limiting angle of brownian motion on certain manifolds suppose that m is a complete simply connected riemannian manifold of non positive sectional curvature with dimension m 3 if outside a fixed compact set the sectional curvatures are bounded above by a negative constant multiple of the inverse of the square of the geodesic distance from a fixed point and below by another negative constant multiple of the square of the geodesic distance then the angular part of brownian motion on m tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire s m 1 this improves a result of hsu and march

conservation laws for conformally invariant variational problems we succeed in writing 2 dimensional conformally invariant non linear elliptic pde harmonic map equation prescribed mean curvature equations etc in divergence form these divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces from this form we can recover without the use of moving frame all the classical regularity results known for 2 dimensional conformally invariant non linear elliptic pde see hel it enables us also to establish new results in particular we solve a conjecture by e heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by s hildebrandt hil1 claiming that critical points of continuously differentiable elliptic conformally invariant lagrangian in two dimensions are continuous

grassmannian central projection and output feedback pole assignment of linear systems a sufficient rank condition for arbitrary pole assignability by static output feedback of m input p output linear systems of mcmillan degree n mp is given this condition is easy to check and is satisfied by the generic system an effective computational method to compute the feedback laws assigning any self conjugate set of closed loop poles is also given

a new sufficient condition for arbitrary pole placement by real constant output feedback abstract a new sufficient condition for generic pole assignment is given based on the height of the first whitney class of a certain grassmannian this result contains the condition m p 1 n and is stronger than lscat p m n this test is equivalent to the factorial test in 7 but it has the advantage that it is testable for most of the cases

exit probability estimates for martingales in geodesic balls using curvature kallenberg and sztencel have recently discovered exponential upper bounds independent of dimension on the probability that a vector martingale will exit from a ball in euclidean space by timet this article extends their results to martingales on riemannian manifolds including brownian motion and shows how exit probabilities depend on curvature using comparison with rotationally symmetric manifolds these estimates are easily computable and are sharp up to a constant factor in certain cases

new results in pole assignment by real output feedback this paper considers the problem of tuning natural frequencies of a linear system by a memoryless controller using algebro geometric methods it is shown how it is possible to improve current sufficiency conditions the main result is an exact combinatorial characterization of the nilpotency index of the bmod 2 cohomology ring of the real grassmannian using this characterization new sufficiency results for generic pole assignment for the linear system with m inputs p outputs and mcmillan degree n are given among other results it is shown that 2 25 cdot max m p min m p 3 geq n is a sufficient condition for generic real pole placement provided min m p geq 4

generic eigenvalue assignment by memoryless real output feedback by extensive use of methods from algebraic geometry x wang proved that arbitrary pole placement by static output feedback is generically possible for strictly proper plants with n states m inputs and p outputs if n mp here we show the same result using no more of algebraic geometry than the definition of genericity

brownian motion and harmonic functions on rotationally symmetric manifolds we consider brownian motion x on a rotationally symmetric manifold m g mathbb r n ds 2 ds 2 dr 2 g r 2 d theta 2 an integral test is presented which gives a necessary and sufficient condition for the nontriviality of the invariant sigma field of x hence for the existence of nonconstant bounded harmonic functions on m g conditions on the sectional curvatures are given which imply the convergence or the divergence of the test integral

some dimension free features of vector valued martingales given any local maringalem inr d orl 2 there exists a local martingalen inr 2 such that m n m n and m n it follows in particular that any inequality for martingales inr 2 which involves only the processes m m and m remains true in arbitrary dimension whenm is continuous the processes m 2 and m satisfy certain sde s which are independent of dimension and yield information about the growth rate ofm this leads in particular to tail estimates of the same order as in one dimension the paper concludes with some new maximal inequalities in continuous time

a necessary condition for pole assignment by constant output feedback in this paper we shall investigate the pole assignment problem by constant output feedback using transfer function techniques in particular a necessary condition is given for the existence of constant output feedback which assigns the eigenvalues of closed loop system to any desired positions

analysis aspects of willmore surfaces a new formulation for the euler lagrange equation of the willmore functional for immersed surfaces in m is given as a nonlinear elliptic equation in divergence form with non linearities comprising only jacobians letting vec h be the mean curvature vector of the surface our new formulation reads mathcal l vec h 0 where mathcal l is a well defined locally invertible self adjoint elliptic operator several consequences are studied in particular the long standing open problem asking for a meaning to the willmore euler lagrange equation for immersions having only l 2 bounded second fundamental form is now solved the regularity of weak willmore immersions with l 2 bounded second fundamental form is also established its proof relies on the discovery of conservation laws which are preserved under weak convergence a weak compactness result for willmore surfaces with energy less than 8 the li yau condition ensuring the surface is embedded is proved via a point removability result established for wilmore surfaces in m thereby extending to arbitrary codimension the main result in ks3 finally from this point removability result the strong compactness of willmore tori below the energy level 8 is proved both in dimension 3 this had already been settled in ks3 and in dimension 4

existence of non trivial harmonic functions on cartan hadamard manifolds of unbounded curvature the liouville property of a complete riemannian manifold m i e the question whether there exist non trivial bounded harmonic functions on m attracted a lot of attention for cartan hadamard manifolds the role of lower curvature bounds is still an open problem we discuss examples of cartan hadamard manifolds of unbounded curvature where the limiting angle of brownian motion degenerates to a single point on the sphere at infinity but where nevertheless the space of bounded harmonic functions is as rich as in the non degenerate case to see the full boundary the point at infinity has to be blown up in a non trivial way such examples indicate that the situation concerning the famous conjecture of greene and wu about existence of non trivial bounded harmonic functions on cartan hadamard manifolds is much more complicated than one might have expected

stabilizability of multivariable systems and the ljusternik snirel mann category of real grassmannians abstract in this paper a numeric criterion k m p n is given for stabilizability by constant gain output feedback of the generic linear multivariable system with m inputs n states and p outputs this criterion is defined in terms of a topological invariant of the space of gains both finite and infinite arising from an interpretation of the problem of stabilizability as a problem concerning open covers of this space this invariant originally considered by ljusternik and snirel mann in the calculus of variations can be estimated from below by using a theorem of eilenberg together with the methods of the schubert calculus thus leading to some very explicit corollaries concerning generic stabilizability as far as i am aware these corollaries cannot be derived from existing results concerning pole assignability after passing to a problem on coverings of the compactified gain space the main technical problem which remains in a high gain stability analysis stated here as a high gain lemma which appears to be of independent interest that is the topological argument implies that the closed loop system is stable for some gain possibly infinite if the root locus map were defined and continuous at all infinite gains then the conclusion that the system is stabilizable by finite gain could be deduced from a simple continuity argument however it is known that if mp n then the root locus map has points of discontinuity at infinity and since mp n is now known to be necessary for generic stabilizability in most cases of interest one requires a far more subtle argument than one would give for example in the scalar case

static output feedback pole placement via a trust region approach this technical note presents two closely related algorithms for the problem of pole placement via static output feedback the algorithms are based on two different trust region methods and utilize the derivatives of the closed loop poles extensive numerical experiments show the effectiveness of the algorithms in practice though convergence to a solution is not guaranteed for either algorithm while desired poles must be distinct strategies for dealing with repeated poles are also presented

multivariable nyquist criteria root loci and pole placement a geometric viewpoint classical control theory is concerned with the topics in our title in the context of single input single output systems there is now a large and growing literature on the extension of these ideas to the multiinput multioutput case this development has posed certain difficulties some due to the intrinsic nature of the problem and some we would argue due to an inadequate reflection on what the multivariable problem calls for in this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory a rather satisfactory generalization of the nyquist criterion will be described and a clear analog of the asymptotic properties of the root locus will be obtained in the multiparameter case however an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains this calls for a new ingredient a compactification of the space of gains and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement this problem is difficult in the multivariable case but by establishing a correspondence with a classical set of problems in geometry we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field

limiting angles of γ martingales suppose that m is a complete simply connected riemannian manifold of non positive sectional curvature with dimension m 3 and that outside a fixed compact set the sectional curvatures are bounded above by c 1 r 2 ln r and below by c 2 r 2 where c 1 and c 2 are two positive constants and r is the geodesic distance from a fixed point we show that when 1 satisfies certain conditions the angular part of a quasi conformal martingale on m tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire sphere at infinity this improves both a result of le for brownian motion and also results concerning the non existence of quasi conformal harmonic maps from certain types of riemannian manifolds into m

on pole placement via eigenstructure assignment approach this note comes back to the hard problem of pole placement by static output feedback let a triplet of matrices a b c be given with n state variables m inputs and p ouputs find a matrix k such that the spectrum of a bkc equals a specified set when mp m p a simple noniterative technique based upon the notion of eigenstructure that in most cases assigns m p roots is proposed it therefore enables the designer to assign the whole of the desired spectrum when m p n mp

the limiting angle of certain riemannian brownian motions let m be a cartan hadamard manifold of dimension d 3 let p m and x exp r x x be geodesic polar coordinates with pole p and let x be the brownian motion on m let sectm x denote the sectional curvature of any plane section in mx we prove that for each c 2 there is a 0 1 such that if l2r x 2 sectm x cr x 2 for all x in the complement of a compact set then limt xt exists a s and defines a nontrivial invariant random variable the dirichlet problem at infinity and a conjecture of greene and wu are also discussed

a sufficient condition for arbitrary eigenvalue assignment in linear descriptor systems by output feedback the problem of eigenvalue assignment in the linear descriptor system e mathdotx ax bu y cx via output feedback is considered it is shown that mp rank e m and p are respectively the numbers of inputs and outputs of the system is a sufficient condition for generic real output feedback eigenvalue assignability the result is the best possible for general m p rank e in general the assignment is achieved not in an exact sense but in arbitrarily small neighborhoods around a given set of eigenvalues

eigenvalue assignment by static output feedback on a new solvability condition and the computation of low gain feedback matrices in this article the static output feedback problem for linear time invariant systems is considered for arbitrary assignability of the roots of the characteristic polynomial by static output feedback a new necessary and sufficient condition is derived although the proof is based on simple analysis the known sufficient conditions derived by techniques of algebraic geometry are directly covered furthermore an algorithm for the calculation of feedback matrices assigning a desired set of eigenvalues is proposed this algorithm does not require the desired eigenvalues to be distinct and it explicitly exploits the available degrees of freedom for reducing the feedback gain the presented approach is illustrated on computational examples