global stabilization for a class of switched nonlinear feedforward systems abstract the problem of global stabilization for a class of switched nonlinear feedforward systems under arbitrary switchings is investigated in this paper based on the integrator forwarding technique and the common lyapunov function method we design bounded state feedback controllers of individual subsystems to guarantee asymptotic stability of the closed loop system a common coordinate transformation of all subsystems is exploited to avoid individual coordinate transformations for subsystems that are required when applying the forwarding recursive design scheme an example is provided to demonstrate the effectiveness of the proposed design method

control lyapunov razumikhin functions and robust stabilization of time delay systems motivated by control lyapunov functions and razumikhin theorems on stability of time delay systems we introduce the concept of control lyapunov razumikhin functions clrf the main reason for considering clrfs is construction of robust stabilizing control laws for time delay systems most existing universal formulas that apply to clfs are not applicable to clrfs it turns out that the domination redesign control law applies achieving global practical stability and under an additional assumption global asymptotic stability this additional assumption is satisfied in the practically important case when the quadratic part of a clrf is itself a clrf for the jacobian linearization of the system the clrf based domination redesign possesses robustness to input unmodeled dynamics including an infinite gain margin while in general construction of clrfs is an open problem we show that for several classes of time delay systems a clrf can be constructed in a systematic way

an influence of unilateral sources and sinks in reaction diffusion systems exhibiting turing s instability on bifurcation and pattern formation abstract we consider a general reaction diffusion system exhibiting turing s diffusion driven instability in the first part of the paper we supplement the activator equation by unilateral integral sources and sinks of the type k u x k d k and k u x k d k these terms measure an average of the concentration over the set k and are active only when this average decreases below or increases above the value of the reference spatially homogeneous steady state which is shifted to the origin we show that the set of diffusion parameters in which spatially heterogeneous stationary solutions can bifurcate from the reference state is smaller than in the classical case without any unilateral integral terms this problem is studied for the case of mixed pure neumann and periodic boundary conditions in the second part of the paper we investigate the effect of both unilateral terms of the type u u and unilateral integral terms on the pattern formation using numerical experiments on the system with well known schnakenberg kinetics

new beams of global bifurcation points for a reaction diffusion system with inequalities or inclusions abstract we consider a reaction diffusion system of activator inhibitor or substrate depletion type which is subject to diffusion driven instability we show that an obstacle e g a unilateral membrane modeled either in terms of inequalities or of inclusions introduces whole beams of new global bifurcation points of spatially non homogeneous stationary solutions which lie in parameter domains which are excluded as bifurcation points for the problem without the obstacle

unilateral sources and sinks of an activator in reaction diffusion systems exhibiting diffusion driven instability abstract a reaction diffusion system exhibiting turing s diffusion driven instability is considered the equation for an activator is supplemented by unilateral terms of the type s x u s x u describing sources and sinks active only if the concentration decreases below and increases above respectively the value of the basic spatially constant solution which is shifted to zero we show that the domain of diffusion parameters in which spatially non homogeneous stationary solutions can bifurcate from that constant solution is smaller than in the classical case without unilateral terms it is a dual information to previous results stating that analogous terms in the equation for an inhibitor imply the existence of bifurcation points even in diffusion parameters for which bifurcation is excluded without unilateral sources the case of mixed dirichlet neumann boundary conditions as well as that of pure neumann conditions is described

hasse principle for hermitian spaces over semi global fields abstract in a recent paper colliot thelene parimala and suresh conjectured that a local global principle holds for projective homogeneous spaces under connected linear algebraic groups over function fields of p adic curves in this paper we show that the conjecture is true for any linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew hermitian spaces over central simple algebras with involutions the proof implements patching techniques of harbater hartmann and krashen as an application we obtain a springer type theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p adic curves

the period index problem for the brauer group of an algebraic surface in this paper we show that the period equals the index for elements of brauer groups of function fields of surfaces a key idea of the proof is that any azumaya algebra over a surface can be transformed into an azumaya algebra that is unobstructed

brief a receding horizon approach to the nonlinear h control problem the receding horizon rh methodology is extended to the design of a robust controller of h type for nonlinear systems using the nonlinear analogue of the fake h algebraic riccati equation we derive an inverse optimality result for the rh schemes for which increasing the horizon causes a decrease of the optimal cost function this inverse optimality result shows that the input output map of the closed loop system obtained with the rh control law has a bounded l 2 gain robustness properties of the nonlinear h control law in face of dynamic input uncertainty are considered

bifurcation for a reaction diffusion system with unilateral and neumann boundary conditions abstract we consider a reaction diffusion system of activator inhibitor or substrate depletion type which is subject to diffusion driven instability if supplemented by pure neumann boundary conditions we show by a degree theoretic approach that an obstacle e g a unilateral membrane modeled in terms of inequalities introduces new bifurcation of spatial patterns in a parameter domain where the trivial solution of the problem without the obstacle is stable moreover this parameter domain is rather different from the known case when also dirichlet conditions are assumed in particular bifurcation arises for fast diffusion of activator and slow diffusion of inhibitor which is the difference from all situations which we know

nonlinear feedback control of highly manoeuvrable aircraft abstract this paper describes the application of nonlinear quadratic regulator nlqr theory to the design of control laws for a typical high performance aircraft the nlqr controller design is performed using truncated solutions of the hamil ton jacobi bellman equation of optimal control theory the performance of the nlqr controller is compared with the performance of a conventional p i gain scheduled controller designed by applying standard frequency response techniques to the equations of motion of the aircraft linearized at various angles of attack both techniques result in control laws which are very similar in structure to one another and which yield similar performance the results of applying both control laws to a high g vertical turn are illustrated by nonlinear simulation

bifurcation points for a reaction diffusion system with two inequalities abstract we consider a reaction diffusion system of activator inhibitor or substrate depletion type which is subject to diffusion driven instability we show that obstacles e g a unilateral membrane for both quantities modeled in terms of inequalities introduce a new bifurcation of spatially non homogeneous steady states in the domain of stability of the trivial solution of the corresponding classical problem without obstacles

integrator forwarding a new recursive nonlinear robust design we consider the global stabilization of nonlinear systems in strict feedforward form we show that these systems while not feedback linearizable can be globally transformed by feedback and diffeomorphism into lower triangular form the transformation is explicit and recursive we employ this transformation to design a globally stabilizing controller with optimality properties and an input to state stability property with respect to matched uncertainties c 1997 elsevier science ltd

dynamic lyapunov functions lyapunov functions are a fundamental tool to investigate stability properties of equilibrium points of linear and nonlinear systems the existence of lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse lyapunov theorems nevertheless the actual computation of the analytic expression of the function may be difficult herein we propose an approach to avoid the computation of an explicit solution of the lyapunov partial differential inequality introducing the concept of dynamic lyapunov function these functions allow to study stability properties of equilibrium points similarly to standard lyapunov functions in the former however a positive definite function is combined with a dynamical system that render dynamic lyapunov functions easier to construct than lyapunov functions moreover families of standard lyapunov functions can be obtained from the knowledge of a dynamic lyapunov function by rendering invariant a desired submanifold of the extended state space the invariance condition is given in terms of a system of partial differential equations similar to the lyapunov pde differently from the latter however in the former no constraint is imposed on the sign of the solution or on the sign of the term on the right hand side of the equation several applications and examples conclude the paper

on optimal nonlinear feedback regulation of linear plants a design technique for optimal nonlinear regulation of linear plants based on a homogeneous feedback law of degree 2p 1 p frac 1 2 is developed

input output to state stability this work explores lyapunov characterizations of the input output to state stability ioss property for nonlinear systems the notion of ioss is a natural generalization of the standard zero detectability property used in the linear case the main contribution of this work is to establish a complete equivalence between the ioss property and the existence of a certain type of smooth lyapunov function as corollaries one shows the existence of norm estimators and obtains characterizations of nonlinear detectability in terms of relative stability and of finite energy estimates

clf based designs with robustness to dynamic input uncertainties abstract the problem of robust stabilization of nonlinear systems in the presence of input uncertainties is of great importance in practical implementation stabilizing control laws may not be robust to this type of uncertainty especially if cancellation of nonlinearities is used in the design by exploiting a connection between robustness and optimality domination redesign of the control lyapunov function clf based sontag s formula has been shown to possess robustness to static and dynamic input uncertainties in this paper we provide a sufficient condition for the domination redesign to apply this condition relies on properties of local homogeneous approximations of the system and of the clf we show that an inverse optimal control law may not exist when these conditions are violated and illustrate how these conditions may guide the choice of a clf which is suitable for domination redesign

brief an approach for solving the hamilton jacobi isaacs equation hjie in nonlinear h control in this paper we present an approach to the solution of the hamilton jacobi isaacs equation hjie arising in the h control problem for nonlinear systems we show that the hjie can be solved analogously to a scalar quadratic equation with some additional side conditions and present a computational procedure for determining symmetric solutions examples are given for second order affine nonlinear systems to illustrate the procedure and the method can be extended to higher order systems

guaranteed robustness properties of multivariable nonlinear stochastic optimal regulators we study the robustness of optimal regulators for nonlinear deterministic and stochastic multiinput dynamical systems under the assumption that all state variables can be measured we show that under mild assumptions such nonlinear regulators have a guaranteed infinite gain margin moreover they have a guaranteed 50 percent gain reduction margin and a 60 degree phase margin in each feedback channel provided that the system is linear in the control and the penalty to the control is quadratic thus extending the well known properties of lq regulators to nonlinear optimal designs these results are also valid for infinite horizon average cost stochastic optimal control problems

adding integral action for open loop exponentially stable semigroups and application to boundary control of pde systems the article deals with the output feedback regulation of exponentially stable systems by an integral controller we propose appropriate lyapunov functionals to prove exponential stability of the closed loop system an example of parabolic partial differential equation pde systems and an example of hyperbolic systems are worked out to show how exponentially stabilizing integral controllers are designed the proof is based on a novel lyapunov functional construction that employs the forwarding techniques

q admissibility of solvable groups abstract a finite group g is called q admissible if there exists a division algebra finite dimensional and central over the rationals q which is a crossed product for g it is proved that a finite solvable group is q admissible if and only if all its sylow subgroups are metacyclic

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

applications of patching to quadratic forms and central simple algebras this paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields in particular we use a patching approach to reprove and generalize a recent result of parimala and suresh in preprint arxiv 0708 3128 2007 on the u invariant of p adic function fields p 2 the strategy relies on a local global principle for homogeneous spaces for rational algebraic groups combined with local computations

admissibility of groups over function fields of p adic curves let k be a field and g a finite group the question of admissibility of g over k was originally posed by schacher who gave partial results in the case k q in this paper we give necessary conditions for admissibility of a finite group g over function fields of curves over complete discretely valued fields using this criterion we give an example of a finite group which is not admissible over qp t we also prove a certain hasse principle for division algebras over such fields

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

some global bifurcation results for elastic plates this paper is concerned with global bifurcation results of some buckling problems for thin elastic plates under von karman s model we consider these problems as applications of a general topological degree analysis for bifurcation of variational inequalities

asymptotic properties of feedback solutions for a class of quantum control problems control of quantum systems described by the schrodinger equation is considered feedback control laws are developed for orbit tracking via controlled hamiltonians and their asymptotic properties are analyzed numerical integration via time splitting is also investigated and used to demonstrate the feasibility of the proposed feedback laws

some polynomials over q t and their galois groups examples of polynomials with galois group over q t corresponding to every transitive group through degree eight are calculated constructively demonstrating the existence of an infinity of extensions with each galois group over q through degree eight the methods used which for the most part have not appeared in print are briefly discussed

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

tamely ramified subfields of division algebras for any number field k it is unknown which finite groups appear as galois groups of extensions l k such that l is a maximal subfield of a division algebra with center k a k division algebra for k q the answer is described by the long standing q admissibility conjecture we extend a theorem of neukirch on embedding problems with local constraints in order to determine for every number field k what finite solvable groups g appear as galois groups of tamely ramified maximal subfields of k division algebras generalizing liedahl s theorem for metacyclic g and sonn s solution of the q admissibility conjecture for solvable groups

global bifurcation for quasivariational inequalities of reaction diffusion type abstract we consider a reaction diffusion system with implicit unilateral boundary conditions introduced by u mosco we show that global continua of stationary spatially nonhomogeneous solutions bifurcate in the domain of parameters where bifurcation in the case of classical boundary conditions is excluded the problem is formulated as a quasivariational inequality and the proof is based on the leray schauder degree

adaptive control of robot manipulators via velocity estimated feedback an approach for designing robot controllers using state space feedback is presented the robot model parameters are assumed to be unknown and velocity measurements are assumed not to be available this asymptotically stable control scheme combines an adaptive control law with a sliding observer and does not need additional assumptions on the variation of the inertia matrix eigenvalues state observation and parameter adaptation are performed simultaneously the adaptation law the observer gains and the control law are designed on the reduced order manifold which results from the invariance of the switching surface

robustness of nonlinear state feedback a survey a survey of robustness of nonlinear state feedback is given for series perturbations there are fairly complete results showing that under mild restrictions an optimal controller can tolerate an infinite increase in gain for gain reductions there are some results for systems linear in the control in particular there is a 50 reduction tolerance if the control penalty is quadratic usually the optimal controller cannot be computed exactly there are some results showing the effects of truncation on the robustness essentially robustness is maintained but in a reduced computable part of the state space

eigenvalues and bifurcation for problems with positively homogeneous operators and reaction diffusion systems with unilateral terms abstract reaction diffusion systems satisfying assumptions guaranteeing turing s instability and supplemented by unilateral terms of type v and v are studied existence of critical points and sometimes also bifurcation of stationary spatially non homogeneous solutions are proved for rates of diffusions for which it is excluded without any unilateral term the main tool is a general result giving a variational characterization of the largest eigenvalue for positively homogeneous operators in a hilbert space satisfying a condition related to potentiality and existence of bifurcation for equations with such operators the originally non variational non symmetric system is reduced to a single equation with a positively homogeneous potential operator and the abstract results mentioned are used

on the principle of linearized stability for variational inequalities on etudie la stabilite des solutions stationnaires d inegalites variationnelles paraboliques dans des espaces de hilbert

bifurcation points and eigenvalues of inequalities of reaction diffusion type on demontre certaines assertions concernant les points de bifurcation et les valeurs propres de systemes de reaction diffusion avec des conditions unilaterales

brief a transformation approach for solving the hamilton jacobi bellman equation in h2 deterministic and stochastic optimal control of affine nonlinear systems in this paper we present a transformation approach for solving the hamilton jacobi bellman equations hjbes arising in h 2 or quadratic cost control of nonlinear deterministic and stochastic systems we show that the hjbe can be solved analogously to a scalar quadratic equation and we give a parameterization of solutions to the hjbe characterizing the solution of the optimal control problem the procedure is also generalized to include nonsmooth or viscosity solutions of the s hjbe

exact nonlinear control of large angle rotational maneuvers the rigid body attitude control problem with external torques is transformed into an equivalent linear form implementable by three double integrators the linearizing transformations themselves are formulated in vector algebra requiring no integrators for implementation it is thereby shown that optimal command generation for fast slewing maneuvers can be carried out exactly in the transformed system together with regulator design without gain scheduling for correction of unmodeled disturbances in particular a general rest to rest maneuver is computed in closed form and coupled with a previously obtained exact detumbling maneuver so that arbitrary initial conditions can be accommodated an illustrative simulation is appended

uncertainty observer based robust tracking control for high order nonlinear systems using forwarding technique based on uncertainty observers we develop a robust tracking control approach for high order nonlinear systems subject to uncertainties satisfying mismatched condition in this study by combining

decentralized robust stabilization for large scale feedforward non linear systems in this paper we consider the decentralized robust stabilization problem for large scale feedforward non linear systems which possess an upper triangular structure while in the decentralized control area the commonly studied large scale feedback non linear systems possess a lower triangular structure our decentralized robust stabilizer for each subsystem takes a nested saturation feedback with the saturation levels satisfying a set of equations and inequalities global asymptotic stability of the closed loop large scale system has been proved and a simulation example is given

optimal control using an algebraic method for control affine non linear systems a deterministic optimal control problem is solved for a control affine non linear system with a non quadratic cost function we algebraically solve the hamilton jacobi equation for the gradient of the value function this eliminates the need to explicitly solve the solution of a hamilton jacobi partial differential equation we interpret the value function in terms of the control lyapunov function then we provide the stabilizing controller and the stability margins furthermore we derive an optimal controller for a control affine non linear system using the state dependent riccati equation sdre method this method gives a similar optimal controller as the controller from the algebraic method we also find the optimal controller when the cost function is the exponential of integral case which is known as risk sensitive rs control finally we show that sdre and rs methods give equivalent optimal controllers for non linear deterministic systems examples demonstrate the proposed methods

division algebras with psl 2 q galois maximal subfields abstract if g is a finite group and k is a field then g is k admissible if there exists a g galois extension l k such that l is a maximal subfield of a k division algebra we prove that psl 2 7 is k admissible for any number field which either fails to contain 1 or which has two primes lying over the dyadic prime in addition psl 2 11 is shown to be admissible over q or any number field k with at least two extensions of the dyadic prime indeed there exist infinitely many linearly disjoint admissible extensions for these groups

on the robustness of nonlinear regulators and its application to nonlinear systems stabilization in this note we develop a new robustness property of optimal regulators of nonlinear dynamic systems the stability condition we establish presents a great advantage when compared to other stability conditions already available in the literature it does not depend explicitly on the optimal control problem solution allowing then a very simple and a priori analysis of the closed loop system stability based on that we propose a methodology for determining linear decentralized controllers which stabilize asymptotically a wide class of nonlinear dynamic systems as an application example we present and discuss in detail the control design for a two pendulum system 4

stability margins of nonlinear receding horizon control via inverse optimality using the nonlinear analog of the fake riccati equation developed for linear systems we derive an inverse optimality result for several receding horizon control schemes this inverse optimality result unifies stability proofs and shows that receding horizon control possesses the stability margins of optimal control laws

reaction diffusion systems destabilizing effect of unilateral conditions on considere un systeme de reaction diffusion dans un domaine r n soit v une solution stationnaire et spatialement homogene avec les conditions aux limites de neumann on etudie la stabilite linearisee de v

the location of noncrossed products in brauer groups of laurent series fields over global fields since amitsur s discovery of noncrossed product division algebras in 1972 their existence over more familiar fields has been an object of investigation brussel s work was a culmination of this effort exhibiting noncrossed products over the rational function field k t and the laurent series field k t over any global field k the smallest possible centers of noncrossed products witt s theorem gives a transparent description of the brauer group of k t as the direct sum of the brauer group of k and the character group of the absolute galois group of k we classify the brauer classes over k t containing noncrossed products by analyzing the fiber over for each character in witt s theorem in this way a picture of the partition of the brauer group into crossed products noncrossed products is obtained which is in principle ruled by a relation between index and number of roots of unity as a side consequence of the result there are crossed products that have a noncrossed product primary component

patching subfields of division algebras given a field f one may ask which finite groups are galois groups of field extensions e f such that e is a maximal subfield of a division algebra with center f this question was originally posed by schacher who gave partial results in the case f using patching we give a complete characterization of such groups in the case that f is the function field of a curve over a complete discretely valued field with algebraically closed residue field of characteristic zero as well as results in related cases

local global principles for torsors over arithmetic curves we consider local global principles for torsors under linear algebraic groups over function fields of curves over complete discretely valued fields the obstruction to such a principle is a version of the tate shafarevich group and for groups with rational components we compute it explicitly and show that it is finite this yields necessary and sufficient conditions for local global principles to hold our results rely on first obtaining a mayer vietoris sequence for galois cohomology and then showing that torsors can be patched we also give new applications to quadratic forms and central simple algebras

on compensating long actuator delays in nonlinear control we are interested in finite escape open loop unstable plants that are globally stabilizable in the absence of actuator delay but require controller redesign in the presence of delay the simplest such plant is z t z t 2 u t d where d is actuator delay of arbitrary length for this system we present a control law that compensates the delay and achieves feedback linearization of the entire ode delay infinite dimensional cascade however even though exponential stability is achieved the result is not global because the plant can have a finite escape with an initial condition z 0 ges 1 d before the feedback control reaches it at t d we prove a stability result whose region of attraction is essentially z 0 1 d and for which we derive an asymptotic stability bound in terms of the system norm z t 2 intt t d u thetas 2dthetas

satisficing a new approach to constructive nonlinear control the main contribution of this paper is a constructive parameterization of the class of almost smooth universal formulas which render a system asymptotically stable with respect to a known control lyapunov function clf and a constructive parameterization of a class of inverse optimal universal formulas having kalman like stability margins the novelty of the parameterization is that it is given in terms of two function which are constrained to be locally lipschitz and satisfy convex constraints the implication of this result is that the clf universal formula approach can be combined with a priori performance objectives to design high performance control strategies two examples illustrate the approach

unilateral regulation breaks regularity of turing patterns we consider a reaction diffusion system undergoing turing instability and augment it by an additional unilateral source term we investigate its influence on the turing instability and on the character of resulting patterns the nonsmooth positively homogeneous unilateral term v has favorable properties but the standard linear stability analysis cannot be performed we illustrate the importance of the nonsmoothness by a numerical case study which shows that the turing instability can considerably change if we replace this term by its arbitrarily precise smooth approximation however the nonsmooth unilateral term and all its approximations yield qualitatively similar patterns although not necessarily developing from small disturbances of the spatially homogeneous steady state further we show that the unilateral source breaks the approximate symmetry and regularity of the classical patterns and yields asymmetric and irregular patterns moreover a given system with a unilateral source produces spatial patterns even for diffusion parameters with ratios closer to 1 than the same system without any unilateral term

forwarding design with prescribed local behavior among the non linear control techniques some lyapunov design methods forwarding backstepping take advantage of the structure of the system feedforward form feedback form to formulate a continuous control law which stabilizes globally and asymptotically the equilibrium in addition to stabilization we focus on the local behaviour of the closed loop system providing conditions under which we can predetermine the behaviour around the origin for feedforward systems

almost locally free fields abstract using the positive solution of the general abhyankar s conjecture we prove that the fundamental group 1 c of the smooth connected affine curve c is almost free that is for each positive integer e and for almost all 1 e 1 c e in the sense of the haar measure the closed subgroup of 1 c generated by 1 e is profinite free on e generators this implies a theorem of harbater stevenson proved by other means that every finite embedding problem for 1 c is solvable if we restrict the problem to a suitable open subgroups

on the gain margin of nonlinear and optimal regulators the robustness of nonlinear regulators for nonlinear systems with respect to variations in gain is investigated it is shown that there exist regulators that produce asymptotically stable closed loop systems but do not tolerate any variation in gain without instability however if the linearized closed loop system is also asymptotically stable then there is always some gain margin for a wide class of optimal regulators it is shown that the gain margin is infinite with respect to increases in gain and that decreases down to 0 5 can be tolerated the robustness properties of linear quadratic control laws are thus generalized

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft

smooth bifurcation branches of solutions for a signorini problem abstract we study a bifurcation problem for the equation u u g u u 0 on a rectangle with signorini boundary conditions on a part of one edge and mixed zero dirichlet and neumann boundary conditions on the rest of the boundary here r is the bifurcation parameter and g is a small perturbation we prove under certain assumptions concerning an eigenfunction u 0 corresponding to an eigenvalue 0 of the linearized equation with the same nonlinear boundary conditions the existence of a local smooth branch of nontrivial solutions bifurcating from the trivial solutions at 0 in the direction of u 0 the contact sets of these nontrivial solutions are intervals which change smoothly along the branch the main tool of the proof is a local equivalence of the unilateral bvp to a system consisting of a corresponding classical bvp and of two scalar equations to this system classical crandall rabinowitz type local bifurcation techniques scaling and implicit function theorem are applied

degree and global bifurcation for elliptic equations with multivalued unilateral conditions abstract a bifurcation problem for an elliptic multivalued boundary value problem with a real parameter is considered the existence of global bifurcation between two eigenvalues of a certain type of the laplacian is proved for a class of abstract inclusions with compact multivalued mappings in a hilbert space it is shown how the degree can be determined near the eigenvalues of a particular type of an associated linear single valued problem and the jump of the degree is proved as a consequence global bifurcation for such abstract inclusions is obtained the weak formulation of the boundary value problem mentioned is a particular case

galerkin approximations of the generalized hamilton jacobi bellman equation in this paper we study the convergence of the galerkin approximation method applied to the generalized hamilton jacobi bellman ghjb equation over a compact set containing the origin the ghjb equation gives the cost of an arbitrary control law and can be used to improve the performance of this control the ghjb equation can also be used to successively approximate the hamilton jacobi bellman equation we state sufficient conditions that guarantee that the galerkin approximation converges to the solution of the ghjb equation and that the resulting approximate control is stabilizing on the same region as the initial control the method is demonstrated on a simple nonlinear system and is compared to a result obtained by using exact feedback linearization in conjunction with the lqr design method