l induced norm analysis of sampled data systems via piecewise constant and linear approximations this paper deals with the l 1 analysis of linear sampled data systems by which we mean the computation of the l induced norm of linear sampled data systems two computation methods based on piecewise constant and piecewise linear approximations are provided through fast lifting by which the sampling interval 0 h is divided into m subintervals with an equal width even though the central part of the method with the former approximation essentially coincides with a conventional method via fast sample fast hold fsfh approximation after all we show that both methods successfully lead to the upper and lower bounds of the l induced norm whose gap converges to 0 at the rate of 1 m in the former approximation and 1 m 2 in the latter extended approximation such achievements are in sharp contrast with an existing result on the former i e fsfh approximation which only shows the convergence rate of the error in the resulting estimate of the l induced norm without providing any readily computable upper and lower bounds a numerical example is given to illustrate the effectiveness of these methods

rational spl lscr sup 1 suboptimal compensators for continuous time systems the persistent disturbance rejection problem spl lscr sup 1 optimal control for continuous time systems leads to nonrational compensators even for single input single output systems as noted in dahleh and pearson 1987 the difficulty of physically implementing these controllers suggests that the most significant application of the continuous time spl lscr sup 1 theory is to furnish achievable performance bounds for rational controllers in this paper the authors use the theory of positively invariant sets to provide a design procedure based upon the use of the discrete euler approximating system for suboptimal rational spl lscr sup 1 controllers with a guaranteed cost the main results of the paper show that i the spl lscr sup 1 norm of a continuous time system is bounded above by the l sup 1 norm of an auxiliary discrete time system obtained by using the transformation z 1 rs and ii the proposed rational compensators yield spl lscr sup 1 cost arbitrarily close to the optimum even in cases where the design procedure proposed in the above mentioned paper fails due to the existence of plant zeros on the stability boundary

twisted fourier mukai functors due to a theorem by orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of fourier mukai type we extend this result to the case of bounded derived categories of twisted coherent sheaves and at the same time we weaken the hypotheses on the functor as an application we get a complete description of the exact functors between the abelian categories of twisted coherent sheaves on smooth projective varieties

on the 0 dimensional cusps of the kahler moduli of a k3 surface let s be a complex algebraic k3 surface it is proved that the 0 dimensional cusps of the kahler moduli of s are in one to one correspondence with the twisted fourier mukai partners of s as a result a counting formula for the 0 dimensional cusps of the kahler moduli is obtained applications to rational maps between k3 surfaces are given when the picard number of s is 1 the bijective correspondence is calculated explicitly by using the fricke modular curve

a sum theorem for fpv operators and normal cones abstract in his 2008 book from hahn banach to monotonicity s simons mentions that the proof of lemma 41 3 which was presented in the previous edition of his book minimax and monotonicity in 1998 is incorrect and one does not know if this lemma and its consequences are true the aim of this short note is not only to give a proof to the mentioned lemma but also to improve upon it by relaxing one of its assumptions

maximal monotonicity criteria for the composition and the sum under weak interiority conditions the main goal of this article is to present several new results on the maximality of the composition and of the sum of maximal monotone operators in banach spaces under weak interiority conditions involving their domains direct applications of our results to the structure of the range and domain of a maximal monotone operator are discussed the last section of this note studies continuity properties of the duality product between a banach space x and its dual x with respect to topologies compatible with the natural duality x x x x

fitzpatrick functions and continuous linear monotone operators the notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex lower semicontinuous and proper function and any not necessarily symmetric continuous linear positive operator it was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing fitzpatrick functions in this paper we study fitzpatrick functions of continuous linear monotone operators defined on a hilbert space a novel characterization of skew operators is presented a result by brezis and haraux is reproved using the fitzpatrick function we investigate the fitzpatrick function of the sum of two operators and we show that a known upper bound is actually exact in finite dimensional and more general settings cyclic monotonicity properties are also analyzed and closed forms of the fitzpatrick functions of all orders are provided for all rotators in the euclidean plane

maximal monotonicity and cyclic monotonicity arising in nonsmooth lur e dynamical systems we study a precomposition of a maximal monotone operator with linear mappings which preserves the maximal monotonicity in the setting of reflexive banach spaces instead of using the adjoint of such linear operators as in the usual precomposition we consider a more general situation involving operators which satisfy the so called passivity condition we also provide similar analysis for the preservation of the maximal cyclic monotonicity these results are applied to derive existence results for nonsmooth lur e dynamical systems

new properties of forward backward splitting and a practical proximal descent algorithm abstract in this paper we discuss a proximal descent algorithm for finding a zero of the sum of two maximal monotone operators in a real hilbert space some new properties of forward backward splitting are given which extend the well known properties of the usual projection then they are used to analyze the weak convergence of the proximal descent algorithm without assuming lipschitz continuity of the forward operator we also give a new technique of choosing trial values of the step length involved in an armijo like condition which returns the not necessarily decreasing step length self adaptively rudimentary numerical experiments show that it is effective in practical implementations

rational suboptimal continuous time controller design we obtain rational suboptimal continuous time solutions to some optimal control problems specified via time domain performance criteria these include spl lscr sub 1 and spl lscr sub spl infin norms and peak overshoot the approach uses linear semi infinite programming to compute weights for a given finite set of rational basis functions and makes the best possible use of the basis the closed loop transfer functions obtained satisfy appropriate interpolation constraints for internal stability and the formulation allows additional linear constraints to be incorporated

fitzpatrick functions cyclic monotonicity and rockafellar s antiderivative several deeper results on maximal monotone operators have recently found simpler proofs using fitzpatrick functions in this paper we study a sequence of fitzpatrick functions associated with a monotone operator the first term of this sequence coincides with the original fitzpatrick function and the other terms turn out to be useful for the identification and characterization of cyclic monotonicity properties it is shown that for any maximal cyclically monotone operator the pointwise supremum of the sequence of fitzpatrick functions is closely related to rockafellar s antiderivative several examples are explicitly computed for the purpose of illustration in contrast to rockafellar s result a maximal 3 cyclically monotone operator need not be maximal monotone a simplified proof of asplund s observation that the rotation in the euclidean plane by n is n cyclically monotone but not n 1 cyclically monotone is provided the fitzpatrick family of the subdifferential operator of a sublinear and of an indicator function is studied in detail we conclude with a new proof of moreau s result concerning the convexity of the set of proximal mappings

l_ 1 discretization for sampled data controller synthesis via piecewise linear approximation this paper develops a new discretization method with piecewise linear approximation for the l 1 optimal controller synthesis problem of sampled data systems which is the problem of minimizing the l infty induced norm of sampled data systems we apply fast lifting on the top of the lifting technique by which the sampling interval 0 h is divided into m subintervals with an equal width the signals on each subinterval are then approximated by linear functions by introducing two types of linearizing operators for input and output which leads to piecewise linear approximation of sampled data systems by using the arguments of preadjoint operators we provide an important inequality that forms a theoretical basis for tackling the l 1 optimal controller synthesis problem of sampled data systems more efficiently than the conventional method more precisely a mathematical basis for the piecewise linear approximation method associated with the convergence rate is shown through this inequality and this suggests that the piecewise linear approximation method may drastically outperform the conventional method in the l 1 optimal controller synthesis problem of sampled data systems we then provide a discretization procedure of sampled data systems by which the l 1 optimal controller synthesis problem is converted to the discrete time l 1 optimal controller synthesis problem finally effectiveness of the proposed method is demonstrated through a numerical example

a new condition for maximal monotonicity via representative functions abstract in this paper we give a weaker sufficient condition for the maximal monotonicity of the operator s a t a where s x x t y y are two maximal monotone operators a x y is a linear continuous mapping and x y are reflexive banach spaces we prove that our condition is weaker than the generalized interior point conditions given so far in the literature this condition is formulated using the representative functions of the operators involved in particular we rediscover some sufficient conditions given in the past using the so called fitzpatrick function for the maximal monotonicity of the sum of two maximal monotone operators and for the precomposition of a maximal monotone operator with a linear operator respectively

maximal monotonicity for the precomposition with a linear operator we give the weakest constraint qualification known to us that ensures the maximal monotonicity of the operator a circ t circ a when a is a linear continuous mapping between two reflexive banach spaces and t is a maximal monotone operator as a special case we get the weakest constraint qualification that guarantees the maximal monotonicity of the sum of two maximal monotone operators on a reflexive banach space then we give a weak constraint qualification assuring the bre zis haraux type approximation of the range of the subdifferential of the precomposition to a of a proper convex lower semicontinuous function in nonreflexive banach spaces extending and correcting in a special case an older result due to riahi

autoconjugate representers for linear monotone operators monotone operators are of central importance in modern optimization and nonlinear analysis their study has been revolutionized lately due to the systematic use of the fitzpatrick function pioneered by penot and svaiter a topic of recent interest has been the representation of maximal monotone operators by so called autoconjugate functions two explicit constructions were proposed the first by penot and z linescu in 2005 and another by bauschke and wang in 2007 the former requires a mild constraint qualification while the latter is based on the proximal average we show that these two autoconjugate representers must coincide for continuous linear monotone operators on reflexive spaces the continuity and the linearity assumption are both essential as examples of discontinuous linear operators and of subdifferential operators illustrate furthermore we also construct an infinite family of autoconjugate representers for the identity operator on the real line

on the maximality of the sum of two maximal monotone operators abstract a wide variety of problems involving nonlinear partial differential equations subject to boundary conditions may be shown to have solutions by establishing that the associated differential operators satisfy a certain technical condition this condition called maximal monotonicity allows the use of a well developed abstract theory which includes results about existence regularity etc of solutions of operator equations it is quite useful therefore to have easily verified conditions which imply that an operator is maximal monotone frequently an operator may be regarded as the sum of simpler components this paper gives a new sufficient condition which guarantees that the sum of two maximal monotone operators is again maximal monotone author

homogenization of a parabolic model of ferromagnetism this work deals with the homogenization of hysteresis free processes in ferromagnetic composites a degenerate quasilinear parabolic equation is derived by coupling the maxwell ohm system without displacement current with a nonlinear constitutive law b t curl a xe curlh curle a b h xe here a is a periodic positive definite matrix y is maximal monotone and periodic in y e a is an applied field and e 0 an associated initial and boundary value problem is represented by a minimization principle via an idea of fitzpatrick as e 0 a two scale problem is obtained via two scale convergence and an equivalent coarse scale formulation is derived this homogenization result is then retrieved via convergence and the continuity of the solution with respect to the operator and the matrix a is also proved this is then extended to some relaxation dynamics

characterizations and continuity properties for maximal monotone operators with non empty domain interior abstract in the context of general banach spaces characterizations for the maximal monotonicity of operators with non empty domain interior as well as stronger continuity properties of such operators are provided

enlargements of positive sets in this paper we introduce the notion of enlargement of a positive set in ssd spaces to a maximally positive set a we associate a family of enlargements e a and characterize the smallest and biggest element in this family with respect to the inclusion relation we also emphasize the existence of a bijection between the subfamily of closed enlargements of e a and the family of so called representative functions of a we show that the extremal elements of the latter family are two functions recently introduced and studied by stephen simons in this way we extend to ssd spaces some former results given for monotone and maximally monotone sets in banach spaces

stability results for doubly nonlinear differential inclusions by variational convergence we present a stability result for a wide class of doubly nonlinear equations featuring general maximal monotone operators and possibly nonconvex and nonsmooth energy functionals the limit analysis consists in the reformulation of the differential evolution as a scalar energy conservation equation with the aid of the so called fitzpatrick theory for the representation of monotone operators in particular our result applies to the vanishing viscosity approximation of rate independent systems

further results on rational approximations of spl lscr sup 1 optimal controllers the continuous time persistent disturbance rejection problem spl lscr sup 1 optimal control leads to nonrational compensators even for siso systems the difficulty of physically implementing these controllers suggests that the most significant applications of the continuous time spl lscr sup 1 theory is to furnish bounds for the achievable performance of the plant previously two different rational approximations of the optimal spl lscr sup 1 controller were developed by ohta et al 1992 and by blanchini and sznaier 1994 in this paper the authors explore the connections between these two approximations the main result of the paper shows that both approximations belong to the same subset spl omega sub t of the set of rational approximations and that the method proposed by blanchini and sznaier gives the best approximation in the sense of providing the tightest upper bound of the approximation error among the elements of this subset additionally the authors exploit the structure of the dual to the spl lscr sup 1 optimal control problem to obtain rational approximations with approximation error smaller than a prespecified bound

convex analysis can be helpful for the asymptotic analysis of monotone operators asymptotic analysis of monotone operators we use representations of maximal monotone operators for studying recession or asymptotic operators associated to maximal monotone operators such a concept is useful for dealing with unboundedness

the maximality of the sum of monotone operators in banach space and an application to hemivariational inequalities abstract in this paper we give some conditions under whicht fis maximal monotone in the banach spacex not necessarily reflexive wheretis a monotone operator fromxintox and fis the subdifferential of a proper lower semicontinuous convex functionf fromxinto r we also give an application to hemivariational inequalities

applications of convex analysis within mathematics in this paper we study convex analysis and its theoretical applications we first apply important tools of convex analysis to optimization and to analysis we then show various deep applications of convex analysis and especially infimal convolution in monotone operator theory among other things we recapture the minty surjectivity theorem in hilbert space and present a new proof of the sum theorem in reflexive spaces more technically we also discuss autoconjugate representers for maximally monotone operators finally we consider various other applications in mathematical analysis

equivalences of twisted k3 surfaces we prove that two derived equivalent twisted k3 surfaces have isomorphic periods the converse is shown for k3 surfaces with large picard number it is also shown that all possible twisted derived equivalences between arbitrary twisted k3 surfaces form a subgroup of the group of all orthogonal transformations of the cohomology of a k3 surface

a priori estimates for the fitzpatrick function abstract new perspectives proofs and some extensions of known results are presented concerning the behavior of the fitzpatrick function of a monotone type operator in the general context of a locally convex space

rational l sub spl infin suboptimal controllers for siso continuous time systems we study the problem of minimizing the weighted amplitude of the time response due to a given fixed input signal for single input single output siso continuous time systems and focus on obtaining rational suboptimal solutions an eas euler approximating system based method is proposed for designing a rational l sub spl infin suboptimal controller for siso systems it is shown that this rational approximation is the best one among a set of rational approximations in the sense of providing the tightest upper bound and that it can approximate the optimal cost arbitrarily close

persistent disturbance rejection via static state feedback in contrast with spl hscr sub spl infin and spl hscr sub 2 control theories the problem of persistent disturbance rejection l sup 1 optimal control leads to dynamic controllers even when the states of the plant are available for feedback using viability theory shamma showed 1993 in a nonconstructive way that in the state feedback case the same performance achieved by any dynamic linear time invariant controller can be achieved using memoryless nonlinear state feedback in this paper we give an alternative constructive proof of these results for discrete and continuous time systems the main result of the paper shows that in both cases the l sup 1 norm achieved by any stabilizing state feedback linear dynamic controller can be also achieved using a memoryless variable structure controller

scale transformations in the homogenization of nonlinear magnetic processes a class of nonlinear first order processes is formulated as a minimization principle in the presence of oscillating data a two scale model is then derived via nguetseng s notion of two scale convergence the dependence on the fine scale variable is eliminated by averaging with respect to the fine scale scale integration or upscaling conversely any two scale solution is retrieved from a coarse scale one scale disintegration or downscaling these results are first developed in a general functional framework and are then applied to the homogenization of a relaxation dynamics in magnetic composites a x e be t be x e he be 0 he j x e 0 here j is a prescribed current density a y is a positive definite symmetric tensor y is a maximal monotone operator both are assumed to depend periodically on the fine scale variable y the homogenized problem consists in coupling the magnetostatic equations with a maximal monotone relation b h that is local in space but not in time this scale transformation procedure is also interpreted in terms of single scale convergence

rational approximation of l sub 1 optimal controllers for siso systems the l sub 1 optimal control problem with rational controllers for continuous time systems is considered in which it is shown that the optimal l sub 1 performance index with rational controllers is equal to that of irrational controllers a sequence of rational controllers that approximates the optimal index is constructed convergence properties of such a sequence are studied that the corresponding sequence of objective transfer functions is shown to converge in weak topology in bv r sub in the time domain and uniformly in a wider sense in the frequency domain

the relevance of convex analysis for the study of monotonicity abstract we introduce new representations for maximal monotone operators we relate them to previous representations given by krauss fitzpatrick martinez legaz and thera we show their usefulness for the study of compositions and sums of maximal monotone operators

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

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

approximation of distributed delays in this paper we address the approximation problem of distributed delays these elements are convolution operators with kernel having bounded support and appear in the control of time delay systems from the rich literature on this topic we propose a general methodology to achieve such an approximation for this we enclose the approximation problem in the graph topology and working on the convolution banach algebra a constructive approximation is proposed analysis in time and frequency domains is provided this methodology is illustrated on the stabilization control for time delay systems

examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by b f svaiter abstract in this paper we give two explicit examples of unbounded linear maximal monotone operators the first unbounded linear maximal monotone operator s on l 2 is skew we show its domain is a proper subset of the domain of its adjoint s and s is not maximal monotone this gives a negative answer to a recent question posed by svaiter the second unbounded linear maximal monotone operator is the inverse volterra operator t on l 2 0 1 we compare the domain of t with the domain of its adjoint t and show that the skew part of t admits two distinct linear maximal monotone skew extensions these unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened and they are simpler than the example given by phelps simons interesting consequences on fitzpatrick functions for sums of two maximal monotone operators are also given

the kernel average for two convex functions and its application to the extension and representation of monotone operators we provide and analyze an average for two convex functions based on a kernel function it covers several known averages such as the arithmetic average epigraphical average and the proximal average when applied to the fitzpatrick function and the conjugate of fitzpatrick function associated with a monotone operator our average produces an autoconjugate also known as selfdual lagrangian which can be used for finding an explicit maximal monotone extension of the given monotone operator this completely settles one of the open problems posed by fitzpatrick in the setting of reflexive banach spaces

minimal antiderivatives and monotonicity abstract we consider settings in convex analysis which give rise to families of convex functions that contain their lower envelope given certain partial data regarding a subdifferential we consider the family of all convex antiderivatives that comply with the given data we prove that this family is not empty and in particular contains a minimal antiderivative under a fairly general assumption on the given data it turns out that the representation of monotone operators by convex functions fits naturally in these settings duality properties of representing functions are also captured by these settings and the gap between the fitzpatrick function and the fitzpatrick family is filled by this broader sense of minimality of the fitzpatrick function

l 1 optimal compensators for continuous time systems previous work has been concerned with minimizing the l 1 norm of an error pulse response for discrete time siso 1 and mimo 2 systems in this paper we study the problem of minimizing the l1 norm of the error impulse response for siso continuous time systems this problem is quite different from the discrete time problem in that irrational solutions are obtained even when the problem data are rational two methods are suggested for the solution of the continuous time problem an exact method which leads to a finite dimensional nonlinear programming problem and an approximate method which leads to a linear programming problem

the sum of a maximal monotone operator of type fpv and a maximal monotone operator with full domain is maximal monotone abstract the most important open problem in monotone operator theory concerns the maximal monotonicity of the sum of two maximal monotone operators provided that rockafellar s constraint qualification holds in this paper we prove the maximal monotonicity of a b provided that a and b are maximal monotone operators such that dom a int dom b a n dom b is of type fpv and dom a dom b dom b the proof utilizes the fitzpatrick function in an essential way

optimal rejection of persistent bounded disturbances in this paper we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude but is otherwise arbitrary the objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance mathematically this corresponds to requiring uniformly good disturbance rejection over all time since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used see 7 sect 5 6 the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose l 2 norm is small as is the case with the currently popular h infty theory the present theory is a natural counterpart to the existing theory of optimal disturbance rejection the so called h infty theory which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square integrable and has unit energy it is shown that the problem studied here has quite different features from its predecessor complete solutions to the problem are given in several important cases including those where the plant is minimum phase or when it has only a single unstable zero in other cases procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers

on the generalized parallel sum of two maximal monotone operators of gossez type d abstract the generalized parallel sum of two monotone operators via a linear continuous mapping is defined as the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the second one with the linear continuous mapping in this article by assuming that the operators are maximal monotone of gossez type d we provide sufficient conditions of both interiority and closedness type for guaranteeing that their generalized sum via a linear continuous mapping is maximal monotone of gossez type d too this result will follow as a particular instance of a more general one concerning the maximal monotonicity of gossez type d of an extended parallel sum defined for the maximal monotone extensions of the two operators to the corresponding biduals

composition duality and maximal monotonicity this paper shows how the attouch thera duality principle for operator inclusions can be extended to compositions of multifunctions so that the primal and dual inclusions may involve operators between different pairs of spaces we first present the extension and give an example from computational economics to demonstrate that it has practical utility then we consider a particular case often found in applications involving real hilbert spaces and maximal monotone operators we show that in this case our duality framework includes those previously developed by rockafellar mosco and gabay finally we demonstrate that in this case under very simple hypotheses the duality transformation preserves the maximal monotonicity of the operators involved this proof uses an apparently new criterion for maximal monotonicity of operators of the form l tl where t is maximal monotone and l is linear and continuous with adjoint l

twisted fourier mukai number of a k3 surface we give a counting formula for the twisted fourier mukai partners of a projective k3 surface as an application we describe all twisted fourier mukai partners of a projective k3 surface of picard number 1

lipschitz regularity for elliptic equations with random coefficients we develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients the main result is a large scale l type estimate for the gradient of a solution the estimate is proved with optimal stochastic integrability under a one parameter family of mixing assumptions allowing for very weak mixing with non integrable correlations to very strong mixing for example finite range of dependence we also prove a quenched l2 estimate for the error in homogenization of dirichlet problems the approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence form equations