analytical solution of spatial elastica and its application to kinking problem abstract an analytical solution is presented for the spatially large deformation of a thin elastic rod spatial elastica which is naturally straight and uniform with equal principal stiffnesses and is subjected to terminal loads the elastica can suffer not only flexure and torsion as in the classical kirchhoff theory but also extension and shear the present solution is expressed in integral form and described in terms of only four parameters this solution clears the difficulty with the polar singularity in the use of euler angles hence the numerical analysis is possible for various boundary value problems with no limitation in this paper we study the post buckling behavior of an elastica under the terminal twist and uniaxial end shortening and give a theoretical explanation to commonly observed phenomena such as secondary bifurcation formation of a kink snap through behavior the contact problem is analyzed in the case where the elastica contacts with itself and forms a kink these results are available for other analysis e g based on finite element approximations

bifurcations with o 3 symmetry including applications to the benard problem this paper has two purposes to study from the singularity theory point of view certain bifurcation problems which commute with the five dimensional irreducible representation of the orthogonal group o 3 and to give some implications of this study for the benard problem in spherical geometry we shall now describe in general terms our results and compare them with the work of chossat 3 whose paper motivated our own interest in the problem the benard problem is concerned with convection in a viscous fluid when it is heated from below the fluid is assumed to be confined in a spherical shell of outer radius r and inner radius qro where q is near 0 3 this choice of q is partially motivated by considering convection within the molten layer of the core of the earth see busse l for further discussion we consider the binard problem in the boussinesq approximation in this model there is a trivial solution representing pure heat conduction radially outward as the temperature on the inner sphere that is the rayleigh number r is increased this trivial solution loses stability say at r r the benard problem is the study of the resulting bifurcation chossat studies this problem using the lyapunov schmidt reduction in the boussinesq approximation the fluid is driven from the pure conduction state by a term in the momentum equation involving the gravity vector g r and a term in the temperature equation involving the gradient of the conduction temperature vt assuming that the production of heat and the density are uniform throughout the shell these vectors have the form

bifurcations of subharmonic solutions in reversible systems soit f r r n r r n une application c periodique cut on dit que l equation differentielle x f t x est reversible s il existe un operateur symetrique s sur r n tel que s 2 i de telles equations differentielles sont aussi appelees systemes avec la propriete e par rapport a s on etudie les bifurcations des solutions sous harmoniques de tels systemes reversibles

constrained control of siso bilinear systems relative degree and nonminimum phase difficulties limit the applicability of input output feedback linearization hence the need for approximations earlier work on predictive control of bilinear systems overcame these problems by means of interpolation between feedback linearization and state feedback the former providing optimality and the latter guaranteeing feasibility and stability through the use of invariant feasible polytopes the current work also makes use of polytopes in preference to ellipsoids but achieves distinctly different objectives first it is shown that feedback linearization can be used over particular polytopes without needing to resort to either approximation or interpolation then it is shown that invariant polytopes based on bilinear controllers can be much larger these two approaches are combined in an algorithm that guarantees stability over much larger initial condition sets and gives much improved closed loop performance

on finite determinacy for matrices of power series let r k x 1 ldots x s be the ring of formal power series with maximal ideal mathfrak m over a field k of arbitrary characteristic on the ring m m n of m times n matrices a with entries in r we consider several equivalence relations given by the action on m m n of a group g g can be the group of automorphisms of r combined with the multiplication of invertible matrices from the left from the right or from both sides respectively we call a finitely g determined if a is g equivalent to any matrix b with a b in mathfrak m k m m n for some finite integer k which implies in particular that a is g equivalent to a matrix with polynomial entries the classical criterion for analytic or differential map germs f k s 0 rightarrow k m 0 k mathbb r mathbb c says that f in m m 1 is finitely determined with respect to various group actions iff the tangent space to the orbit of f has finite codimension in m m 1 we extend this criterion to arbitrary matrices in m m n if the characteristic of k is 0 or more general if the orbit map is separable in positive characteristic however the problem is more subtle since the orbit map is in general not separable as we show by an example this fact had been overlooked in previous papers our main result is a general sufficient criterion for finite g determinacy in m m n in arbitrary characteristic in terms of the tangent image of the orbit map which we introduce in this paper this criterion provides a computable bound for the g determinacy of a matrix a in m m n which is new even in characteristic 0

a group theoretic approach to the global bifurcation analysis of an axially compressed cylindrical shell abstract the accurate prediction of the buckling load of thin shell structures is an important yet elusive goal it is particularly important in the aerospace industry where thin shell members are commonly used as structural elements due to a lack of adequate analytical results current practices in industry put heavy reliance on experimental testing and empirical data to supplement theoretical analysis see d bushnell computerized buckling analysis of shells martinus nijhoff dordrecht 1985 this paper focuses on recent results of the group theoretic approach to a numerical global postbuckling analysis of a perfect axially compressed cylindrical shell with built in end conditions the built in end conditions obviate the existence of a trivial membrane solution branch the example of an axially compressed cylindrical shell was chosen because it is well known that for thin shells the primary axisymmetric solution branch is riddled with closely spaced symmetry breaking bifurcation points in a numerical arc length continuation scheme the close proximity of the bifurcation points on the primary path manifests itself in severe ill conditioning of the tangent stiffness matrix group theory helps one systematically find an optimal set of basis vectors or symmetry modes which reflect the symmetry of a given solution path the immediate payoff in using these symmetry modes as basis vectors is that the tangent stiffness matrix block diagonalizes and the numerical ill conditioning is avoided thus an efficient and accurate technique for computing solution branches of a specific type and a subsequent diagnosis for symmetry breaking bifurcations is made relatively simple understanding the global behavior of the perfect structure is crucial in identifying critical imperfections and will ultimately diminish the heavy reliance on expensive experimental verification

unconstrained and constrained stabilisation of bilinear discrete time systems using polyhedral lyapunov functions the constrained and unconstrained stabilisation problem of discrete time bilinear systems is investigated using polyhedral lyapunov functions conditions for a polyhedral set to be both positively invariant and domain of attraction for systems with second order polynomial nonlinearities are first established then systematic methods for the determination of stabilising linear feedback for both constrained and unconstrained bilinear systems are presented attention is drawn to the case where no linear control law rendering the pre specified desired domain of attraction positively invariant exists for this case an approach guaranteeing the existence of a possibly suboptimal solution is established

finite determinacy of matrices and ideals abstract the main aim of this paper is to characterize ideals i in the power series ring r k x 1 x s that are finitely determined up to contact equivalence by proving that this is the case if and only if i is an isolated complete intersection singularity provided dim r i 0 and k is an infinite field of arbitrary characteristic here two ideals i and j are contact equivalent if the local k algebras r i and r j are isomorphic if i is minimally generated by a 1 a m we call i finitely contact determined if it is contact equivalent to any ideal j that can be generated by b 1 b m with a i b i x 1 x s k for some integer k we give also computable and semicontinuous determinacy bounds the above result is proved by considering left right equivalence on the ring m m n of m n matrices a with entries in r and we show that the fitting ideals of a finitely determined matrix in m m n have maximal height a result of independent interest the case of ideals is treated by considering 1 column matrices fitting ideals together with a special construction are used to prove the characterization of finite determinacy for ideals in r some results of this paper are known in characteristic 0 but they need new and more sophisticated arguments in positive characteristic partly because the tangent space to the orbit of the left right group cannot be described in the classical way in addition we point out several other oddities including the concept of specialization for power series where the classical approach due to krull does not work anymore we include some open problems and a conjecture

on families of square matrices in this paper we classify families of square matrices up to the following natural equivalence thinking of these families as germs of smooth mappings from a manifold to the space of square matrices we allow arbitrary smooth changes of co ordinates in the source and pre and post multiply our family of matrices by generally distinct families of invertible matrices all dependent on the same variables we obtain a list of all the corresponding simple mappings that is those that do not involve adjacent moduli this is a non linear generalisation of the classical notion of linear systems of matrices we also make a start on an understanding of the associated geometry 2000 mathematics subject classification 58k40 58k50 58k60 32s25

finite dimensional approximation of bifurcation problems in presence of symmetries we give here a brief summary of the results of 4 14 the aim of our work was not only the finite dimensional approximation of the specific problem studied in 3 but we also wanted to show how to adapt to discrete problems and how to use in this case some classical mathematical tools e g the splitting lemma of gromoll meyer magnus the singularity theory etc such considerations are developed in 5 in particular we prove in 5 that in most cases e g in the examples given in 6 7 9 10 the discrete bifurcation problem behaves as an imperfect bifurcation problem and we improve the methods of 12 another way to use universal unfoldings in the approxima tion of bifurcation problems is suggested by 2

pattern selection in long wavelength convection abstract the long wavelength of the first instability in rayleigh benard convection between nearly thermally insulating horizontal plates is typical of a variety of physical systems the evolution of such an instability is described by the equation t 2 4 2 2 2 2 where is the planform function is the scaled rayleigh number and 1 the quantities represent the effects of finite biot number asymmetry in the boundary conditions at top and bottom and departures from the boussinesq approximation respectively the quantity when the original problem is self adjoint but otherwise planform selection is studied for 1 using equivariant bifurcation theory on the square lattice both rolls and squares can be stable depending on the parameters and the possible secondary bifurcations located near various codimension two singularities are analyzed on the hexagonal lattice the primary bifurcation is always degenerate when 0 of the six primary solution branches possible in this case the hexagon branch is stable when 0 both rolls and hexagons bifurcate supercritically but rolls are stable finally when and or 0 a hysteretic transition to h or h occurs depending on sgn k 2 c if stable h h coexist at larger amplitudes but if k 2 c 0 1 a further hysteretic transition takes place with increasing amplitude in which the hexagons are replaced by stable rolls

receding horizon control of nonlinear systems the receding horizon control strategy provides a relatively simple method for determining feedback control for linear or nonlinear systems the method is especially useful for the control of slow nonlinear systems such as chemical batch processes where it is possible to solve sequentially open loop fixed horizon optimal control problems online the method has been shown to yield a stable closed loop system when applied to time invariant or time varying linear systems it is shown that the method also yields a stable closed loop system when applied to nonlinear systems

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

a multiplicity of localised buckling modes for twisted rod equations the kirchhoff love equations governing the spatial equilibria of long thin elastic rods subject to end tension and moment are reviewed and used to examine the existence of localized buckling solutions the effects of shear and axial extension are not considered but the model does additionally allow for nonlinear constitutive laws under the assumption of infinite length the dynamical phase space analogy allows one to use techniques from dynamical systems theory to characterize many possible equilibrium paths localizing solutions correspond to homoclinic orbits of the dynamical system under non dimensionalization the twisted rod equations are shown to depend on a single load parameter and the bifurcation behaviour of localizing solutions of this problem is investigated using analytical and numerical techniques first in the case of a rod with equal principal bending stiffnesses where the equilibrium equations are completely integrable a known one parameter family of localizing solutions is computed for a variety of subcritical loads load deflection diagrams are computed for this family and certain materially nonlinear constitutive laws are shown to make little difference to the qualitative picture the breaking of the geometrical circular symmetry destroys complete integrability and in particular breaks the non transverse intersection of the stable and unstable manifolds of the trivial steady state the resulting transverse intersection which is already known to lead to spatial chaos is explicitly demonstrated to imply a multitude of localized buckling modes a sample of primary and multi modal solutions are computed numerically aided by the reversibility of the differential equations finally parallels are drawn with the conceptually simpler problem of a strut resting on a nonlinear elastic foundation for which much more information is known about the global behaviour of localized buckling modes

robust receding horizon control of constrained nonlinear systems we present a method for the construction of a robust dual mode receding horizon controller which can be employed for a wide class of nonlinear systems with state and control constraints and model error the controller is dual mode in a neighborhood of the origin the control action is generated by a linear feedback controller designed for the linearized system outside this neighborhood receding horizon control is employed existing receding horizon controllers for nonlinear continuous time systems which are guaranteed to stabilize the nonlinear system to which they are applied require the exact solution at every instant of an optimal control problem with terminal equality constraints these requirements are considerably relaxed in the dual mode receding horizon controller presented in this paper stability is achieved by imposing a terminal inequality rather than an equality constraint only approximate minimization is required a variable time horizon is permitted robustness is achieved by employing conservative state and stability constraint sets thereby permitting a margin of error the resultant dual mode controller requires considerably less online computation than existing receding horizon controllers for nonlinear constrained systems

systematic description of imperfect bifurcation behavior of symmetric systems abstract a systematic method is presented for describing experimental curves of force vs strain of a system with regular polygonal dihedral group symmetry subject to bifurcation behavior with an aim toward overcoming the following problems 1 it is difficult to judge whether the system is undergoing bifurcation or not 2 the perfect behavior of the system cannot be known due to the presence of initial imperfections 3 those curves are often qualitatively different from bifurcation diagrams predicted by mathematics the tools employed are the asymptotic theory for imperfect bifurcation such as the koiter law and the stochastic theory of initial imperfections the former theory is extended in this paper to the system with regular polygonal symmetry to present asymptotic laws for recovering perfect curves with reference to the experimental ones these laws are formulated for physically observable displacements instead of the variables in the mathematical bifurcation diagrams in order to make them readily applicable to the experimental curves the stochastic theory is combined with an asymptotic law to develop a means to identify the multiplicity of the bifurcation point the systematic method for describing the experimental curves developed in this manner is applied to the bifurcation analysis of regular polygonal truss domes to testify its validity furthermore this method is applied to the shear behavior of cylindrical sand specimens to show that they in fact are undergoing bifurcation and in turn to demonstrate the importance of a viewpoint of bifurcation in the study of shear behavior of materials the need of a dual viewpoint of bifurcation and plasticity in the study of constitutive relationship of materials is emphasized to conclude the paper

classification and unfolding of degenerate hopf bifurcations with o 2 symmetry no distinguished parameter abstract the hopf bifurcation in the presence of o 2 symmetry is considered when the bifurcation breaks the symmetry the critical imaginary eigenvalues have multiplicity two and generically there are two primary branches of periodic orbits which bifurcate simultaneously in applications these correspond to rotating traveling waves and standing waves using equivariant singularity theory a classification of all such bifurcations up to and including codimension three is presented no distinguished parameter is assumed the universal unfoldings reveal the existence of both 2 tori and 3 tori corresponding to quasiperiodic waves with two and three independent frequencies respectively

pattern selection in oscillatory rotating convection abstract three dimensional pattern selection in a low prandtl number boussinesq fluid with stress free boundaries where the onset of convection is oscillatory is explored restricting the problem to a square lattice the normal form coefficients are calculated as functions of the square root of the taylor number and the prandtl number there is a large region of the plane where a heteroclinic cycle connecting four travelling roll states is stable as is decreased the cycle undergoes a transverse loss of stability creating quasiperiodic orbits which may themselves become chaotic all these stable dynamics occur at onset although conjectured on the basis of general results from symmetric bifurcation theory and well known for steady convection as the kuppers lortz instability j fluid mech 35 1969 609 cycling behaviour has not previously been demonstrated directly from the hydrodynamic equations in the oscillatory case a second region of the plane contains stable travelling roll solutions we examine their stability to perturbations at varying angles and demonstrate the existence of small angle instabilities of travelling rolls

feedback control of linear discrete time systems under state and control constraints in this paper the problem of stabilizing linear discrete time systems under state and control linear constraints is studied based on the concept of positive invariance existence conditions of linear state feedback control laws respecting both the constraints are established these conditions are then translated into an algorithm of linear programming

brief robust receding horizon predictive control for systems with uncertain dynamics and input saturation a receding horizon predictive control method which assures stability for systems with model uncertainty and input saturation is derived by extending earlier work in two important respects i ellipsoidal invariant sets are replaced by polyhedral invariant sets and ii the constraint that the state lie in an invariant set is invoked n steps ahead the new algorithm allows for a considerable enlargement of the stabilizable set and provides extra degrees of freedom with which to improve performance

event location for ordinary differential equations abstract an initial value problem for y f t y may have an associated event function g t y an event is said to occur at t when g t y t 0 we consider problems for which the definition of f t y changes at the time of an event a number of solvers locate events and restart the integration there so as to deal with the changes in f but there is little theoretical support for what is done here we prove that with reasonable assumptions about the problem and the solver the error of the numerical solution is qualitatively the same whether or not events occur numerical results obtained with a wide range of solvers confirm the theory developed here

computational methods for bifurcation problems with symmetries with special attention to steady state and hopf bifurcation points we show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations we extend the approach by werner 1988 by presenting methods for the detection of bifurcation points and the computation of multiple hopf points the essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group the group theoretical tool is an a priori knowledge of the interaction of certain subgroups 0 and of at possibly multiple steady state or hopf bifurcation points minimal 0 breaking bifurcation we introduce a bifurcation graph which shows graphically this a priori information its edges represent possible symmetry breaking bifurcations our analysis follows the lines of golubitsky stewart and schaeffer 1988 but it is aimed to numerical applications we have chosen a 4 box brusselator model in order to explain our notions and ideas and to discuss the numerical procedure

normal forms for driven surface waves boundary conditions symmetry and genericity abstract the parametric excitation of standing waves in a vertically oscillated fluid can be studied by analyzing the period doubling bifurcation in the associated stroboscopic map when the fluid container has symmetry the stroboscopic map will exhibit this symmetry and this in turn significantly affects the structure of the bifurcation problem for both square and circular containers the normal forms appropriate to the reduced maps for the critical mode amplitudes are discussed the experiments of simonelli and gollub with square containers strongly suggest that the effective symmetry of the problem is larger than the obvious geometric symmetry of the boundaries for the ideal fluid model of benjamin and ursell with neumann boundary conditions nbc one can show that the problem does indeed have a larger symmetry because it embeds in a model on a larger domain with periodic boundary conditions pbc the original nbc are then realized as a symmetry constraint on the solutions of this extended model the additional symmetries introduced in this way lead to a large number of effects that would otherwise be non generic including critical eigenspaces carrying reducible representations of and reduced maps on these eigenspaces that have symmetries larger than these extra symmetries relate in phase and out of phase mixed modes and lead to new branches to pure modes the case of circular geometry is also discussed as it does not allow this extension to pbc and thus provides an instructive contrast with the subtleties found for square geometry

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

brief nonlinear model predictive control with polytopic invariant sets ellipsoidal invariant sets have been widely used as target sets in model predictive control mpc these sets can be computed by constructing appropriate linear difference inclusions together with additional constraints to ensure that the ellipsoid lies within a given inclusion polytope the choice of inclusion polytope has a significant effect on the size of the target ellipsoid but the optimal inclusion polytope cannot in general be computed systematically this paper shows that use of polytopic invariant sets overcomes this difficulty allowing larger stabilizable sets without loss of performance in the interests of online efficiency consideration is focused on interpolation based mpc

symmetries and pattern selection in rayleigh benard convection abstract this paper describes the process of pattern selection between rolls and hexagons in rayleigh benard convection with reflectional symmetry in the horizontal midplane this symmetry is a consequence of the boussinesq approximation provided the boundary conditions are the same on the top and bottom plates all possible local bifurcation diagrams assuming certain non degeneracy conditions are found using only group theory the results are therefore applicable to other systems with the same symmetries rolls hexagons or a new solution regular triangles can be stable depending on the physical system rolls are stable in ordinary rayleigh benard convection the results are compared to those of buzano and golubitsky 1 without the midplane reflection symmetry the bifurcation behavior of the two cases is quite different and a connection between them is established by considering the effects of breaking the reflectional symmetry finally the relevant experimental results are described

ultimate boundedness control for uncertain discrete time systems via set induced lyapunov functions in this note linear discrete time systems affected by both parameter and input uncertainties are considered the problem of the synthesis of a feedback control assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned rate of convergence is investigated it is shown that the problem has a solution if and only if there exists a certain lyapunov function which does not belong to a preassigned class of functions e g the quadratic ones but it is determined by the target set in which ultimate boundedness is desired one of the advantages of this approach is that we may handle systems with control constraints no matching assumptions are made for systems with linearly constrained uncertainties it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets the resulting compensator may be implemented as a linear variable structure control

the classification of bifurcations with hidden symmetries we set up a singularity theoretic framework for classifying one parameter steady state bifurcations with hidden symmetries this framework also permits a non trivial linearization at the bifurcation point many problems can be reduced to this situation for instance the bifurcation of steady or periodic solutions to certain elliptic partial differential equations with neumann or dirichlet boundary conditions we formulate an appropriate equivalence relation with its associated tangent spaces so that the usual methods of singularity theory become applicable we also present an alternative method for computing those matrix valued germs that appear in the equivalence relations employed in the classification of equivariant bifurcation problems this result is motivated by hidden symmetries appearing in a class of partial differential equations defined on an n dimensional rectangle under neumann boundary conditions

on simple families of functions and their legendrian mappings we study germs of n parameter families of functions that is function germs of the type f mathbb r n times mathbb r 0 to mathbb r 0 defined on the total space of the trivial bundle mathbb r n times mathbb r to mathbb r n there is a natural notion of v equivalence for such function germs we introduce the young diagram of n parameter families satisfying a non degeneracy condition we classify all such simple n parameter families and give their versal deformations this result has direct applications to contact and projective geometry

technical communique extended invariance and its use in model predictive control the use of time invariant linear state feedback control laws for the definition of terminal invariant regions can be conservative thereby reducing the efficacy of predictive control in terms of size of stabilisable sets and closed loop performance this difficulty which is particularly pronounced in the case of nonlinear and or uncertain dynamics can be remedied through the use of time varying control laws and terminal invariant sets in existing mpc schemes however these have to be computed online thereby rendering implementation impracticable for anything other than low dimensional systems here the definition of invariance is extended to apply over n predicted control moves thereby enabling the use of pre determined offline time varying state feedback gains more importantly this extension allows for the use of local uncertainty or linear difference inclusion sets and thus affords significant improvements e g in terms of the size of terminal regions

a quasi infinite horizon nonlinear model predictive control scheme with guaranteed stability we present in this paper a novel nonlinear model predictive control scheme that guarantees asymptotic closed loop stability the scheme can be applied to both stable and unstable systems with input constraints the objective functional to be minimized consists of an integral square error ise part over a finite time horizon plus a quadratic terminal cost the terminal state penalty matrix of the terminal cost term has to be chosen as the solution of an appropriate lyapunov equation furthermore the setup includes a terminal inequality constraint that forces the states at the end of the finite prediction horizon to lie within a prescribed terminal region if the jacobian linearization of the nonlinear system to be controlled is stabilizable we prove that feasibility of the open loop optimal control problem at time t 0 implies asymptotic stability of the closed loop system the size of the region of attraction is only restricted by the requirement for feasibility of the optimization problem due to the input and terminal inequality constraints and is thus maximal in some sense

on the positive invariance of polyhedral sets for discrete time systems abstract in this note necessary and sufficient conditions for a polyhedral set to be a positively invariant set of a linear discrete time system are established

admissible sets and feedback control for discrete time linear dynamical systems with bounded controls and states for linear discrete time dynamical systems necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set a variable structure linear state feedback controller given in terms of the controls at the vertices of the polyhedral state constraint set is presented

a numerical method for the calculation of traveling wave solutions of a quench front problem a numerical method is described for the solution of the following quench front problem find u x y and such that xk u u x vq u u x yk u 0 u yy 0 f u u yy 1 0 u y 0 u y 1 the method is based on the idea of isotherm migration the resulting problem is an eigenvalue problem for a system of nonlinear cauchy riemann equations the method is very efficient in comparison with previous methods for this problem

more on the determinacy of smooth map germs they don t in this situation and i r determinacy will for notational convenience be written approximation and determinacy this notation was introduced in 4 together with some motivation brief ly ranges of determinacy give a useful way of dexcribing which map germs are equivalent to a given one

geometric quotients of unipotent group actions let g be a unipotent algebraic group over k a field of characteristic 0 which acts rationally on an affine scheme x spec a over k where a is a commutative k algebra the problem of finding sufficient and manageable conditions to guarantee that the geometric quotient x g exists is of fundamental interest in the theory of moduli spaces for local objects such as isolated singularities or cohen macaulay modules over the local ring of a singularity cf l p g p2 g h p h

flow induced bifurcations to three dimensional oscillatory motions in continuous tubes three dimensional motions of a cantilever tube carrying an incompressible fluid and having rotational symmetry about the vertical axis are examined for bifurcating oscillatory solutions the system behavior depends on two parameters rho the flow velocity and beta the mass ratio of the fluid and the tube as the flow rate is increased past a critical value the zero solution becomes unstable by a double pair of complex conjugate eigenvalues of the linearized operator crossing the imaginary axis using ideas from the center manifold theory and the method of integral averaging the governing partial differential equations are reduced to a set of four nonlinear ordinary differential equations it is then shown that the system bifurcates into two qualitatively distinct oscillatory motions these oscillatory solutions themselves for some values of the mass ratio beta bifurcate into more complex amplitude modulated oscillations

enlargement of polytopic terminal region in nmpc by interpolation and partial invariance this paper uses the concept of partial invariance to derive a sequence of linear programs in order to maximize offline the volume of an invariant polytopic set with an arbitrary predefined number of vertices subject to a bound on closed loop performance interpolation techniques are used to determine a nonlinear control law which is optimal with respect to a closed loop cost bound through the on line solution of a linear program the invariant polytope is also used to define a receding horizon control law through an appropriate terminal constraint and cost

invariance under the dihedral group and application to bifurcation problems soit un groupe d ordre agissant lineairement sur r s et soient 1 2 t des generateurs homogenes de r x 1 x s l algebre de tous les polynomes invariants a s variables si f est une fonction invariante de classe c m sur r s avec 0 m alors il existe une fonction f c p r t telle que f f 1 t avec p sup 0 m s 1 on considere ici le cas s 2 d n le groupe diedre d ordre 2n avec n 2

piecewise linear lq control for systems with input constraints abstract a piecewise linear control plc law based on lq theory is derived ths law raises an lq gain as the controlled error converges towards the origin it can be computed off line and it guarantees that input bounds are never exceeded without causing input saturation a succession of positively invariant sets are defined of diminishing size and associated with each is the corresponding highest lq gain possible in the presence of the input bounds the formulation gives rise to an iteration function whose solution is a fixed point fp the fp defines the optimal gain and the invariant set which comprise ellipsoids fp existence uniqueness and convergence properties are proven we explain the algorithms for the generation and subsequent implementation of the plc law the notion of a switching function is introduced consideration is given to the computational aspects of the fp the plc scheme is demonstrated for the linearized dynamics of the simple pendulum and the puma 560 robot a discrete time plc law can also be derived using the discrete time algebraic riccati equation

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

bounds for the model quench front the model quench front problem is the following nonlinear eigenvalue problem determine a number p 0 and a function u x y such that u x u xx pu yy 0 in the domain 0 leqq y leqq 1 infty u c u 0 0 u c where 0 0 the parameters b and u c are supposed given upper and lower bounds are obtained for p using a variant of the method of separation of variables the method of lines for this problem is also discussed

an efficient and stabilizing model predictive control of switched systems model predictive control mpc of switched systems typically requires an on line solution of a mixed integer program mip since the worst case complexity of the optimization problem increases exponentially with respect to the number of integer variables an on line implementation of the mpc for problems with large number of sub systems and or large horizons is usually expensive in this technical note we propose a stabilizing mpc formulation for state dependent switched systems that enables a tradeoff between the computational complexity of the mpc controller and the optimal performance of the closed loop system the proposed approach uses a pre terminal set in addition to the positively invariant terminal set which aids in reducing the on line complexity although at the expense of optimality examples are presented to illustrate the computational benefits of the proposed mpc strategy over existing mpc for switched systems