time invariant systems with uncertainty in the state space model we consider uncertain systems with norm bounded parameter uncertainty in the input matrix the paper presents a state feedback control design which stabilizes the plant for all admissible uncertainties and also guarantees an h norm bound constraint on disturbance attenuation paralleling to the theory of robust control the robust h control problem is solved via the notion of quadratic stabilization with an h norm bound constraint necessary and sufficient conditions for quadratic stabilization with an h norm bound are derived it is shown that the solution to this problem involves solving a parameter dependent algebraic riccati equation the results can be regarded as extensions of existing results on robust stabilization of linear uncertain systems and h optimal control

robust sliding mode control for uncertain time delay systems an lmi approach this note is devoted to robust sliding mode control for time delay systems with mismatched parametric uncertainties a delay independent sufficient condition for the existence of linear sliding surfaces is given in terms of linear matrix inequalities based on which the corresponding reaching motion controller is also developed the results are illustrated by an example

logarithmically singular parabolic equations as limits of the porous medium equation abstract let u m be a local weak solution to the porous medium equation u m t w m 0 where w m u m m 1 m it is shown that if u m is locally in l loc r for r 1 2 n uniformly in m and if w m is in l loc p for p n 2 in the space variables uniformly in time then u m contains a subsequence converging in c loc 1 2 to a local weak solution to the logarithmically singular equation u t ln u the result is based on local upper and lower bounds on u m uniform in m the uniform local lower bounds are realized by a harnack type inequality