ricci collineations of static spherically symmetric spacetimes the ricci collineations of static spherically symmetric spacetimes are classified and their relationship with isometries is discussed a general theorem about this relationship is stated and its extension to all spherically symmetric spacetimes is discussed

brief paper output regulation for linear distributed parameter systems using finite dimensional dual observers in this article the solution of the output regulation problem is considered for linear infinite dimensional systems where the outputs to be controlled cannot be measured it is shown that this problem can be solved by a finite dimensional dual observer that is directly implementable so that the separation principle can be applied for the stabilization as in finite dimensions a parametric design of these dual observers is proposed for riesz spectral systems that allows to achieve a low controller order and a desired control performance for the closed loop system the presented results are illustrated by determining a finite dimensional regulator for an euler bernoulli beam with kelvin voigt damping that achieves tracking for steplike reference inputs and that asymptotically rejects sinusoidal disturbances

on the link between umbilic geodesics and soliton solutions of nonlinear pdes in this paper we describe a new class of soliton solutions called umbilic solitons for certain nonlinear integrable pdes these umbilic solitons have the property that as the space variable x tends to infinity the solution tends to a periodic wave and as x tends to minus infinity it tends to a phase shifted wave of the same shape the equations admitting solutions in this new class include the dym equation and equations in its hierarchy the methods used to find and analyse these solutions are those of algebraic and complex geometry we look for classes of solutions by constructing associated finite dimensional integrable hamiltonian systems on riemann surfaces in particular in this setting we use geodesics on n dimensional quadrics to find the spatial or x flow which together with the commuting t flow given by the equation itself defines new classes of solutions amongst these geodesics particularly interesting ones are the umbilic geodesics which then generate the class of umbilic soliton solutions this same setting also enables us to introduce another class of solutions of dym like equations which are related to elliptic and umbilic billiards

traveling wave solutions of the camassa holm equation all weak traveling wave solutions of the camassa holm equation are classified we show that in addition to smooth solutions there are a multitude of traveling waves with singularities peakons cuspons stumpons and composite waves

lq optimal boundary control of infinite dimensional systems with yosida type approximate boundary observation a class of boundary control systems with boundary observation is considered for which the unbounded operators often lead to technical difficulties an extended model for this class of systems is described and analyzed which involves no unbounded operator except for the dynamics generator a method for the resolution of the lq optimal control problem for this model is described and the solution provides a stabilizing feedback for the nominal system with unbounded operators in the sense that in closed loop the state trajectories converge to zero exponentially fast the model consists of an extended abstract differential equation whose state components are the boundary input the state up to an affine transformation and a yosida type approximation of the output of the nominal system it is shown that under suitable conditions the model is well posed and in particular that the dynamics operator is the generator of a c 0 semigroup moreover the model is shown to be observable and to carry controllability stabilizability and detectability properties from the nominal system a general method of resolution based on the problem of spectral factorization of a multi dimensional operator valued spectral density is described in order to solve a lq optimal control problem for this model it is expected that this approach will lead hopefully to a good trade off between the cost of modeling and the efficiency of methods of resolution of control problems for such systems

on a nonlinear hyperbolic variational equation i global existence of weak solutions we study the nonlinear hyperbolic partial differential equation u t uux x 1 2u x 2 this partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles in particular it describes waves in a massive director field of a nematic liquid crystal global smooth solutions of the partial differential equation do not exist since their derivatives blow up in finite time while weak solutions are not unique we therefore define two distinct classes of admissible weak solutions which we call dissipative and conservative solutions we prove the global existence of each type of admissible weak solution provided that the derivative of the initial data has bounded variation and compact support these solutions remain continuous despite the fact that their derivatives blow up there are no a priori estimates on the second derivatives in any l p space so the existence of weak solutions cannot be deduced by using sobolev type arguments instead we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation we also describe the qualitative properties of the partial differential equation including a comparison with the burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions we show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics and we prove that the large time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink wave

generalized hunter saxton equation and the geometry of the group of circle diffeomorphisms we study an equation lying mid way between the periodic hunter saxton and camassa holm equations and which describes evolution of rotators in liquid crystals with external magnetic field and self interaction we prove that it is an euler equation on the diffeomorphism group of the circle corresponding to a natural right invariant sobolev metric we show that the equation is bihamiltonian and admits both cusped and smooth traveling wave solutions which are natural candidates for solitons we also prove that it is locally well posed and establish results on the lifespan of its solutions throughout the paper we argue that despite similarities to the kdv ch and hs equations the new equation manifests several distinctive features that set it apart from the other three

dynamics of director fields an asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed the equation is used to study a nonlinear instability in the director field of a nematic liquid crystal it is shown that smooth solutions of the asymptotic equation break down in finite time also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up

robust multivariable pi controller for infinite dimensional systems a robust multivariable controller is introduced for a class of distributed parameter systems the system to be controlled is given as dot x ax bu y cx in a banach space the purpose of the control which is based on the measurement y is to stabilize and regulate the system so that y t rightarrow y r as t rightarrow infty where y r is a constant reference vector under the assumptions that operator a generates a holomorphic stable semigroup b is linear and bounded c is linear and a bounded and the input and output spaces are of the same dimension a necessary and sufficient condition is found for the existence of a robust multivariable controller this controller appears to be a multivariable pi controller also a simple necessary criterion for the existence of a decentralized controller is derived the tuning of the controller is discussed and it is shown that the i part of the controller can be tuned on the basis of step responses without exact knowledge of the system s parameters the presented theory is then used as an example to control the temperature profile of a bar with the dirichlet boundary conditions

finite dimensional compensators for parabolic distributed systems with unbounded control and observation it is proved that for a class of parabolic distributed parameter systems with unbounded control and observation there exists a finite dimensional compensator using dynamic output feedback finite dimensional means here that the dynamics relating the input to the output is finite dimensional a constructive design algorithm is presented and several examples are considered

addendum symmetries of the energy momentum tensor in recent papers j math phys 44 5142 2003 45 1518 2003 45 1532 2004 we have discussed matter symmetries of nonstatic spherically symmetric space times static plane symmetric space times and cylindrically symmetric static space times these have been classified for both cases when the energy momentum tensor is nondegenerate and also when it is degenerate here we add up some consequences and the missing references about the ricci tensor

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

the classification of static plane symmetric space times according to their ricci collineations static plane symmetric space times are classified according to their ricci collineations rcs their relation with isometries of the space times is established unlike the spherically symmetric space times some metrics are found for a nondegenerate ricci tensor which have more rcs than isometries

on the blow up structure for the generalized periodic camassa holm and degasperis procesi equations considered herein are the generalized camassa holm and degasperis procesi equations in the spatially periodic setting the precise blow up scenarios of strong solutions are derived for both of equations several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established the exact blow up rates are also determined finally geometric descriptions of these two integrable equations from non stretching invariant curve flows in centro equiaffine geometries pseudo spherical surfaces and affine surfaces are given

on a completely integrable nonlinear hyperbolic variational equation abstract we show that the nonlinear partial differential equation u t uu x xx 1 2 u x 2 x is a completely integrable bi variational bi hamiltonian system the corresponding equation for w u xx belongs to the harry dym hierarchy this equation arises in two different physical contexts in two nonequivalent variational forms it describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field and it is the high frequency limit of the camassa holm equation which is an integrable model equation for shallow water waves using the bi hamiltonian structure we derive a recursion operator a lax pair and an infinite family commuting hamiltonian flows together with the associated conservation laws we also give the transformation to action angle coordinates smooth solutions of the partial differential equation break down because their derivative blows up in finite time nevertheless the hamiltonian structure and complete integrability appear to remain valid globally in time even after smooth solutions break down we show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions we compute the bi hamiltonian structure on this invariant manifold which is obtained by restricting the bi hamiltonian structure of the partial differential equation

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

on a nonlinear hyperbolic variational equation ii the zero viscosity and dispersion limits we consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation u t uux x 1 2u x 2 with the simplest initial data such that u x blows up in finite time we prove that the zero viscosity limit selects a unique global weak solution of the partial differential equation without viscosity we also present numerical experiments which indicate that the zero dispersion limit selects a different global weak solution of the same initial value problem

finite dimensional dual state feedback control of linear boundary control systems in this article the finite dimensional control of distributed parameter systems with boundary control and unbounded observation is considered this problem is solved by extending the concept of dual observer based compensators to infinite dimensions these compensators have the advantage that on the one hand the separation principle can be used to directly determine a finite dimensional compensator and on the other hand they can be directly applied to boundary control systems with a time derivative of the input in order to achieve a low controller order as well as a desired control performance a parametric approach is proposed to parameterise the degrees of freedom contained in the controller consequently desired design specifications can be achieved by using a parameter optimisation the proposed design of finite dimensional dual observer based compensators is demonstrated by means of a heat conducting system with neumann boundary control and boundary measurements

euler equations on homogeneous spaces and virasoro orbits abstract we show that the following three systems related to various hydrodynamical approximations the korteweg de vries equation the camassa holm equation and the hunter saxton equation have the same symmetry group and similar bihamiltonian structures it turns out that their configuration space is the virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right invariant metrics on this group or on appropriate homogeneous spaces in particular we describe how arnold s approach to the euler equations as geodesic flows of one sided invariant metrics extends from lie groups to homogeneous spaces we also show that the above three cases describe all generic bihamiltonian systems which are related to the virasoro group and can be integrated by the translation argument principle they correspond precisely to the three different types of generic virasoro orbits finally we discuss interrelation between the above metrics and kahler structures on virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits

analyticity of the cauchy problem for two component hunter saxton systems abstract this paper is mainly concerned with the periodic cauchy problem for a generalized two component hunter saxton system with analytic initial data the analyticity of its solutions is proved in both variables globally in space and locally in time the obtained result can be also applied to its special cases the classical integrable two component hunter saxton system the generalized hunter saxton equation and the classical hunter saxton equation

idea of invariant subspace combined with elementary integral method for investigating exact solutions of time fractional npdes abstract in this paper inspired by the idea of invariant subspace method and combined with elementary integral method we introduced a novel approach for investigating exact solutions of a time fractional nonlinear partial differential equation npde based on hypothetical structure of solution of separated variable a time fractional npde defined by time and space variables can be reduced to a nonlinear ordinary differential equation node or nodes defined by space variable alone and then using the elementary integral method to solve the node or nodes different kinds of exact solutions of a time fractional npde are obtained finally as examples the time fractional hunter saxton equation and time fractional li olver equation were studied different kinds of exact solutions of these equations were obtained and their dynamical properties were illustrated

conservative weak solutions of the periodic cauchy problem for μhs equation we prove the existence and uniqueness of conservative weak solutions of the periodic cauchy problem for an integrable evolution equation from mathematical physics our method is to first prove the existence and uniqueness of solutions for all time in lagrangian coordinates and construct the maps to carry this result to the eulerian coordinates

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

solitary wave solutions of the generalized two component hunter saxton system abstract the existence of solitary wave solutions of the generalized two component hunter saxton system is determined it is also shown that there are peaked and cusped solitary waves with singularities among those smooth solitary wave solutions

backstepping observers for a class of parabolic pdes in this paper we design exponentially convergent observers for a class of parabolic partial integro differential equations p i des with only boundary sensing available the problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system observer gain output injection function is shown to satisfy a well posed hyperbolic pde that is closely related to the hyperbolic pde governing backstepping control gain for the state feedback problem for several physically relevant problems the observer gains are obtained in closed form the observer gains are then used for an output feedback design in both collocated and anti collocated setting of sensor and actuator the order of the resulting compensator can be substantially lowered without affecting stability explicit solutions of a closed loop system are found in particular cases

integrable evolution equations on spaces of tensor densities and their peakon solutions we study a family of equations defined on the space of tensor densities of weight on the circle and introduce two integrable pde one of the equations turns out to be closely related to the inviscid burgers equation while the other has not been identified in any form before we present their lax pair formulations and describe their bihamiltonian structures we prove local wellposedness of the corresponding cauchy problem and include results on blow up as well as global existence of solutions moreover we construct peakon and multi peakon solutions for all 0 1 and shock peakons for 3 we argue that there is a natural geometric framework for these equations that includes other well known integrable equations and which is based on v arnold s approach to euler equations on lie groups

a parametric approach to finite dimensional control of linear distributed parameter systems this article considers the design of finite dimensional compensators for distributed parameter systems using eigenvalue assignment the proposed compensator consists of an observer estimating additional outputs and a static feedback of the measurable and the estimated outputs since the additional outputs can be asymptotically reconstructed the compensator can be designed using the separation principle i e the closed loop eigenvalues are given by the observer eigenvalues and the eigenvalues resulting from the static output feedback control in order to solve the corresponding eigenvalue assignment problem the parametric approach for the design of static output feedback controllers in finite dimensions is extended to distributed parameter systems by using a parameter optimisation it is possible to assign all closed loop eigenvalues within specified regions of the complex plane in order to stabilise the system and to assure a desired control performance a heat conductor is used to demonstrate the prop