survey paper transfer functions of distributed parameter systems a tutorial in this tutorial article the rich variety of transfer functions for systems described by partial differential equations is illustrated by means of several examples under various boundary conditions an important feature is the strong influence of the choice of boundary conditions on the dynamics and on system theoretic properties such as pole and zero locations properness relative degree and minimum phase it is sometimes possible to design a controller using the irrational transfer function and several such techniques are outlined more often the irrational transfer function is approximated by a rational one for the purpose of controller design various approximation techniques and their underlying theory are briefly discussed

strong q convexity in uniform neighborhoods of subvarieties in coverings of complex spaces the main result is that for any projective compact analytic subset y of dimension q 0 in a reduced complex space x there is a neighborhood of y such that for any covering space upsilon colon widehat x to x in which widehat y equiv upsilon 1 y has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components there exists a c exhaustion function varphi on widehat x which is strongly q convex on widehat omega upsilon 1 omega outside a uniform neighborhood of the q dimensional compact irreducible components of widehat y

admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

integral control of infinite dimensional linear systems subject to input saturation closing the loop around an exponentially stable single input single output regular linear system subject to a globally lipschitz and nondecreasing actuator nonlinearity and compensated by an integral controller is shown to ensure asymptotic tracking of constant reference signals provided that a the steady state gain of the linear part of the plant is positive b the positive integrator gain is sufficiently small and c the reference value is feasible in a very natural sense the class of actuator nonlinearities under consideration contains standard nonlinearities important in control engineering such as saturation and deadzone

stability of a heat exchanger feedback control system using the circle criterion the standard linearized model of a controlled heat exchanger in the form of a system of two first order pdes is interpreted as an abstract semigroup system on an appropriately selected hilbert state space we analyse some its basis properties exponential stability admissibility of an observation functional and regularity of the system impulse response next we give a proof of a version of the circle criterion which bases on the use of the leray schauder fixed point theorem jointly with a uniqueness result the circle criterion is then implemented to a feedback control system of the heat exchanger with a proportional controller and a non linear valve in the feedback loop all considerations are illustrated by a numerical example of the heat exchanger installed in central heating system at the agh university of science and technology cracow poland for which we estimate feasible settlements of the proportional controller

smoothing q convex functions in the singular case definition 1 let f2 be a complex space f 2 r a continuous function and q 1 an integer then a q is called q convex in the c sense if for each point p c f2 there is a local coordinate patch u on 62 with a holomorphic embedding z u t for some open set u c c n and a c strongly q convex function on v such that 1u o i b 0 is called q convex with corners if for each p c f2 there is an open neighborhood u c f2 and on u finitely many q convex functions in the c sense q1 q such that qi u max 01 qz

some q convexity properties of coverings of complex manifolds let c be a nowhere dense compact analytic subset of a 2 dimensional complex manifold x a result of napier is the construction of a neighborhood v of c such that for any covering space open image in new window in which open image in new window is holomorphically convex there is a c exhaustion function on open image in new window which is strictly plurisubharmonic on 1 v away from the compact irreducible components of open image in new window col oiu and v j itu obtained a q convex version of this result for c a q dimensional fiber of a proper surjective submersion x y the goal of this paper is a similar version for cprojective

frequency domain results for systems with multiple time scales the mathematical treatment of systems with multiple time scales has traditionally employed the use of singular perturbation theory for differential equations this paper treats such systems from a frequency domain viewpoint previously existing two frequency scale results involving approximation and algebraic characterization are extended in addition an exact frequency scale decomposition is presented

clarification of a footnote of on the generalized nyquist stability criterion prof e a francis pointed out to the original authors that footnote 31 on p 189 of the above named article ibid vol ac 25 pp 181 196 april 1980 the footnote stated incorrectly that the discriminant could only be zero at a finite number of points in c this is false a correct reasoning is presented

convexity of complements of tropical varieties and approximations of currents the goal of this note is to affirm a local version of conjecture of nisse sottile 19 on higher convexity of complements of tropical varieties while providing a family of counter examples for the global nisse sottle conjecture in any codimension and dimension higher than one moreover it is shown that surprisingly this family also provides a family of counter examples for the generalized hodge conjecture for positive currents in these dimensions and gives rise to further approximability obstruction

q completeness of subsets in complex projective space the importance of exhaustion functions in complex analysis arises mainly from the theorems of andreott i and grauert 2 and the applications of morse theory the first give vanishing theorems for cohomology groups of coherent sheaves the second imply topological statements if x is a compact complex manifold and y an analytic subset of x the vanishing of the relative homotopy groups rh x y for i d i m x q follows from the q completeness of x y which is by definition the same as the existence of a q convex exhaustion function these theorems are called lefschetz theorems the first of this kind was the lefschetz theorem on hyperplane sections which can be proved by using the fact that the complement of a hypersurface in projective space is stein because there is a 1 convex exhaustion function it is natural to ask whether there is an exhaustion function of p a with certain properties if one has a lefschetz theorem if a is a q codimensional complex submanifold of p we have h p a 7z 0 4 and even i p a 0 9 for i n 2 q 1 so we are led to the question if p a is 2 q 1 complete and the proof of this assertion is the aim of this paper

multivariable singularly perturbed feedback systems with time delay multivariable singularly perturbed systems with both small and large time delays in the feedback loop are considered conditions for stability of the full order closed loop system are given in terms of properties of the slow and fast subsystems

robust control of flexible structures a case study a comparison is made between three partial differential equation models for a flexible beam with different types of damping and varying parameter values robust controllers can be designed to stabilize all linear systems whose transfer functions lie within a ball in the l norm given a nominal model for our flexible beam we calculate which sets of models can be stabilized by the same finite dimensional robust controller by stabilize we mean here that the poles of the system are moved to the left of the line re s 0

the linear quadratic control problem for infinite dimensional systems with unbounded input and output operators this paper establishes a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators

analytic controllability of time dependent quantum control systems the question of controllability is investigated for a quantum control system in which the hamiltonian operator components carry explicit time dependence which is not under the control of an external agent we consider the general situation in which the state moves in an infinite dimensional hilbert space a drift term is present and the operators driving the state evolution may be unbounded however considerations are restricted by the assumption that there exists an analytic domain dense in the state space on which solutions of the controlled schrodinger equation may be expressed globally in exponential form the issue of controllability then naturally focuses on the ability to steer the quantum state on a finite dimensional submanifold of the unit sphere in hilbert space and thus on analytic controllability a relatively straightforward strategy allows the extension of lie algebraic conditions for strong analytic controllability derived earlier for the simpler time independent system in which the d

analytic cohomology groups in top degrees of zariski open sets in mathbb p n we show that if a is a closed analytic subset of mathbb p n of pure codimension q then h i mathbb p n setminus a mathcal f are finite dimensional for every coherent algebraic sheaf mathcal f and every i geq n left frac n 1 q right if n 1 geq 2q rm we show that h n 2 mathbb p n setminus a mathcal f 0

circle criteria small gain conditions and internal stability for infinite dimensional systems abstract an internal version of the small gain theorem is given for a class of plants including time delay and distributed systems more precisely it is shown that for a large class of infinite dimensional state space systems the small gain condition in l 2 is sufficient for internal stability furthermore it is proved that there exist unstable feedback systems of loop gain equal to one the internal version of the small gain theorem is used in order to establish that the circle criterion is sufficient for global exponential stability

a nyquist type stability test for multivariable distributed systems a multivariable nyquist criterion is given that applies to a wide class of open loop transfer matrices including some that are improper or have infinitely many poles in the right half plane application of the test to specific examples requires only elementary knowledge of complex variable theory the actual test requires operations similar to those used in the standard nyquist test the simplicity of the frequency domain description of the class t analyticity and polynomial boundedness makes it relatively easy to verify if the test is applicable to a given transfer matrix

absolute stability via boundary control of a semilinear parabolic pde we consider the problem of achieving global absolute stability of an unstable equilibrium solution of a semilinear dissipative parabolic partial differential equation pde through boundary control the state space of the system is extended in order to write the action of the boundary control as an unbounded operator in an abstract evolution equation absolute stability via boundary control is accomplished by analyzing a control lyapunov function based on the infinite dimensional dynamics and applying a finite dimensional linear quadratic regulator lqr controller sufficient conditions for absolute stability of the infinite dimensional system are established by the feasibility of two finite dimensional linear matrix inequalities lmis numerical results are presented for a dirichlet boundary controlled system however the analysis in this work applies to nuemann and robin type boundary controllers as well

online packing of arbitrary sized items into designated and multipurpose bins abstract we consider an online multitype bin packing problem with two item types this setting can be motivated by transport applications in which some items may be shipped either in dry shipping containers or more costly refrigerated ones while other items can only be transported in refrigerated containers the problem was introduced by goldberg and karhi omega 71 85 92 2017 who focused on the case of up to two item types three bin types and only two item sizes for this special case a tight also known as optimal absolute competitive ratio was shown of approximately 1 618 here we consider the general problem of arbitrary item sizes and show a lower bound on the absolute competitive ratio of any online algorithm that is a function of the bin costs this bound in the worst case is approximately 1 781 we then extend the first fit method to our problem setting and prove an absolute competitive ratio bound that is a function of the bin costs in the worst case this upper bound is approximately 1 930 in addition an upper bound of 1 750 is established on the worst case asymptotic as the number of items grows large competitive ratio

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

the levi problem for riemann domains over stein spaces with isolated singularities we consider the following question let p y rightarrow x be an unbranched riemann domain and assume that x is a stein space and p is a stein morphism does it follow that y is stein we show that the answer is affirmative if x has isolated singularities this generalizes a result of andreotti and narasimhan

frequency domain results for systems with slow and fast dynamics the mathematical treatment of systems with slow and fast dynamics has traditionally involved singular perturbation theory for differential equations this paper suggests a set of conditions to be placed on a frequency domain description of a system to guarantee two time scale behavior some basic stability and approximation results are presented

deformation retracts of stein spaces let x be an n dimensional stein space it was proved in 4 5 3 that x has the homotopy type of a cw complex of dimension5n in the algebraic case this was proved in 8 with the additional conclusion that the cw complex is nite it is therefore natural to ask if there exists a subset q of x with the same topological properties as x for instance q is a strong deformation retract of x and q is a cw complex of dimension5 n the main result of the paper is

multivariable nyquist criteria root loci and pole placement a geometric viewpoint classical control theory is concerned with the topics in our title in the context of single input single output systems there is now a large and growing literature on the extension of these ideas to the multiinput multioutput case this development has posed certain difficulties some due to the intrinsic nature of the problem and some we would argue due to an inadequate reflection on what the multivariable problem calls for in this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory a rather satisfactory generalization of the nyquist criterion will be described and a clear analog of the asymptotic properties of the root locus will be obtained in the multiparameter case however an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains this calls for a new ingredient a compactification of the space of gains and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement this problem is difficult in the multivariable case but by establishing a correspondence with a classical set of problems in geometry we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field

plurisubharmonic functions in calibrated geometry and q convexity let m be a kahler manifold an integrable function varphi on m is called q plurisubharmonic if the current dd c varphi wedge omega q 1 is positive we prove that varphi is q plurisubharmonic if and only if varphi is subharmonic on all q dimensional complex subvarieties we prove that a q plurisubharmonic function is q convex and admits a local approximation by smooth q plurisubharmonic functions for any closed subvariety z subset m dim mathbb c z leq q 1 there exists a strictly q plurisubharmonic function in a neighbourhood of z this result is known for q convex functions this theorem is used to give a new proof of sibony s lemma on integrability of positive closed p p forms which are integrable outside of a complex subvariety of codimension p 1

on the n concavity of covering spaces with parameters x t where x and t are connected complex manifolds of dimensions n m and m respectively is a proper and surjective holomorphic submersion and is a covering map thus x can be regarded as a family of n dimensional complex manifolds over t we will use the same definition of q convexity as in 1 for the precise formulation see the next section r green and h wu 7 proved that a connected non compact complex manifold is n complete in 5 m col oiu and v v j itu proved the following

robustness of linear systems abstract in this paper we consider linear systems which are stable and examine the robustness of this property the perturbations are assumed to have unbounded structure and we determine the smallest perturbation which destroys stability we begin with an abstract analysis of the problem but in later sections specialize to three types of systems those whose mild solutions are given in terms of 1 a strongly continuous semigroups 2 b resolvent operators integrodifferential systems 3 c evolution operators time varying systems for b a banach space l2 t0 t b denotes the space of all square integrable functions defined on t0 t with values in b for b1 b2 banach spaces t0 t denotes the space of strongly measurable functions f t0 t l b1 b2 with ess sup f b 1 t0 l b1 b2 denotes the space of strongly measurable functions f t0 l b1 b2 with f t 0 and k l1 t0

linear state space systems in infinite dimensional space the role and characterization of joint stabilizability detectability several fundamental results from the theory of linear state space systems in finite dimensional space are extended to encompass a class of linear state space systems in infinite dimensional space the results treated are those pertaining to the relationship between input output and internal stability the problem of dynamic output feedback stabilization and the concept of joint stabilizability detectability a complete structural characterization of jointly stabilizable detectable systems is obtained the generalized theory applies to a large class of linear state space systems assuming only that i the evolution of the state is governed by a strongly continuous semigroup of bounded linear operators ii the state space is hilbert space iii the input and output spaces are finite dimensional and iv the sensing and control operators are bounded general conclusions regarding the fundamental structure of control theoretic problems in infinite dimensional space can be drawn from these results

q complete domains with corners in mathbb p n and extension of line bundles we show that if a compact set x in mathbb p n is laminated by holomorphic submanifolds of dimension q then mathbb p n setminus x is q 1 complete with corners consider a manifold u q complete with corners let mathcal n be a holomorphic line bundle in the complement of a compact in u we study when mathcal n extends as a holomorphic line bundle in u we give applications to the non existence of some levi flat foliations in open sets in mathbb p n the results apply in particular when u is a stein manifold of dimension n 3 then every holomorphic line bundle in the complement of a compact extends holomorphically to u

convexity properties of coverings of 1 convex manifolds we show the following result let x be a 1 convex manifold a its exceptional set k dima and p y x any covering then y can be exhausted by an increasing sequence y nu nu in mathbb n of smoothly bounded k convex domains

on the center problem for ordinary differential equations we study the center problem for equations v i 1 a i x v i 1 this problem is closely connected to the classical poincare center focus problem about the characterization of planar polynomial vector fields whose integral trajectories are closed curves with interiors containing a fixed point a center

algorithms for the variable sized bin packing problem in this paper we consider the variable sized bin packing problem where the objective is to minimize the total cost of used bins when the cost of unit size of each bin does not increase as the bin size increases two greedy algorithms are described and analyzed in three special cases a the sizes of items and bins are divisible respectively b only the sizes of bins are divisible and c the sizes of bins are not divisible here we say that a list of numbers a1 a2 am are divisible when aj exactly divides aj 1 for each 1 j m in the case of a the algorithms give optimal solutions and in the case of b each algorithm gives a solution whose value is less than 119c b 4119 where c b is the optimal value in the case of c each algorithm gives a solution whose value is less than 32c b 1

stability radius for structured perturbations and the algebraic riccati equation abstract in this paper we generalize the notion of stability radius introduced in 1 to allow for structured perturbations we then relate the stability radius to the existence of hermitian solutions of an algebraic riccati equation and give some applications of this result

oka properties of ball complements let n 1 be an integer we prove that holomorphic maps from stein manifolds x of dimension n to the complement mathbb c n setminus l of a compact convex set l subset mathbb c n satisfy the basic oka property with approximation and interpolation if l is polynomially convex then the same holds when 2 dim x le n we also construct proper holomorphic maps immersions and embeddings x rightarrow mathbb c n with additional control of the range thereby extending classical results of remmert bishop and narasimhan

the class constrained bin packing problem with applications to video on demand in this paper we present approximation results for the class constrained bin packing problem that has applications to video on demand systems in this problem we are given bins of size b with c compartments and n items of q different classes each item i 1 n with class c i and size s i the problem is to pack the items into bins where each bin contains at most c different classes and has total items size at most b we present several approximation algorithms for offline and online versions of the problem

on convex hulls and pseudoconvex domains generated by q plurisubharmonic functions part iii we study the properties of hulls of compact sets in mathbb c n that are generated by certain subfamilies of q plurisubharmonic functions we consider in particular those functions that are plurisubharmonic on the level sets of holomorphic mappings defined from mathbb c n into mathbb c q we also compare the above mentioned hulls against the generalised polynomially and rationally convex hulls already defined in the literature our main result yields that the hulls defined by q plurisubharmonic functions or q pseudoconvex sets are all q maximum so that their complement is relatively n text q text 2 pseudoconvex

finite dimensional compensators for infinite dimensional systems with unbounded input operators this paper contains a design procedure for constructing finite dimensional compensators for a class of infinite dimensional systems with unbounded input operators applications to retarded functional differential systems with delays in the input or the output variable and to partial differential equations with boundary input operators are discussed

nyquist type stability test for undamped elastic structures a transfer function with possibly infinitely many imaginary axis poles is used to represent a linear multiresonant elastic structure with negligible damping a stable coprime factorization of this transfer function is presented this factorization is the basis for a graphical nyquist type feedback stability test that does not require counting encirclements of the critical point the test can be used to assess the input output stability of a damped or undamped structure with stable feedback compensators using displacement and velocity sensors the test is applied to a continuous slewing beam example and it is shown that contrary to the finite dimensional case this feedback system is unstable whenever either the velocity feedback compensator transfer function is strictly proper or a certain feedback delay which can be arbitrarily small is present

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods