nonlinear evolution equations with nonmonotonic perturbations soit v un espace de banach reflexif densement et continument plonge dans un espace de hilbert reel h on considere l existence des solutions du probleme aux valeurs initiales du dt au g u f 0 t t u 0 u 0 ou a est un operateur monotone de v dans v g v h est une application continue et f 0 t v est une fonction mesurable

identification problem of two operators for nonlinear systems in banach spaces abstract we consider the identification problem of two operators having different properties for the systems governed by nonlinear evolution equations for the identification problem we show the existence of optimal solutions and present necessary optimality conditions we illustrate the approach on two examples

relaxation in nonconvex optimal control problems described by fractional differential equations abstract we consider the minimization problem of an integral functional with integrand that is not convex in the control on solutions of a control system described by fractional differential equation with mixed nonconvex constraints on the control a relaxation problem is treated along with the original problem it is proved that under general assumptions the relaxation problem has an optimal solution and that for each optimal solution there is a minimizing sequence of the original problem that converges to the optimal solution with respect to the trajectory the control and the functional in appropriate topologies simultaneously

the eigenstructure of finite field trigonometric transforms in this paper we discuss the eigenstructure of finite field trigonometric transforms fftt this transform family includes different types of cosine and sine transforms that correspond in the finite field case to the well known discrete cosine and sine transforms defined over the real numbers more specifically we determine the eigenvalues of the fftt matrices and study their multiplicities we propose procedures for constructing the associated eigenvectors applications of the developed theory to multiuser communication systems and error correcting codes are also suggested

jordan structures of alternating matrix polynomials alternating matrix polynomials that is polynomials whose coefficients alternate between symmetric and skew symmetric matrices generalize the notions of even and odd scalar polynomials we investigate the smith forms of alternating matrix polynomials showing that each invariant factor is an even or odd scalar polynomial necessary and sufficient conditions are derived for a given smith form to be that of an alternating matrix polynomial these conditions allow a characterization of the possible jordan structures of alternating matrix polynomials and also lead to necessary and sufficient conditions for the existence of structure preserving strong linearizations most of the results are applicable to singular as well as regular matrix polynomials

bogolyubov type theorem with constraints induced by a control system with hysteresis effect abstract we consider the problem of minimization of an integral functional with nonconvex with respect to the control integrand we minimize our functional over the solution set of a control system described by two ordinary differential equations subject to a control constraint given by a multivalued mapping with closed nonconvex values the coefficients of the equations and the constraint depend on the phase variables one of the equations contains the subdifferential of the indicator function of a closed convex set depending on the unknown phase variable the equation containing the subdifferential describes an input output relation of hysteresis type along with the original problem we also consider the problem of minimizing the integral functional with the convexified with respect to the control integrand over the solution set of the same system with the convexified control constraint under sufficiently general assumptions we prove that this relaxed problem has an optimal solution which is the limit of a minimizing sequence of the original problem the convergence takes place simultaneously with respect to the trajectory the control and the functional and is uniform in appropriate topologies

optimal control for an adsorbate induced phase transition model this paper is concerned with the optimal control problem for an adsorbate induced phase transition model that is we show the existence of the optimal control and derive the optimality conditions by showing the differentiability of the cost functional with respect to the control

optimal processes governed by integral equation equations with unilateral constraint abstract an optimal process with unilateral constraint is considered using relaxed controls existence of optimal control is automatic further problems with linear and certain quadratic cost functions with linear integral and state constraints are such that the controls are actually ordinary i e bounded measurable functions in addition if the control set is a convex polyhedron the optimal controls take their values in the vertices of the polyhedron control processes governed by integral equations have been considered by v r vinokurov siam 7 1969 324 355 however there existence is not shown for the general problem and optimality conditions are not available for the problem when state constraints are present furthermore the above work has some errors as pointed out by l w neustadt and j warga siam 8 1970 572 finally we point out that the basic idea in our approach here is the same as in a forthcoming paper j optim theory appl in press

structured strong linearizations from fiedler pencils with repetition ii abstract in this paper we give strong linearizations of a matrix polynomial p preserving the skew symmetry or t alternating structure of p the linearizations obtained are of the form s l where l is a block symmetric fiedler pencil with repetition and s is a direct sum of blocks of the form i or i with i the identity matrix this paper is a continuation of 2 where the corresponding problem for p with a symmetric structure was studied and as a consequence the block symmetric fiedler pencils with repetition were characterized

on seminormality of integral functionals and their integrands we present a definition of seminormality which extends classical notions due to tonelli fondamenti di calcolo delle variazioni 1921 mcshane ann scuola norm sup pisa 2 3 1934 pp 181 211 287 315 and cesari traps amer math soc 124 1966 pp 369 412 j optim theory appl 6 1970 pp 114 137 siam j control optim 9 1971 pp 287 315 and applies to both integrands seminormality in the small and their integral functionals seminormality in the large our main result states that under very general hypotheses seminormality in the small and large are equivalent by introducing a notion called nagumo tightness we show that the usual sufficient conditions for the lower semicontinuity of an integral functional also imply a form of seminormality of the integral functional necessary conditions for the lower semicontinuity of an integral functional can also be obtained from our results

a simplified approach to fiedler like pencils via block minimal bases pencils abstract the standard way of solving the polynomial eigenvalue problem associated with a matrix polynomial is to embed the matrix coefficients of the polynomial into a matrix pencil transforming the problem into an equivalent generalized eigenvalue problem such pencils are known as linearizations many of the families of linearizations for matrix polynomials available in the literature are extensions of the so called family of fiedler pencils these families are known as generalized fiedler pencils fiedler pencils with repetition and generalized fiedler pencils with repetition or fiedler like pencils for simplicity the goal of this work is to unify the fiedler like pencils approach with the more recent one based on strong block minimal bases pencils introduced in f m dopico et al 2017 17 to this end we introduce a family of pencils that we have named extended block kronecker pencils whose members are under some generic nonsingularity conditions strong block minimal bases pencils and show that with the exception of the non proper generalized fiedler pencils all fiedler like pencils belong to this family modulo permutations as a consequence of this result we obtain a much simpler theory for fiedler like pencils than the one available so far moreover we expect this simplification to allow for further developments in the theory of fiedler like pencils such as global or local backward error analyses and eigenvalue conditioning analyses of polynomial eigenvalue problems solved via fiedler like linearizations

a note on companion matrices abstract we show that the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices n 1 of them being the identity matrix in which a 2 2 identity submatrix in two consecutive rows and columns is replaced by an appropriate 2 2 matrix the remaining being the identity matrix with the last entry replaced by possibly different entry by a certain similarity transformation we obtain a simple new companion matrix in a pentadiagonal form some generalizations are also possible

skew symmetric matrix polynomials and their smith forms we characterize the smith form of skew symmetric matrix polynomials over an arbitrary field f showing that all elementary divisors occur with even multiplicity restricting the class of equivalence transformations to unimodular congruences a smith like skew symmetric canonical form for skew symmetric matrix polynomials is also obtained these results are used to analyze the eigenvalue and elementary divisor structure of matrices expressible as products of two skew symmetric matrices as well as the existence of structured linearizations for skew symmetric matrix polynomials by contrast with other classes of structured matrix polynomials e g alternating or palindromic polynomials every regular skew symmetric matrix polynomial is shown to have a structured strong linearization while there are singular skew symmetric polynomials of even degree for which a structured linearization is impossible for each odd degree we develop a skew symmetric companion form that uniformly provides a structured linearization for every regular and singular skew symmetric polynomial of that degree finally the results are applied to the construction of minimal symmetric factorizations of skew symmetric rational matrices

eigenvalues and equivalent transformation of a trigonometric matrix associated with filter design the n n trigonometric matrix p whose entries are p i j 1 i j 2 cos i j appears in connection with the design of finite impulse response fir digital filters with real coefficients we prove several results about its eigenvalues in particular assuming n 4 we prove that p has one positive and one negative eigenvalue when is an integer while it has two positive and two negative eigenvalues when is not an integer we also show that for not being an integer and a sufficiently large n the two positive eigenvalues converge to n 2 and the two negative eigenvalues to n 2 where 1 2 p 3 8 furthermore an equivalent transformation diagonalizing p is described

optimal design of complex fir filters with arbitrary magnitude and group delay responses this paper presents a method for the frequency domain design of digital finite impulse response filters with arbitrary magnitude and group delay responses the method can deal with both the equiripple design problem and the peak constrained least squares pcls design problem consequently the method can also be applied to the equiripple passbands and pcls stopbands design problem as a special case of the pcls design both the equiripple and the pcls design problems are converted into weighted least squares optimization problems they are then solved iteratively with appropriately updated error weighting functions a novel scheme for updating the error weighting function is developed to incorporate the design requirements design examples are included in order to compare the performance of the filters designed using the proposed scheme and several other existing methods

optimal control of quasilinear parabolic equations this paper deals with optimal control problems governed by quasilinear parabolic equations in divergence form whose cost functional is of lagrangian type our aim is to prove the existence of solutions and derive some optimality conditions to attain this second objective we accomplish the sensitivity analysis of the state equation with respect to the control proving that under some assumptions this relation is g teaux differentiable finally a regularising procedure along with ekeland s variational principle allow us to treat some other problems for which this differentiability property cannot be stated

state constrained relaxed problems for semilinear elliptic equations in this paper we study optimal control problems governed by semilinear elliptic equations in the presence of pointwise state constraints since no convexity condition is assumed on data of the problem we define a relaxed control problem prove the existence of relaxed solutions and give some relaxation results by adapting the penalty method of berkovitz we prove a pontryagin s minimum principle for relaxed solutions in nonqualified form and in qualified form under a stability condition

palindromic companion forms for matrix polynomials of odd degree the standard way to solve polynomial eigenvalue problems p x 0 is to convert the matrix polynomial p into a matrix pencil that preserves its spectral information a process known as linearization when p is palindromic the eigenvalues elementary divisors and minimal indices of p have certain symmetries that can be lost when using the classical first and second frobenius companion linearizations for numerical computations since these linearizations do not preserve the palindromic structure recently new families of pencils have been introduced with the goal of finding linearizations that retain whatever structure the original p might possess with particular attention to the preservation of palindromic structure however no general construction of palindromic linearizations valid for all palindromic polynomials has as yet been achieved in this paper we present a family of linearizations for odd degree polynomials p which are palindromic whenever p is and which are valid for all palindromic polynomials of odd degree we illustrate our construction with several examples in addition we establish a simple way to recover the minimal indices of the polynomial from those of the linearizations in the new family

block minimal bases l ifications of matrix polynomials abstract the standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil known as a strong linearization this process transforms the problem into an equivalent generalized eigenvalue problem however there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials this has motivated the idea of a strong l ification of a matrix polynomial which is a matrix polynomial of degree at most l having the same finite and infinite elementary divisors and the same numbers of left and right minimal indices as the original matrix polynomial we present in this work a novel method for constructing strong l ifications of matrix polynomials of size m n and grade d when l d and l divides nd or md this method is based on a family called strong block minimal bases matrix polynomials and relies heavily on properties of dual minimal bases we show how strong block minimal bases l ifications can be constructed from the coefficients of a given matrix polynomial p we also show that these l ifications satisfy many desirable properties for numerical applications they are strong l ifications regardless of whether p is regular or singular the minimal indices of the l ifications are related to those of p via constant uniform shifts and eigenvectors and minimal bases of p can be recovered from those of any of the strong block minimal bases l ifications in the special case where l divides d we introduce a subfamily of strong block minimal bases matrix polynomials named block kronecker matrix polynomials which is shown to be a fruitful source of companion l ifications

large vector spaces of block symmetric strong linearizations of matrix polynomials abstract given a matrix polynomial p i 0 k i a i of degree k where a i are n n matrices with entries in a field f the development of linearizations of p that preserve whatever structure p might posses has been a very active area of research in the last decade most of the structure preserving linearizations of p discovered so far are based on certain modifications of block symmetric linearizations the block symmetric linearizations of p available in the literature fall essentially into two classes linearizations based on the so called fiedler pencils with repetition which form a finite family and a vector space of dimension k of block symmetric pencils called dl p such that most of its pencils are linearizations one drawback of the pencils in dl p is that none of them is a linearization when p is singular in this paper we introduce new vector spaces of block symmetric pencils most of which are strong linearizations of p the dimensions of these spaces are o n 2 which for n k are much larger than the dimension of dl p when k is odd many of these vector spaces contain linearizations also when p is singular the coefficients of the block symmetric pencils in these new spaces can be easily constructed as k k block matrices whose n n blocks are of the form 0 i n a i or arbitrary n n matrices where is an arbitrary nonzero scalar

optimal control problems governed by some semilinear parabolic equations abstract this paper is concerned with the optimal control problem for the keller segel equations that is as a generalization of ryu and yagi j math anal appl 256 2001 45 66 we derive the optimality conditions for the optimal control problem governed by a semilinear abstract equation of nonmonotone type moreover we prove the uniqueness of the optimal control under some conditions

automatic recovery of eigenvectors and minimal bases of matrix polynomials from generalized fiedler pencils with repetition abstract linearization is a widely used method for solving polynomial eigenproblems in which a matrix polynomial is transformed to a matrix pencil of larger size fiedler pencils generalized fiedler pencils fiedler pencils with repetition fprs and generalized fiedler pencils with repetition gfprs are well known classes of strong linearizations of matrix polynomials the class gfprs is also a rich source of structure preserving strong linearizations of structured matrix polynomials the recovery of eigenvectors minimal bases and minimal indices of matrix polynomials from those of the linearizations is an important task it is well known that eigenvectors and minimal bases of matrix polynomials can be easily recovered from those of the fiedler and generalized fiedler pencils by contrast barring a small subclass of fprs the recovery of eigenvectors and minimal bases of matrix polynomials from those of the fprs and gfprs is still an open problem the purpose of this paper is to fill this gap we consider the class of gfprs which subsumes the class of fprs and describe the recovery of eigenvectors minimal bases and minimal indices of matrix polynomials from those of the gfprs and show that the recovery is operation free

spectral equivalence of matrix polynomials and the index sum theorem abstract the concept of linearization is fundamental for theory applications and spectral computations related to matrix polynomials however recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive because some structured polynomials do not have any linearization with the same structure this phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations motivated by this fact we introduce equivalence relations that allow the possibility of matrix polynomials with coefficients in an arbitrary field to be equivalent with the same spectral structure but have different sizes and degrees these equivalence relations are directly modeled on the notion of linearization and consequently inherit the simplicity applicability and most relevant properties of linearizations simultaneously though they are much more flexible in the possible degrees of equivalent polynomials this flexibility allows us to define in a unified way the notions of quadratification and l ification to introduce the concept of companion form of arbitrary degree and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second frobenius companion forms the properties of l ifications are studied in depth in this process a fundamental result on matrix polynomials the index sum theorem is recovered and extended to arbitrary fields although this result is known in the systems theory literature for real matrix polynomials it has remained unnoticed by many researchers it establishes that the sum of the finite and infinite partial multiplicities together with the left and right minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial the index sum theorem turns out to be a key tool for obtaining a number of significant results on the possible sizes and degrees of l ifications and companion forms on the minimal index preservation properties of companion forms of arbitrary degree as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree this paper presents many new results blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields therefore we have written the paper in an expository and self contained style that makes it accessible to a wide variety of readers

on the optimal control of strongly nonlinear evolution equations abstract in this paper we study optimal control problems governed by nonlinear evolution equations with nonmonotone nonlinearities in the state first we prove an existence theorem then using the penalty method we obtain necessary conditions for optimality from them we derive a bang bang characterization of optimal controls finally an example of a nonlinear parabolic optimal control system is worked out in detail

necessary and sufficient conditions for l 1 strong weak lower semicontinuity of integral functionals soit t 7b f un espace fini de mesure nonatomique x ii ii un espace de banach non separable et v i i un espace de banach reflexif separable dont v est le dual soit f txxxv une fonction 7b f 7b b x v mesurable on etudie l integrale fonctionnelle associee i f l x 1 l v 1 definie par if x v f r x t v t dt

counting fiedler pencils with repetitions abstract we introduce a new notation based on diagrams to deal with fiedler pencils with repetitions fpr and use it to solve several counting problems in particular we give explicit recurrences to count the number of fprs of a given degree d the number of symmetric palindromic and antipalindromic ones where the latter two structures are intended in the sense of 5 we relate these structures to the presence of symmetries in the associated diagrams

structured strong linearizations from fiedler pencils with repetition i abstract in many applications the polynomial eigenvalue problem p x 0 arises with p being a structured matrix polynomial for example palindromic symmetric skew symmetric in order to solve a structured polynomial eigenvalue problem it is convenient to use strong linearizations with the same structure as p to ensure that the symmetries in the eigenvalues due to that structure are preserved in numerical computations in this paper we characterize all the pencils in the family of the fiedler pencils with repetition introduced by vologiannidis and antoniou 25 associated with a square matrix polynomial p that are block symmetric for every matrix polynomial p we show that this family of pencils is precisely the set of all fiedler pencils with repetition that are symmetric when p is when some generic nonsingularity conditions are satisfied these pencils are strong linearizations of p in particular our family strictly contains the standard basis for dl p a k dimensional vector space of symmetric pencils introduced by mackey mackey mehl and mehrmann 20

on eigenvalues and equivalent transformation of trigonometric matrices abstract this paper discusses some issues related to trigonometric matrices arising from the design of finite impulse response fir digital filters a conjecture on the eigenvalues of a trigonometric matrix is posed with a partial proof given a new result is also presented on the related equivalent transformation of this trigonometric matrix into a diagonal matrix