brief paper global stabilization of semilinear systems using switching controls this paper considers feedback stabilization of distributed semilinear systems using switching controls the semilinear processes together with the switched controllers constitute a type of switched nonlinear system three kinds of stabilizabilities namely weak strong and exponential stabilizabilities are investigated one by one sets of sufficient conditions are obtained for each case a necessary condition for strong stabilizability is given the stabilizing control is characterized using a minimization problem applications to parabolic and hyperbolic like equations are considered

the number of small amplitude limit cycles in arbitrary polynomial systems abstract in this paper we study the number of small amplitude limit cycles in arbitrary polynomial systems it is found that almost all the results for the number of small amplitude limit cycles are obtained by calculating lyapunov constants and determining the order of the corresponding hopf bifurcation it is well known that the difficulty in calculating the lyapunov constants increases with the increasing of the degree of polynomial systems so it is necessary and valuable for us to achieve some general results about the number of small amplitude limit cycles in arbitrary polynomial systems with degree m which is denoted by m m in this paper by applying the method developed by c christopher and n lloyd in 1995 and m han and j li in 2012 we first obtain the lower bounds for m 6 m 14 and then prove that m m m 2 if m 23 finally we obtain that m m grows as least as rapidly as 18 25 1 2 ln 2 m 2 2 ln m 2 for all large m it is proved by m han j li lower bounds for the hilbert number of polynomial systems j differential equations 252 2012 3278 3304 that the number of all limit cycles in arbitrary polynomial systems with degree m grows as least as rapidly as 1 2 ln 2 m 2 2 ln m 2

sliding mode control in indefinite dimensional systems abstract a mathematical description of motion in a sliding mode is considered for systems with discontinuous controls described by differential equations in banach space applying the regularization principle to such systems the sliding mode equations are obtained for a set of interconnected heat processes the control system with the prescribed dynamics is designed through deliberate introduction of sliding modes

boundary optimal lq control of coupled hyperbolic pdes and odes this contribution addresses the development of a linear quadratic lq regulator for a set of hyperbolic pdes coupled with a set of odes through the boundary the approach is based on an infinite dimensional hilbert state space description of the system and the well known operator riccati equation ore in order to solve the optimal control problem the ore is converted to a set of matrix riccati equations the feedback operator is found by solving the resulting matrix riccati equations the performance of the designed control policy is assessed by applying it to a system of interconnected continuous stirred tank reactor cstr and a plug flow reactor pfr through a numerical simulation

exponential stability for a class of boundary conditions on a euler bernoulli beam subject to disturbances via boundary control abstract this paper outlines the procedure for applying a sliding mode backstepping boundary control technique to a euler bernoulli beam subject to unknown bounded disturbances on the boundary for four different types of boundary conditions referred to in this paper as a class of boundary conditions these boundary conditions include all combinations of pinned and sliding types with control and disturbances on one boundary i e pinned pinned pinned sliding sliding pinned and sliding sliding with input on the second boundary the technique was developed for the specific case of a pinned pinned beam in the literature and in this paper the technique is generalized to the degree that it is possible to cover all four of the aforementioned cases furthermore a proof of the fully arbitrary exponential stability of the closed loop system is provided in this paper to this point only asymptotic stability was shown for the closed loop system for one case the technique outlined in this paper provides for a class of boundary conditions a controller that exponentially stabilizes the vibrations in a beam in the presence of bounded unknown disturbances the beam displacement returns to the origin and its decay is bounded by a known exponential decay function after a known finite time reaching phase

bifurcation of limit cycles in a cubic order planar system around a nilpotent critical point abstract in this paper bifurcation of limit cycles is considered for planar cubic order systems with an isolated nilpotent critical point normal form theory is applied to compute the generalized lyapunov constants and to prove the existence of at least 9 small amplitude limit cycles in the neighborhood of the nilpotent critical point in addition the method of double bifurcation of nilpotent focus is used to show that such systems can have 10 small amplitude limit cycles near the nilpotent critical point these are new lower bounds on the number of limit cycles in planar cubic order systems near an isolated nilpotent critical point moreover a set of center conditions is obtained for such cubic systems

on bounded query machines abstract simple proofs are given for each of the following results a p pspace if and only if for every set a p a pquery a selman et al 1983 b np pspace if and only if for every set a np a npquery s book 1981 c ph pspace if and only if for every set a ph a pqh a book and wrathall 1981 c ph pspace if and only if for every set set s ph s pqh s pspace s balcazar et al 1986 long and selman 1986

strong and robustly strong polynomial time reducibilities to sparse sets reducibility defined by oracle strong nondeterministic machines is studied two definitions of relativized strength are presented and separated the corresponding reduction classes of the sparse sets are characterized in terms of nonuniform complexity classes an oracle restricted positive relativization of the probabilistic class zpp is developed

generating limit cycles from a nilpotent critical point via normal forms abstract it is well known that the normal form theory can be applied to solve the center focus problem for monodromic planar nilpotent singularities in this paper we see how this theory can also be applied to generate limit cycles from this type of singularities

exponential stability of nondissipative linear system in banach space and application to unbounded bilinear systems abstract we consider linear control systems of the form d z t d t a z t b z t where is a real parameter a is the infinitesimal generator of a linear c 0 semigroup of contractions s 0 t on a banach space x and the linear perturbation b is a relatively bounded operator with respect to a we provide necessary and sufficient conditions for exponential stability of the above system our stability results are then applied to establish uniform exponential stabilization of linear and bilinear systems with unbounded control operators applications to hyperbolic like equations are considered

positive relativizations for log space computability we consider some open questions about log space computability deterministic non deterministic and alternating and polynomial time computability deterministic and nondeterministic in particular the questions dl nl dl p and nl np are considered where dl nl denotes the class of sets accepted by log space bounded deterministic nondeterministic turing machines and p np denotes the class of sets accepted by polynomial time bounded deterministic nondeterministic turing machines we develop positive relativizations for the above questions in two different ways first dl and nl are relativized such that the oracle tape is not subject to the space bound and such that no other restrictions are placed on the oracle machines in this case we prove that c d iff ct dt for all tally sets t where c d is a pair of classes from dl nl p np and conp next both nl and al alternating log space are relativized with the restriction introduced by ruzzo simon and tompa j comput system sci 28 1984 216 230 we call this restriction the rst restriction and prove that c d iff carst darst for all sets a where c d is a pair of classes from dl nl conl and al and crst denotes the corresponding relativized class with the rst restriction

nondeterministic functions and the existence of optimal proof systems we provide new characterizations of two previously studied questions on nondeterministic function classes q1 do nondeterministic functions admit efficient deterministic refinements q2 do nondeterministic function classes contain complete functions we show that q1 for the class npmv t is equivalent to the question whether the standard proof system for sat is p optimal and to the assumption that every optimal proof system is p optimal assuming only the existence of a p optimal proof system for sat we show that every set with an optimal proof system has a p optimal proof system under the latter assumption we also obtain a positive answer to q2 for the class npmv t an alternative view on nondeterministic functions is provided by disjoint sets and tuples we pursue this approach for disjoint np pairs and its generalizations to tuples of sets from np and conp with disjointness conditions of varying strength in this way we obtain new characterizations of q2 for the class npsv question q1 for npsv is equivalent to the question of whether every disjoint np pair is easy to separate in addition we characterize this problem by the question of whether every propositional proof system has the effective interpolation property again these interpolation properties are intimately connected to disjoint np pairs and we show how different interpolation properties can be modeled by np pairs associated with the underlying proof system

the boolean hierarchy ii applications the boolean hierarchy i structural properties j cai et al siam j comput 17 1988 pp 1232 252 explores the structure of the boolean hierarchy the closure of np with respect to boolean

quantitative relativizations of complexity classes consider the following open problems i text p text np ii text np text co np iii text p text pspace iv text np text pspace in this paper we study these four problems from a particular point of view to illustrate our approach consider the first problem it is known that there exist recursive sets a and b such that text p a text np a and text p b ne text np b we study restrictions r on both the deterministic and also the nondeterministic polynomial time bounded oracle machines such that the following holds text p text np if and only if for every set a text p r a text np r a the restrictions are quantitative in the sense that the size of the set of strings queried by the oracle in computations of a machine on an input is bounded by a polynomial in the length of the input we study several different ways of specifying such quantitative restrictions each of which has the desire

discontinuous feedback stabilization of minimum phase semilinear infinite dimensional systems with application to chemical tubular reactor this paper develops discontinuous control methods for minimum phase semilinear infinite dimensional systems driven in a hilbert space the control algorithms presented ensure asymptotic stability global or local accordingly as state feedback or output feedback is available as well as robustness of the closed loop system against external disturbances with the a priori known norm bounds the theory is applied to stabilization of chemical processes around prespecified steady state temperature and concentration profiles corresponding to a desired coolant temperature two specific cases a plug flow reactor and an axial dispersion reactor governed by hyperbolic and parabolic partial differential equations of first and second order respectively are under consideration to achieve a regional temperature feedback stabilization around the desired profiles with the region of attraction containing a prescribed set of interest a component concentration observer is constructed and included into the closed loop system so that there is no need for measuring the process component concentration which is normally unavailable in practice performance issues of the discontinuous feedback design are illustrated in a simulation study of the plug flow reactor

the center focus problem and reversibility in this paper we discuss the relation between time reversibility and the center focus problem we show the following for any analytic planar system x if j1x the one jet of x is conjugated to y x then x is analytically time reversible if and only if it is a center if j1x is conjugated to y 0 then some sufficient and necessary conditions for x to have a center are given in particular the reversibility of certain types of polynomial vector fields is studied

the boolean hierarchy i structural properties in this paper we study the complexity of sets formed by boolean operations union intersection and complement on np sets these are the sets accepted by trees of hardware with np predicates as leaves and together these form the boolean hierarchy we present many results about the structure of the boolean hierarchy separation and immunity results natural complete languages and structural asymmetries between complementary classes we show that in some relativized worlds the boolean hierarchy is infinite and that for every k there is a relativized world in which the boolean hierarchy extends exactly k levels we prove natural languages variations of vertex cover complete for the various levels of the boolean hierarchy we show the following structural asymmetry though no set in the boolean hierarchy is text d text p immune there is a relativized world in which the boolean hierarchy contains text cod text p immune sets thus this paper explores the structural properties of the

discontinuous unit feedback control of uncertain infinite dimensional systems control systems driven by a discontinuous unit feedback in a hilbert space are studied the equation which describes a system motion taking place in the discontinuity manifold and further referred to as a sliding mode is derived by means of a special regularization technique based on the sliding mode equation the procedure of synthesis of a discontinuous unit control signal is developed restricted to a class of infinite dimensional systems with finite dimensional unstable part this procedure generates the control law which ensures desired dynamic properties as well as robustness of the closed loop system with respect to matched disturbances as an illustration of the capabilities of the procedure proposed a scalar unit controller of an uncertain exponentially minimum phase dynamic system is constructed and applied to heat processes and distributed mechanical oscillators

computation of normal forms via a perturbation technique considering qualitative behavior of a non linear dynamical system often leads to first simplifying the differential equations or finding their normal forms a perturbation technique for computing normal forms is presented this technique associated with the method of multiple scales can be used to systematically find a unique form for a given set of differential equations the technique is discussed in detail through the analysis of hopf bifurcation it is shown that for hopf bifurcation the method only requires solving two dimensional matrix systems for any higher order normal forms of a generaln dimensional system with the aid of the symbolic language maple this approach is straightforward and is computationally efficient and fast furthermore a simple verification scheme is given for verifying the normal forms and associated non linear transformations obtained using any methodology examples are presented to demonstrate the applicability of the perturbation technique and the computation efficiency of the maple program

continuous state feedback tracking of an uncertain heat diffusion process a spatially distributed form of a second order sliding mode control algorithm is suggested for the feedback control of a heat process subject to unmeasurable sufficiently smooth persistent disturbances of arbitrary shape in order to obtain a continuous control input while retaining similar properties of robustness as those usually endowing the discontinuous control systems a dynamic input extension technique is exploited by adding an integrator at the input side and by applying a discontinuous distributed input to the integrator we present a constructive lyapunov analysis of the closed loop performance that yields simple controller tuning conditions guaranteeing the asymptotic tracking of a sufficiently smooth reference trajectory despite the presence of norm bounded persistent disturbances numerical simulations confirm the effectiveness of the proposed approach

nine limit cycles around a singular point by perturbing a cubic hamiltonian system with a nilpotent center in this paper we study bifurcation of limit cycles in planar cubic near hamiltonian systems with a nilpotent center we use normal form theory to compute the generalized lyapunov constants and show that there exist at least 9 limit cycles around the nilpotent center this is a new lower bound on the number of limit cycles in planar cubic near hamiltonian systems with a nilpotent center

spectral sequences and parametric normal forms abstract this paper extends recent developments on spectral sequence approach on the simplest normal form theory to parametric cases in this paper the method is first applied to two simple examples parametric single zero and parametric resonant saddle singularities and then to parametric generalized hopf singularity we provide the simplest parametric normal form for these cases

tracking control of the uncertain heat and wave equation via power fractional and sliding mode techniques in the present paper preliminary results towards the generalization to the infinite dimensional setting of some well known robust finite dimensional control algorithms are illustrated more specifically we deal with the tracking problem for some classes of linear uncertain infinite dimensional systems evolving in hilbert spaces we design distributed variable structure stabilizers that are shown to be effective in the presence of external disturbances the main focus of the present paper is on the rejection of nonvanishing external disturbances the generalization to the infinite dimensional setting of the well known finite dimensional controllers namely the power fractional controller s p bhat and d s bernstein siam j control optim 38 2000 pp 751 766 and two second order sliding mode control algorithms the twisting and supertwisting algorithms l fridman and a levant higher order sliding modes as a natural phenomenon in control theory in robust control via variable structure and lyapunov techniques springer verlag berlin 1996 pp 107 133 a levant internat j control 58 1993 pp 1247 1263 is the main contribution of the present investigation first the distributed twisting control algorithm is developed to address the asymptotic state tracking of the perturbed wave equation next the finite time state tracking of the unperturbed heat equation is provided by means of a distributed power fractional controller finally the distributed supertwisting controller is suggested to address the asymptotic state tracking of the heat equation in spite of the presence of persistent disturbances constructive proofs of stability are developed via the lyapunov functional technique which leads to simple tuning rules for the controller parameters simulation results are discussed to verify the effectiveness of the proposed schemes

bounded query machines on np and npquery abstract the np and npquery operators are studied in order to develop necessary and sufficient conditions for the class pspace to be equal to the union of the polynomial time hierarchy

a sliding mode observer for a linear reaction convection diffusion equation with disturbances abstract a sliding mode observer is designed and analyzed for a linear reaction convection diffusion equation subject to external disturbances based on a discontinuous input the proposed observer ensures both state estimation and disturbance rejection using only one boundary measurement an appropriate lyapunov function is used to prove the asymptotic convergence of the observation error the performance of the proposed observer is illustrated by some numerical simulation results

exponential stabilization of conservation systems with interior disturbance abstract in this paper we consider the stabilization problem of the conservation systems with interior disturbance employing the idea of sliding mode control we design a nonlinear distributed feedback controller we prove the solvability of the resulted closed loop system by the maximal monotone operator theory further we prove the exponential stability of the closed loop system in particular we prove that the requirement of a classical solution in the lyapunov approach is unnecessary

a cubic system with thirteen limit cycles abstract we construct a planar cubic system and demonstrate that it has at least 13 limit cycles the construction is essentially based on counting the number of zeros of some abelian integrals

bi center problem and bifurcation of limit cycles from nilpotent singular points in z2 equivariant cubic vector fields abstract in this paper bi center problem and bifurcation of limit cycles from nilpotent singular points in z 2 equivariant cubic vector fields are studied first the system is simplified by using some proper transformations and the first five lyapunov constants at a nilpotent singular point are calculated by applying the inverse integrating factor method then sufficient and necessary conditions are obtained for two nilpotent singular points of the system being centers a new perturbation scheme is present to prove the existence of 12 small amplitude limit cycles in cubic z 2 equivariant vector fields which bifurcate from two nilpotent singular points this is a new lower bound of the number of limit cycles bifurcating in such systems

on truth table reducibility to sat we show that polynomial time truth table reducibility via boolean circuits to sat is the same as logspace truth table reducibility via boolean formulas to sat and the same as logspace turing reducibility to sat in addition we prove that a constant number of rounds of parallel queries to sat is equivalent to one round of parallel queries we give an oracle relative to which 2p is not equal to the class of predicates polynomial time truth table reducible to sat

exponential stabilization of the uncertain wave equation via distributed dynamic input extension the wave equation is studied under the effect of a persistent external perturbation a dynamical distributed controller is suggested based on an infinite dimensional generalization of second order sliding mode control techniques that provides the exponential attainment of a sufficiently smooth arbitrary reference of the state trajectory the control system comprises of both feedforward and feedback parts the latter being a discontinuous term directly connected to the plant input through a dynamical filter that augments the system state smoothing out the discontinuity of the feedback control loop as a result a continuous input is applied to the plant a constructive lyapunov based proof of convergence of the proposed control algorithm is carried out and supporting numerical results are presented

on the query complexity of selecting minimal sets for monotone predicates propositional satisfiability sat solvers are routinely used for solving many function problems a natural question that has seldom been addressed is what is the number of calls to a sat solver for solving some target function problem this article improves upper bounds on the query complexity of solving several function problems defined on propositional formulas these include computing the backbone of a formula and computing the set of independent variables of a formula for the general case of monotone predicates the article improves upper bounds on the query complexity of computing a minimal set when the number of minimal sets is constant this applies for example to the computation of a minimal unsatisfiable subset mus for cnf formulas but also to the computation of prime implicants and implicates with immediate application in a number of ai settings

computing solutions uniquely collapses the polynomial hierarchy is there an np function that when given a satisfiable formula as input outputs one satisfying assignment uniquely that is can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments we show that if there is such a nondeterministic function then the polynomial hierarchy collapses to zppnp and thus in particular to npnp as the existence of such a function is known to be equivalent to the statement every np function has an np refinement with unique outputs our result provides the strongest evidence yet that np functions cannot be refined we prove our result via a result of independent interest we say that a set a is npsv selective npmv selective if there is a 2 ary partial np function with unique values a 2 ary partial np function that decides which of its inputs if any is more likely to belong to a this is a nondeterministic analog of the recursion theoretic notion of the semi recursive sets and the extant complexity theoretic notion of p selectivity our hierarchy collapse result follows by combining foo the easy observation that every set in np is npmv selective with the following result if a in np is npsv selective then a in np cap conp poly relatedly we prove that if a in np is npsv selective then a is low 2 we prove that the polynomial hierarchy collapses even further namely to np if all conp sets are npmv selective this follows from a more general result we prove every self reducible npmv selective set is in np

brief paper sliding mode boundary control of a parabolic pde system with parameter variations and boundary uncertainties this paper considers the stabilization problem of a one dimensional unstable heat conduction system rod modeled by a parabolic partial differential equation pde powered with a dirichlet type actuator from one of the boundaries by applying the volterra integral transformation a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties the associated lyapunov function is used for designing an infinite dimensional sliding manifold on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances it is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero a continuous control law satisfying the reaching condition is obtained by passing a discontinuous signum signal through an integrator

bifurcation of ten small amplitude limit cycles by perturbing a quadratic hamiltonian system with cubic polynomials abstract this paper contains two parts in the first part we shall study the abelian integrals for oladek s example 13 in which the existence of 11 small amplitude limit cycles around a singular point in a particular cubic vector field is claimed we will show that the bases chosen in the proof of 13 are not independent which leads to failure in drawing the conclusion of the existence of 11 limit cycles in this example in the second part we present a good combination of melnikov function method and focus value or normal form computation method to study bifurcation of limit cycles an example by perturbing a quadratic hamiltonian system with cubic polynomials is presented to demonstrate the advantages of both methods and 10 small amplitude limit cycles bifurcating from a center are obtained by using up to 5th order melnikov functions

a hierarchy based on output multiplicity abstract the class np k v consists of those partial multivalued functions that can be computed by a nondeterministic polynomial time bounded transducer that has at most k distinct values on any input we define the output multiplicity hierarchy to consist of the collection of classes np k v for all positive integers k 1 in this paper we investigate the strictness of the output multiplicity hierarchy and establish three main results pertaining to this 1 1 if for any k 1 the class np k v collapses into the class np k 1 v then the polynomial hierarchy collapses to 2 p 2 2 if the converse of the above result is true then any proof of this converse cannot relativize we exhibit an oracle relative to which the polynomial hierarchy collapses to p np but the output multiplicity hierarchy is strict 3 3 relative to a random oracle the output multiplicity hierarchy is strict this result is in contrast to the still open problem of the strictness of the polynomial hierarchy relative to a random oracle in introducing the technique for the third result we prove a related result of interest relative to a random oracle up np

erratum the polynomial time hierarchy collapses if the boolean hierarchy collapses it is shown that if the boolean hierarchy bh collapses then there exists a sparse set s such that text co np subseteq text np s and therefore the polynomial time hierarchy ph col

bounded query machines on np and pspace abstract it is shown that np is equal to pspace if and only if for every oracle set a np a is equal to the class npquery a of languages accepted by nondeterministic polynomial space bounded oracle machines that are allowed to query the oracle for a only a polynomial number of times further it is shown that there is an oracle set b such that npquery b is not equal to the class pquery b of languages accepted by deterministic polynomial space bounded oracle machines that are allowed to query the oracle for b only a polynomial number of times

relativizing time space and time space consider the classes of formal languages specified by nondeterministic acceptors that operate simultaneously within time bounds from a set mathcal t and space bounds from a set mathcal s how large must the time bounds be in order to obtain all of the languages specified by nondeterministic acceptors that operate within space bounds from mathcal s how large must the space bounds be in order to obtain all of the languages specified by nondeterministic acceptors that operate with time bounds from mathcal t the first question is shown to be equivalent with appropriate restrictions on mathcal s and mathcal t to the question of whether it matters if the time bounds apply to all of the steps or only to the steps which query the oracle the second question is shown to be equivalent to the question of whether it matters if the space bounds apply to all of the configurations or only to the configurations in which the oracle is queried these results generalize a more specific result 6

lq optimal control of hyperbolic pdaes the linear quadratic control synthesis for a set of coupled first order hyperbolic partial differential and algebraic equations is presented by using the infinite dimensional hilbert state space representation of the system and the well known operator riccati equation ore method solving the algebraic equations and substituting them into the partial differential equations pdes results in a model consisting of a set of pure hyperbolic pdes the resulting pde system involves a hyperbolic operator in which the velocity matrix is spatially varying non symmetric and its eigenvalues are not necessarily negative through of the domain the c0 semigroup generation property of such an operator is proven and it is shown that the generated c0 semigroup is exponentially stable and consequently the ore has a unique and non negative solution conversion of the ore into a matrix riccati differential equation allows the use of a numerical scheme to solve the control problem

an analysis and design method for uncertain variable structure systems with bounded controllers we present a method for estimating the asymptotic stability region asr of uncertain variable structure systems with bounded switching feedback controllers using linear matrix inequalities lmis we estimate the asr and we show the exponential stability of the closed loop control system in the estimated asr we also give a simple lmi based method for designing switching surfaces that will make the estimated asr big finally we give numerical examples in order to show the effectiveness of our method

on the structure of poincare lyapunov constants for the weak focus of polynomial vector fields in this paper generalized poincare lyapunov constants vi i 1 2 are defined and an explicit formula is given for the lyapunov functions it is shown that vi is determined modulo v1 vi 1 this is used for the investigation and creation of limit cycles

asymptotic stabilization of minimum phase nonlinear systems how a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown more precisely conditions under which for every compact set of initial states it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described the theory includes the development of globally defined transformations of the system equations to their global normal form

boundary second order sliding mode control of an uncertain heat process with unbounded matched perturbation the primary concern of the present paper is the regulation of an uncertain heat process with collocated boundary sensing and actuation the underlying heat process is governed by an uncertain parabolic partial differential equation pde with neumann boundary conditions it exhibits an unknown constant diffusivity parameter and it is affected by a smooth boundary disturbance which is not available for measurements and which is possibly unbounded in magnitude the proposed robust synthesis is formed by the linear feedback design and by the twisting second order sliding mode control algorithm suitably combined and re worked in the infinite dimensional setting a non standard lyapunov functional is invoked to establish the global asymptotic stability in a sobolev space involving spatial state derivatives of the same order as that of the plant equation the stability proof is accompanied by a set of simple tuning rules for the controller parameters the effectiveness of the developed control scheme is supported by simulation results

strong nondeterministic polynomial time reducibilities abstract building on the work of adleman and manders 1 we define a new class of reducibilities the strong nondeterministic polynomial time reducibilities we show that several reducibilities in this class differ over the recursive sets and assuming np conp that they each determine different classes of np hard sets comparisons in strength are also made between the strong nondeterministic polynomial time reducibilities and the classes of deterministic and non deterministic polynomial time reducibilities in ladner lynch and selman 6

lower bounds for the hilbert number of polynomial systems let h m denote the maximal number of limit cycles of polynomial systems of degree m it is called the hilbert number the main part of hilbert s 16th problem posed in 1900 is to find its value the problem is still open even for m 2 however there have been many interesting results on the lower bound of it for m 2 in this paper we give some new lower bounds of this number the results obtained in this paper improve all existing results for all m 7 based on some known results for m 3 4 5 6 in particular we obtain that h m grows at least as rapidly as 12ln2 m 2 2ln m 2 for all large m

the center problem for a family of systems of differential equations having a nilpotent singular point abstract we study the analytic system of differential equations in the plane x y t i 0 f q p 2 i s where p q n p q s n 1 p q 0 n n and f i p i q i t are quasi homogeneous vector fields of type t p q and degree i with f q p y 0 t and q q p 2 s 1 0 0 the origin of this system is a nilpotent and monodromic isolated singular point we prove for this system the existence of a lyapunov function and we solve theoretically the center problem for such system finally as an application of the theoretical procedure we characterize the centers of several subfamilies

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

sliding mode boundary control of an euler bernoulli beam subject to disturbances in this paper a control technique is presented for an undamped pinned pinned euler bernoulli beam with control inputs and bounded disturbances on one boundary the control strategy drives the system to its origin at an arbitrary exponential rate in the presence of the disturbances this is achieved in two main steps first a backstepping transformation is used to convert the marginally stable euler bernoulli beam system to a new form that has an exponentially stable homogeneous form control inputs are needed to fully convert the system to this form however since they are distorted by unknown bounded disturbances the next step implements a sliding mode controller to account for them the proposed sliding manifolds require a combination of classical and second order techniques in order to avoid discontinuous chattering on the physical system therefore the continuous sliding mode controllers developed return the beam to its origin at an arbitrary exponential rate and do so in the presence of unknown bounded disturbances on the boundary the main contributions of this paper with respect to previous backstepping designs for the euler bernoulli beam are that all three of the following goals are accomplished together i steady state position is the origin ii decay rate has no theoretical restrictions and iii is robust to bounded disturbances

bounded queries to sat and the boolean hierarchy we study the complexity of decision problems that can be solved by a polynomial time turing machine that makes a bounded number of queries to an np oracle depending on whether we allow some queries to depend on the results of other queries we obtain two probably different hierarchies we present several results relating the bounded np query hierarchies to each other and to the boolean hierarchy we also consider the similarly defined hierarchies of functions that can be computed by a polynomial time turing machine that makes a bounded number of queries to an np oracle we present relations among these two hierarchies and the boolean hierarchy in particular we show for all k that there are functions computable with 2k parallel queries to an np set that are not computable in polynomial time with k serial queries to any oracle unless p np as a corollary k 1 parallel queries to an np set allow us to compute more functions than are computable with only k parallel queries to an np set unless p np the same is true of serial queries similar results hold for all tt self reducible sets using a mind change technique we show that 2k 1 parallel queries to an np set allow us to accept in polynomial time exactly the same sets as can be accepted in polynomial time with k serial queries to an np set in fact the same is true for any class in place of np that is closed under polynomial time positive bounded truth table reductions this contrasts with the expected result for function computations with an np oracle beigel 1988 in addition we show that the boolean hierarchy and the bounded query hierarchies of languages either stand or collapse together finally we show that if the boolean hierarchy collapses to any level but the zeroth deterministic polynomial time then for all k there are functions computable in polynomial time with k parallel queries to an np set that are not computable in polynomial time with k 1 serial queries to any set np complete sets are p superterse

oracles that compute values this paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in npmv the class of all multivalued partial functions that are computable nondeterministically in polynomial time concerning deterministic polynomial time reducibilities it is shown that a multivalued partial function is polynomial time computable with k adaptive queries to npmv if and only if it is polynomial time computable via 2k 1 nonadaptive queries to npmv a characteristic function is polynomial time computable with k adaptive queries to npmv if and only if it is polynomial time computable with k adaptive queries to np unless the boolean hierarchy collapses for every k k adaptive nonadaptive queries to npmv are different than k 1 adaptive nonadaptive queries to npmv nondeterministic reducibilities lowness and the difference hierarchy over npmv are also studied the difference hierarchy for partial functions does not collapse unless the boolean hierarchy collapses but surprisingly the levels of the difference and bounded query hierarchies do not interleave as is the case for sets unless the polynomial hierarchy collapses