a note on a conjecture on permanents abstract let kn denote the set of all n n nonnegative matrices whose entries have sum n and let be a real function on kn defined by x ni 1 nj 1xij nj 1 ni 1xij per x for x xij kn a matrix a kn is called a maximizing matrix on kn if a x for all x kn it is conjectured that jn 1 n n n is the unique maximizing matrix on kn in this note the following are proved i if a is a positive maximizing matrix then a jn ii if a is a row stochastic maximizing matrix then a jn iii every row sum and every column sum of a maximizing matrix lies between 1 2 n nn and 1 n 1 2 n nn iv for any p s d symmetric a kn a 2 n nn with equality iff a jn v attains a strict local maximum on kn at jn

maximization of a matrix function related to the dittert conjecture abstract let k n denote the set of all nonnegative n n matrices whose entries have sum n and let j n 1 n n n for k 1 n and for a k n with row sum s r 1 r n and column sums c 1 c n let k be defined by k a q k i r i j c j per a we propose a problem of maximizing k of which the dittert conjecture is a special case and obtain some results related to this problem many of known theorems for n are shown to hold for k k 1 n

separable and localized system level synthesis for large scale systems a major challenge faced in the design of large scale cyber physical systems such as power systems the internet of things or intelligent transportation systems is that traditional distributed optimal control methods do not scale gracefully neither in controller synthesis nor in controller implementation to systems composed of a large number e g on the order of billions of interacting subsystems this paper shows that this challenge can now be addressed by leveraging the recently introduced system level approach sla to controller synthesis in particular in the context of the sla we define suitable notions of separability for control objective functions and system constraints such that the global optimization problem or iterate update problems of a distributed optimization algorithm can be decomposed into parallel subproblems we then further show that if additional locality i e sparsity constraints are imposed then these subproblems can be solved using local models and local decision variables the sla is essential to maintain the convexity of the aforementioned problems under locality constraints as a consequence the resulting synthesis methods have o 1 complexity relative to the size of the global system we further show that many optimal control problems of interest such as localized lqr and lqg mathcal h 2 optimal control with joint actuator and sensor regularization and localized mixed mathcal h 2 mathcal l 1 optimal control problems satisfy these notions of separability and use these problems to explore tradeoffs in performance actuator and sensing density and average versus worst case performance for a large scale power inspired system

the monotonicity of and the đokovic conjectures on permanents of doubly stochastic matrices abstract let n denote the set of all n n doubly stochastic matrices and let j n 1 n n n for a n and any integer k 1 k n let pk a denote the sum of all subpermanents of order k of a and let d k a p k a n k 1 2 nk p k 1 a for k 2 n a n is called a dk minimizing matrix on n if d k a d k x x n the okovic conjecture asserts that if a is a dk minimizing matrix on n then dk a 0 k 2 n in this paper we prove that if a is a positive dk minimizing matrix on n then dk a 0 and a jn k 2 n and we settle the conjecture for an n 2 dimensional face of n we also prove the monotonicity of the permanent for a x y u v n where each of the blocks of a is a matrix of equal entries and for any a n with n 1 identical rows

on a neutral functional differential equation in a fading memory space abstract the linear autonomous neutral system of functional differential equations are studied in a fading memory space conditions are given which imply that solutions of the functional differential equation can be decomposed into a stable part and an unstable part these conditions are of frequency domain type the results can be used to decompose the semigroup generated by the functional differential equation into a stable part and an unstable part

on a conjecture of e dittert abstract let kn denote the set of all n x n nonnegative matrices whose entries have sum n and let be a real valued function defined on kn by x in 1 n j 1cj n per x for x e kn with row sum vector r1 rn and column sum vector cl hellip cn for the same x let ij x k irk 1 jc1 per x i j a kn is called a maximizing matrix if a x for all x kn dittert s conjecture asserts that jn 1 n n n is the unique maximizing matrix on kn in this paper the following are proved i if a aij is a maximizing matrix on kn then ij a a if aij 0 and ij a a if aij 0 ii the conjecture is true for n 3

the null space and the range of a convolution operator in a fading memory space we study the convolution equation p x t v x t oo 0 instead we permit and to be of fredholm type i e and v need not vanish on 00 0 we restrict the growth rate of x and at plus and minus infinity and we look at the problem of the existence and uniqueness of solutions of and on the whole real line satisfying conditions like x i ctj i 00 t 00 where c is a constant depending on x and ij is a predefined function some authors use the word admissible when discussing problems of this type in the case when the homoge neous version of has nonzero solutions we decompose the solutions into components with different exponential growth rates and give a priori bounds on the growth rates of the solutions as an application of the basic theory we look at the cauchy problem for a neutral functional differential equation and prove the existence of stable and unstable manifolds

spectral decomposition and invariant manifolds for some functional partial differential equations we study the integrodifferential convolution equationddt x x ax x fon 0 x on 0 as well as a nonlinear perturbation of the corresponding homogeneous equation hereais the generator of an analytic semigroup on a hilbert spaceh and and are operator valued dominated measures with values inl h andl d a h respectively under the assumption that the operator given by the laplace transform of the left hand side of the equation is boundedly invertible on some right half plane and on a line in the left half plane parallel to the imaginary axis we decompose the solutions into components with different exponential growth rates we construct projectors onto the stable and unstable subspaces which are then used for the construction of stable and unstable manifolds for the nonlinear equation which can have a fully nonlinear character the results are applied to two equations of parabolic type moreover the spectrum of the generator of the translation semigroup in various weighted spaces is determined including the stable and unstable subspaces of our problem

generalized linear quadratic control we consider the problem of stochastic finite and infinite horizon linear quadratic control under power constraints the calculations of the optimal control law can be done off line as in the classical linear quadratic gaussian control theory using dynamic programming which turns out to be a special case of the new theory developed in this technical note a numerical example is solved using the new methods

a characterization of convex problems in decentralized control ast we consider the problem of constructing optimal decentralized controllers we formulate this problem as one of minimizing the closed loop norm of a feedback system subject to constraints on the controller structure we define the notion of quadratic invariance of a constraint set with respect to a system and show that if the constraint set has this property then the constrained minimum norm problem may be solved via convex programming we also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback these results are developed in a very general framework and are shown to hold in both continuous and discrete time for both stable and unstable systems and for any norm this notion unifies many previous results identifying specific tractable decentralized control problems and delineates the largest known class of convex problems in decentralized control as an example we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate we also show that symmetric synthesis is included in this classification and provide a test for sparsity constraints to be quadratically invariant and thus amenable to convex synthesis

some results towards the dittert conjecture on permanents abstract let k n denote the convex set consisting of all real nonnegative n n matrices whose entries have sum n for a k n with row sums r 1 r n and column sums c 1 c n define a i 1 n r i j 1 n c j per a dittert s conjecture asserts that the maximum of on k n occurs uniquely at j n 1 n n n in this paper we prove i if a k n is partly decomposable then a j n ii if the zeroes in a k n form a block then a is not a maximising matrix iii a j n unless per j n per a o n 4 e 2 n and k i r i 2 k k i c i 2 k and i j a ij k k for all sets of k integers chosen from 1 2 n

integral equations as evolution equations abstract we study a class of time dependent linear integrodifferential equations ve with the evolution equation approach we determine the generators of a time dependent evolution equation de which is equivalent to the given integrodifferential equation under very general assumptions we prove the well posedness and continuity of ve from the stability of de the related question of convergence of a family of approximate solutions is examined as an application we include an example of hyperbolic integro partial differential equation to illustrate the theory

an update on minc s survey of open problems involving permanents abstract we summarise the progress which has been made since 1986 on the conjectures and open problems listed in h minc s survey articles on the theory of permanents

explicit solutions to separable problems in optimal cooperative control this paper presents explicit state space solutions for a class of networked cooperative control problems we consider the optimal control problem for a class of systems with dynamics defined at the vertices of a directed acyclic graph and dynamic propagation along the edges of the graph the controller is also constrained to use information flowing along the edges of the graph with the additional assumption of transitive closure such problems have linear optimal controllers we show that under the factorization condition that every disturbance non sink takes state feedback these problems separate into several independent subproblems and hence they can be explicitly solved this technique generalizes and unifies several existing approaches for optimal decentralized control and it further provides the optimal controller for many cases in which the optimal control was previously unknown the optimal solutions are expressed in terms of a series of riccati equations whose sizes are bounded by the system order the separated problems highlight the structure of the optimal controller we demonstrate our approach on a salvo guidance problem for a team of networked missiles

resolvent operators for integral equations in a banach space conditions are given which ensure the existence of a resolvent operator for an integrodifferential equation in a banach space the resolvent operator is similar to an evolution operator for nonautonomous differential equations in a banach space as in the finite dimensional case this operator is used to obtain a variation of parameters formula which can be used to obtain results concerning the asymptotic behaviour of solutions and weak solutions

regularization for design when designing controllers for large scale systems the architectural aspects of the controller such as the placement of actuators sensors and the communication links between them can no longer be taken as given the task of designing this architecture is now as important as the design of the control laws themselves by interpreting controller synthesis in a model matching setup as the solution of a particular linear inverse problem we view the challenge of obtaining a controller with a desired architecture as one of finding a structured solution to an inverse problem building on this conceptual connection we formulate and analyze a framework called regularization for design rfd in which we augment the variational formulations of controller synthesis problems with convex penalty functions that induce a desired controller architecture the resulting regularized formulations are convex optimization problems that can be solved efficiently these convex programs provide a unified computationally tractable approach for the simultaneous co design of a structured optimal controller and the actuation sensing and communication architecture required to implement it further these problems are natural control theoretic analogs of prominent approaches such as the lasso the group lasso the elastic net and others that are employed in structured inference in analogy to that literature we show that our approach identifies optimally structured controllers under a suitable condition on a signal to noise type ratio

optimal decentralized state feedback control with sparsity and delays this work presents the solution to a class of decentralized linear quadratic state feedback control problems in which the plant and controller must satisfy the same combination of delay and sparsity constraints using a novel decomposition of the noise history the control problem is split into independent subproblems that are solved using dynamic programming the approach presented herein both unifies and generalizes many existing results

combinatorial verification of the elementary divisors of tensor products abstract using combinatorial arguments we determine the determinantal divisors for the tensor product of two matrices thereby obtaining its elementary divisors and jordan canonical form

a neutral fde with stable d operator is retarded abstract a neutral functional differential equation with a linear autonomous and stable d operator can be rewritten as a retarded functional differential equation with infinite delay this fact can be used e g in the study of the stability of the neutral equation and to simplify the proofs of some standard asymptotic results

on neutral functional differential equations with nonatomic difference operator where the difference operator d is nonatomic at zero here x denotes the function defined by x s x t s y d s d 0 the case of d l being bounded linear functionals c r 0 w r with d atomic at zero has been studied extensively in the literature cf e g s introducing a state which in the continuous case corresponds to dx x instead of x t x as in the retarded case burns herdman and stech proved in 3 that problem 1 1 gives rise to a c semigroup in r x lp a 0 fp 1 dp co provided d is a continuous functional on c r 0 w atomic at zero and l is continuous from w r 0 w into r in 3 it was also shown by an example that atomicity of d at zero is not necessary for 1 1 to generate a c semigroup on r x lp r 0 w

local analyticity in weighted spaces and applications to stability problems for volterra equations we study the qualitative properties of the solutions of linear convolution equations such as x x t f x a x f and x t f we are especially concerned with finding conditions which ensure that these equations have resolvents which belong to or are determined up to a term belonging to certain weighted l1 spaces our results are obtained as consequences of more general banach algebra results on functions that are locally analytic with respect to the elements of a weighted l1 space in particular we derive a proposition of wiener levy type for weighted l1 spaces which underlies all subsequent results our method applies equally well to equations more general than those mentioned above we unify and sharpen the results of several recent papers on the asymptotic behavior of volterra convolution equations of the types mentioned above and indicate how many of them can be extended to the fredholm case in addition we give necessary and sufficient conditions on the perturbation term f for the existence of bounded or integrable solutions x in some critical cases when the corresponding limit equations have nontrivial solutions

some well posed functional equations which generate semigroups consider the abstract linear functional equation fe dx t f t t 0 x t t t 0 in a banach space b a theorem is proven which contains the following result as a special case let y r b be a lp space or c0 space on r t8 with a suitable weight function and with values in b let d be a closed unbounded causal linear operator in y r b which commutes with translations suppose that d i has a continuous causal inverse for some complex and that d restricted to those functions in y r b which vanish on r 0 has a continuous causal inverse then fe generates a strongly continuous semigroup of translation type on a banach space which is essentially the cross product of the restriction of the domain of d to r and y r b examples with b cn on how the theory applies to a neutral functional differential equation a difference equation a volterra integrodifferential equation with nonintegrable kernel but integrable resolvent and a fractional order functional differential equation are given also an abstract neutral functional differential equation in a hilbert space is studied and applications to an abstract volterra integrodifferential equation in a banach space are indicated

neutral functional integro differential equations with weakly singular kernels the well posedness of a general functional differential equation of neutral type is considered where the difference operator does not have an atom at zero the results are applicable to certain classes of singular integro differential equations that occur in aeroelasticity the equations are considered as dynamical systems on product spaces