affine parameter dependent lyapunov functions and real parametric uncertainty this paper presents new tests to analyze the robust stability and or performance of linear systems with uncertain real parameters these tests are extensions of the notions of quadratic stability and performance where the fixed quadratic lyapunov function is replaced by a lyapunov function with affine dependence on the uncertain parameters admittedly with some conservatism the construction of such parameter dependent lyapunov functions can be reduced to a linear matrix inequality lmi problem and hence is numerically tractable these lmi based tests are applicable to constant or time varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations they also avoid the frequency sweep needed in real spl mu analysis and numerical experiments indicate that they often compare favorably with spl mu analysis for time invariant parameter uncertainty

a less conservative lmi condition for the robust stability of discrete time uncertain systems abstract in this paper a less conservative condition for the robust stability of uncertain discrete time linear systems is proposed the uncertain parameters assumed to be time invariant are supposed to belong to convex bounded domains polytope type uncertainty the stability condition is formulated in terms of a set of linear matrix inequalities involving only the vertices of the polytope domain a simple and numerically efficient feasibility test provides a set of lyapunov matrices whose convex combination can be used to assess the stability of any dynamic matrix inside the uncertainty domain examples illustrate the results

central configurations of the n body problem via equivariant morse theory in this paper we use the equivariant morse theory to give an estimate of the minimal number of central configurations in the n body problem in 3 in the case of equal masses we prove that the planar central configurations are saddle points for the potential energy from this we deduce the presence of non planar central configurations for every n 4

mixed h 2 h control for discrete time systems via convex optimization abstract a mixed h 2 h control problem for discrete time systems is considered where an upper bound on the h 2 norm of a closed loop transfer matrix is minimized subject to an h constraint on another closed loop transfer matrix both state feedback and output feedback cases are considered it is shown that these problems are equivalent to finite dimensional convex programming problems in the state feedback case nearly optimal controllers can be chosen to be static gains in the output feedback case nearly optimal controllers can be chosen to have a structure similar to that of the central single objective h controller in particular the state dimension of nearly optimal output feedback controllers need not exceed the plant dimension

equivariant morse theory for flows and an application to the n body problem a l aide de l indice de conley et de la cohomologie equivariante on deduit certaines inegalites de type morse pour un flot equivariant par rapport a l action d un groupe topologique compact

on a convex parameter space method for linear control design of uncertain systems this paper presents a new procedure for continuous and discrete time linear control systems design it consists of the definition of a convex programming problem in the parameter space that when solved provides the feedback gain one of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation yielding solutions to problems that have raised a great deal of interest within the last few years this is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices in this last case necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and to the authors knowledge this is one of the first numerical procedures able to handle and solve this interesting design problem for high order continuous time or discrete time linear models the theory is illustrated by examples

lmi characterization of structural and robust stability the discrete time case abstract this paper extends to the discrete time case some robust stability conditions recently obtained for continuous time systems those conditions are expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable as in the continuous time case parameter dependent lyapunov functions can be constructed and consequently the new approach can yield much sharper and less conservative results than the simultaneous stability approach in particular well known stability problems namely d stability and robust stability in the presence of diagonally structured uncertainty can be more efficiently addressed numerical examples are included to illustrate the advantages of the new stability conditions

comet like periodic orbits in the n body problem abstract take any nonresonant relative equilibrium solution of the n body problem then there is a periodic solution of the n 1 body problem where n of the particles remain close to the relative equilibrium solution and the remaining particle is close to a circular orbit of the kepler problem encircling the center of mass of the n particle system

finiteness of central configurations of five bodies in the plane we prove there are finitely many isometry classes of planar central configurations also called relative equilibria in the newtonian 5 body problem except perhaps if the 5 tuple of positive masses belongs to a given codimension 2 subvariety of the mass space

piecewise lyapunov functions for robust stability of linear time varying systems in this paper we investigate the use of two term piecewise quadratic lyapunov functions for robust stability of linear time varying systems by using the so called s procedure and a special variable reduction method we provide numerically efficient conditions for the robust asymptotic stability of the linear time varying systems involving the convex combinations of two matrices an example is included to demonstrate the usefulness of our results

piecewise linear quadratic optimal control the use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control lower bounds on the optimal control cost are obtained by semidefinite programming based on the bellman inequality this also gives an approximation to the optimal control law an upper bound to the optimal cost is obtained by another convex optimization problem using the given control law a compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy

morse theory and relative equilibria in the planar n vortex problem morse theoretical ideas are applied to the study of relative equilibria in the planar n vortex problem for the case of positive circulations we prove that the morse index of a critical point of the hamiltonian restricted to a level surface of the angular impulse is equal to the number of pairs of real eigenvalues of the corresponding relative equilibrium periodic solution the morse inequalities are then used to prove the instability of some families of relative equilibria in the four vortex problem with two pairs of equal vorticities we also show that for positive circulations relative equilibria cannot accumulate on the collision set

new wavelet based approach for time domain simulations a new numerical method for the time domain solution of maxwell s equations in linear media is proposed in this paper the field quantities derived from the spatial discretization of maxwell s equations are expanded in the time domain by wavelets on the interval this choice yields a new arrangement of the unknowns into a matrix instead of the usual vector and transforms the differential equations in time in a sylvester 1983 matrix equation the memory requirements are proportional to the number of spatial unknowns and the time evolution of the space quantities is obtained with better accuracy than in conventional marching on time techniques

symmetric trajectories for the 2n body problem with equal masses we consider the problem of 2n bodies of equal masses in mathbb r 3 for the newtonian like weak force potential r and we prove the existence of a family of collision free nonplanar and nonhomographic symmetric solutions that are periodic modulo rotations in addition the rotation number with respect to the vertical axis ranges in a suitable interval these solutions have the hip hop symmetry a generalization of that introduced in 19 for the case of many bodies and taking account of a topological constraint the argument exploits the variational structure of the problem and is based on the minimization of lagrangian action on a given class of paths

the topology of reduces phase spaces of the motion of vortices on a sphere abstract the motion of point vortices in a perfect incompressible fluid on a two dimensional sphere can be represented by a hamiltonian flow which is invariant under the action of the special orthogonal group so 3 following the reduction method of marsden and weinstein this motion can be described by hamiltonian flows on reduced phase spaces some topological invariants of these reduced phase spaces are calculated for suitable vorticities the results obtained can be used to give lower bounds on numbers of relative equilibria of the original system

on central configurations this paper concerns an old problem which arose in celestial mechanics and has since taken on a life of its own the probem of central configurations of the n body problem

a robust stability problem for discrete time systems subject to an uncertain parameter abstract in this paper a linear discrete time uncertain system depending in norm bounded form on an uncertain real time varying parameter whose magnitude is between the quadratic stability radius and the real stability radius is considered in this case stability of the system depends on the time rate of variation of the parameter by utilizing parameter dependent lyapunov functions one can take the parameter rate of change into account and obtain sufficient conditions for stability which are less restrictive than those imposed by parameter independent lyapunov functions

a new discrete time robust stability condition a new robust stability condition for uncertain discrete time systems with convex polytopic uncertainty is given it enables to check stability using parameter dependent lyapunov functions which are derived from lmi conditions it is shown that this new condition provides better results than the classical quadratic stability besides the use of a parameter dependent lyapunov function this condition exhibits a kind of decoupling between the lyapunov and the system matrices which may be explored for control synthesis purposes a numerical example illustrates the results

morse theory for symmetric functionals on the sphere and an application to a bifurcation problem abstract this paper uses conley s index to study the cortical points of a function f on a finite dimensional sphere in presence of a symmetry group the authors prove a theorem which leads to a lower bound on the number of critical points of f when the group is finite even if the action is not free this investigation has been motivated by the following bifurcation problem au lambda where a is a variational g equivariant operator an estimated on the number of branches bifurcating from an eigenvalue of a 0 is given author

a note on quadratic stability of uncertain linear discrete time systems in this paper we consider a linear discrete time system depending on a vector of uncertain parameters assuming that the system matrix depends on parameters as the ratio of a multiaffine matrix valued function and a multiaffine polynomial and that the parameters range in a hyper rectangle we show that the uncertain system is quadratically stable if and and only if the set of vertex systems is quadratically stable this allows us to state a necessary and sufficient condition for quadratic stability in terms of the solvability of a feasibility problem involving linear matrix inequalities

analysis and synthesis of robust control systems via parameter dependent lyapunov functions in this paper the problem of robust stability of systems subject to parametric uncertainties is considered sufficient conditions for the existence of parameter dependent lyapunov functions are given in terms of a criterion which is reminiscent of but less conservative than popov s stability criterion an equivalent frequency domain criterion is demonstrated the relative sharpness of the proposed test and existing stability criteria is then discussed the use of parameter dependent lyapunov functions for robust controller synthesis is then considered it is shown that the search for robustly stabilizing controllers may be limited to controllers with the same order as the original plant a possible synthesis procedure and a numerical example are then discussed

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

ℋ2 guaranteed cost control for uncertain discrete time linear systems the 2 guaranteed cost control problem is analysed for uncertain discrete time linear systems it consists on the determination of a constant state feedback stabilizing matrix gain and an 2 norm upper bound which holds for all feasible models two different approaches are considered the first is based on the optimality conditions provided by bellmans s dynamic programming equation the second is based on the geometric properties of the aforementioned problem it is shown that it is possible to parametrize all stabilizing gains over a convex set in this special parameter space the 2 cost is found to be a linear function of the free parameters finally some relationships between these approaches are presented and numerical examples given