conewise linear systems non zenoness and observability conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation this class of dynamical systems represents a large number of piecewise linear systems most notably linear complementarity systems with the p property and their generalizations to affine variational systems which have many applications in engineering systems and dynamic optimization the challenges of dealing with this type of hybrid system are due to two major characteristics mode switchings are triggered by state evolution and states are constrained in each mode in this paper we first establish the absence of zeno states in such a system based on this fundamental result we then investigate and relate several state observability notions short time and t time or finite time local global observability for the short time observability notions constructive finitely verifiable algebraic both sufficient and necessary conditions are derived due to their long time mode transitional behavior which is very difficult to predict only partial results are obtained for the t time observable states nevertheless we completely resolve the t time local observability for the bimodal conewise linear system for finite t and provide numerical examples to illustrate the difficulty associated with the long time observability

when is a linear multi modal system disturbance decoupled in this paper we study the question under which conditions a linear multi modal system is disturbance decoupled we establish necessary and sufficient geometric conditions from which the existing results on switched linear systems and conewise linear systems can be recovered as special cases also we apply these conditions to a class of linear complementarity systems in order to obtain a more crisp characterization

discovering multiple lyapunov functions for switched hybrid systems in this paper we analyze local asymptotic stability of switched hybrid systems whose subsystems have polynomial vector fields by discovering multiple lyapunov functions in quadratic forms we start with an algebraizable sufficient condition for the existence of quadratic multiple lyapunov functions then since different discrete modes are considered we apply real root classification together with a projection operator to underapproximate this sufficient condition step by step arriving at a set of semialgebraic sets which only involve the coefficients of the pre assumed multiple lyapunov function note that for each step we additionally use the information on the different discrete modes to optimize our intermediate computation results afterward we compute a sample point in the corresponding semialgebraic set for the coefficients resulting in a multiple lyapunov function finally we test our approach on some examples using a prototypical implementation and compare it with the generic quantifier e

strongly regular differential variational systems a differential variational system is defined by an ordinary differential equation ode parameterized by an algebraic variable that is required to be a solution of a finite dimensional variational inequality containing the state variable of the system this paper addresses two system theoretic topics for such a nontraditional nonsmooth dynamical system namely non zenoness and local observability of a given state satisfying a blanket strong regularity condition for the former topic which is of contemporary interest in the study of hybrid systems we extend the results in our previous paper where we have studied zeno states and switching times in a linear complementarity system lcs as a special case of the differential variational inequality dvi the lcs consists of a linear time invariant ode and a linear complementarity problem the extension to a nonlinear complementarity system ncs with analytic inputs turns out to be non trivial as we need to use the lie derivatives of analytic functions in order to arrive at an expansion of the solution trajectory near a given state further extension to a differential variational inequality is obtained via its equivalent karush kuhn tucker formulation for the second topic which is classical in system theory we use the non zenoness result and the recent results in a previous paper pertaining to the b differentiability of the solution operator of a nonsmooth ode to obtain a sufficient condition for the short time local observability of a given strongly regular state of an ncs refined sufficient conditions and necessary conditions for local observability of the lcs satisfying the p property are obtained

toric surfaces k stability and calabi flow let x be a toric surface and u be a normalized symplectic potential on the corresponding polygon p suppose that the riemannian curvature is bounded by a constant c 1 and int partial p u d sigma 0 depending only on c 1 c 2 and p such that donaldson s m condition holds for u as an application we show that if x p is analytic relative k stable then the modified calabi flow converges to an extremal metric exponentially fast by assuming that the calabi flow exists for all time and the riemannian curvature is uniformly bounded along the calabi flow

differential variational inequalities this paper introduces and studies the class of differential variational inequalities dvis in a finite dimensional euclidean space the dvi provides a powerful modeling paradigm for many applied problems in which dynamics inequalities and discontinuities are present examples of such problems include constrained time dependent physical systems with unilateral constraints differential nash games and hybrid engineering systems with variable structures the dvi unifies several mathematical problem classes that include ordinary differential equations odes with smooth and discontinuous right hand sides differential algebraic equations daes dynamic complementarity systems and evolutionary variational inequalities conditions are presented under which the dvi can be converted either locally or globally to an equivalent ode with a lipschitz continuous right hand function for dvis that cannot be so converted we consider their numerical resolution via an euler time stepping procedure which involves the solution of a sequence of finite dimensional variational inequalities borrowing results from differential inclusions dis with upper semicontinuous closed and convex valued multifunctions we establish the convergence of such a procedure for solving initial value dvis we also present a class of dvis for which the theory of dis is not directly applicable and yet similar convergence can be established finally we extend the method to a boundary value dvi and provide conditions for the convergence of the method the results in this paper pertain exclusively to systems with index not exceeding two and which have absolutely continuous solutions

nonsmooth lyapunov pairs for differential inclusions governed by operators with nonempty interior domain the general theory of lyapunov stability of first order differential inclusions in hilbert spaces has been studied by the authors in the previous paper adly et al in nonlinear anal 75 3 985 1008 2012 this new contribution focuses on the case when the interior of the domain of the maximally monotone operator governing the given differential inclusion is nonempty this includes in a natural way the finite dimensional case the current setting leads to simplified more explicit criteria and permits some flexibility in the choice of the generalized subdifferentials some consequences of the viability of closed sets are given our analysis makes use of standard tools from convex and variational analysis

positive invariance of constrained affine dynamics and its applications to hybrid systems and safety verification motivated by long time dynamic analysis of hybrid systems and safety verification problems this paper addresses fundamental positive invariance issues of an affine dynamical system on a general polyhedron and their applications necessary and sufficient algebraic conditions are established for the existence of a positively invariant set of an affine system on a polyhedron using the tools of lexicographic relation and long time oscillatory dynamic analysis a linear program based algorithm is proposed to verify these conditions and its computational complexity is analyzed the positive invariance results are applied to obtain an explicit characterization of global switching behaviors of piecewise affine systems further the positive invariance techniques developed in this paper are exploited to show the decidability of safety verification of a class of affine dynamics on semialgebraic sets

long time existence of minimizing movement solutions of calabi flow abstract we recast the calabi flow in degiorgi s language of minimizing movements we establish the long time existence of minimizing movements for k energy with arbitrary initial condition furthermore we establish some a priori regularity results for these solutions and that sufficiently regular minimizing movements are smooth solutions to calabi flow

on the regularization of zeno hybrid automata fundamental properties of hybrid automata such as existence and uniqueness of executions are studied particular attention is devoted to zeno hybrid automata which are hybrid automata that take infinitely many discrete transitions in finite time it is shown that regularization techniques can be used to extend the zeno executions of these automata to times beyond the zeno time different types of regularization may however lead to different extensions a water tank control problem and a bouncing ball system are used to illustrate the results

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

cardinality constrained linear quadratic optimal control as control implementation often incurs not only a variable cost associated with the magnitude or energy of the control but also a setup cost we consider a discrete time linear quadratic lq optimal control problem with a limited number of control implementations termed in this technical note the cardinality constrained linear quadratic optimal control cclq we first derive a semi analytical feedback policy for cclq problems using dynamic programming dp due to the exponential growth of the complexity in calculating the action regions however dp procedure is only efficient for cclq problems with a scalar state space recognizing this fact we develop then two lower bounding schemes and integrate them into a branch and bound bnb solution framework to offer an efficient algorithm in solving general cclq problems adopting the devised solution algorithm for cclq problems we can solve efficiently the linear quadratic optimal control problem with setup costs

solution dependence on initial conditions in differential variational inequalities in the first part of this paper we establish several sensitivity results of the solution x t to the ordinary differential equation ode initial value problem ivp dx dt f x x 0 as a function of the initial value for a nondifferentiable f x specifically we show that for xi t equiv x t xi 0 0 leq t leq t a if f is b differentiable on xi t then so is the solution operator x t at 0 b if f is semismooth on xi t then so is x t at 0 c if f has a linear newton approximation on xi t then so does x t at 0 moreover the linear newton approximation of the solution operator can be obtained from the solution of a linear differential inclusion in the second part of the paper we apply these ode sensitivity results to a differential variational inequality dvi and discuss a the existence uniqueness and lipschitz dependence of solutions to subclasses of the dvi subject to boundary conditions via an implicit function theorem for semismooth equations and b the convergence of a nonsmooth shooting method for numerically computing such boundary value solutions

observability and controllability of piecewise affine and hybrid systems we prove in a constructive way the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics automata and propositional logic by focusing our investigation on the former class we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems instead we propose practical numerical tests based on mixed integer linear programming

algebraic necessary and sufficient conditions for the controllability of conewise linear systems the problem of checking certain controllability properties of even very simple piecewise linear systems is known to be undecidable this paper focuses on conewise linear systems i e systems for which the state space is partitioned into conical regions and a linear dynamics is active on each of these regions for this class of systems we present algebraic necessary and sufficient conditions for controllability we also show that the classical results of controllability of linear systems and input constrained linear systems can be recovered from our main result our treatment employs tools both from geometric control theory and mathematical programming

on the existence uniqueness and nature of caratheodory and filippov solutions for bimodal piecewise affine dynamical systems abstract in this paper we deal with the well posedness in the sense of existence and uniqueness of solutions and nature of solutions for discontinuous bimodal piecewise affine systems in a differential inclusion setting first we show that the conditions guaranteeing uniqueness of filippov solutions in the context of general differential inclusions are quite restrictive when applied to bimodal piecewise affine systems later we present a set of necessary and sufficient conditions for uniqueness of filippov solutions for bimodal piecewise affine systems we also study the so called zeno behavior possibility of infinitely many switchings within a finite time interval for filippov solutions

hamiltonian 2 forms in kahler geometry iii extremal metrics and stability this paper concerns the existence and explicit construction of extremal kahler metrics on total spaces of projective bundles which have been studied in many places we present a unified approach motivated by the theory of hamiltonian 2 forms as introduced and studied in previous papers in the series but this paper is largely independent of that theory

observer design for lur e systems with multivalued mappings a passivity approach this paper considers the design of state observers for lur e systems with multivalued mappings in the feedback path in particular we focus on maximal monotone mappings that that do not require any compactness and local boundedness properties and include various models for relays friction characteristics and complementarity conditions we propose two types of observers that are constructed by rendering a suitable operator passive in an appropriate sense the well posedness properties of the observer dynamics are carefully analyzed and the global asymptotic stability of the observation error is formally proven

non zenoness of a class of differential quasi variational inequalities the zeno phenomenon of a switched dynamical system refers to the infinite number of mode switches in finite time the absence of this phenomenon is crucial to the numerical simulation of such a system by time stepping methods and to the understanding of the behavior of the system trajectory extending a previous result for a strongly regular differential variational inequality this paper establishes that a certain class of non strongly regular differential variational inequalities is devoid of the zeno phenomenon the proof involves many supplemental results that are of independent interest specialized to a frictional contact problem with local compliance and polygonal friction laws this non zenoness result is of fundamental significance and the first of its kind

extremal metrics on toric surfaces a continuity method the paper develops an existence theory for solutions of the abreu equation which include extremal metrics on toric surfaces the technique employed is a continuity method combined with blow up arguments general existence results are obtained assuming a hypothesis the m condition on the solutions which is shown to be related to the injectivity radius

shape restricted smoothing splines via constrained optimal control and nonsmooth newton s methods shape restricted smoothing splines receive considerable attention motivated by many important applications in science and engineering in this paper we consider smoothing splines subject to general linear dynamics and control constraints and formulate them as finite horizon constrained linear optimal control problems with unknown initial state and control by exploring techniques from functional and variational analyses optimality conditions are developed in terms of variational inequalities due to the control constraints the optimality conditions give rise to a nonsmooth b differentiable equation of an optimal initial condition whose unique solution completely determines the shape restricted smoothing spline a modified nonsmooth newton s algorithm with line search is used to solve this equation detailed convergence analysis of the proposed algorithm is presented using techniques from nonsmooth analysis and polyhedral theory we show the global convergence of the algorithm for shape restricted smoothing splines subject to general polyhedral control constraints

convergence of a class of semi implicit time stepping schemes for nonsmooth rigid multibody dynamics in this work we present a framework for the convergence analysis in a measure differential inclusion sense of a class of time stepping schemes for multibody dynamics with contacts joints and friction this class of methods solves one linear complementarity problem per step and contains the semi implicit euler method as well as trapezoidal like methods for which second order convergence was recently proved under certain conditions by using the concept of a reduced friction cone the analysis includes for the first time a convergence result for the case that includes joints an unexpected intermediary result is that we are able to define a discrete velocity function of bounded variation although the natural discrete velocity function produced by our algorithm may have unbounded variation

well posedness conditions for bimodal piecewise affine systems abstract this paper considers well posedness the existence and uniqueness of the solutions of bimodal piecewise affine systems in r n it is assumed that both modes are observable but only one of the modes is in observable canonical form this allows the vector field to be discontinuous when the trajectories change mode necessary and sufficient conditions for well posedness are given as a set of algebraic conditions and sign inequalities it is shown that these conditions induce a joint structure for the system matrices of the two modes this structure can be used for the classification of well posed bimodal piecewise affine systems furthermore it is also shown that under certain conditions well posed bimodal piecewise affine systems in r n may have one or two equilibrium points or no equilibrium points

cone copositive piecewise quadratic lyapunov functions for conewise linear systems in this technical note cone copositive piecewise quadratic lyapunov functions pwq lfs for the stability analysis of conewise linear systems are proposed the existence of a pwq lf is formulated as the feasibility of a cone copositive programming problem which is represented in terms of linear matrix inequalities a constructive procedure for its solution is provided examples show the effectiveness of the approach also in the case of uncertain conewise linear systems

ricci deformation of the metric on complete noncompact riemannian manifolds soit m une variete de riemann complete non compacte a n dimensions n 3 pour c1 c2 0 et 0 il existe une constante e e n c1 c2 0 telle que si la courbure de m satisfait a vol b x y c 1 n x m 0 et b rm 2 er 2 0 r c 2 x 0 x 2 x m alors l equation d evolution g ij t t 2r ij t g ij 0 g ij a une solution pour tout temps 0 t et la metrique g ij t converge vers une metrique lisse g ij quand t telle que r ijkl 0 sur m

piecewise linear identification of non linear systems methods of identifying a series of piecewise linear models which approximate to a non linear system over some operating range are investigated spatial linear models and models with signal dependent parameters are considered

the global existence and convergence of the calabi flow on cn zn izn in this note we study the long time existence of the calabi flow on x cn zn izn assuming the uniform bound of the total energy we establish the non collapsing property of the calabi flow by using donaldson s estimates and streets regularity theorem next we show that the curvature is uniformly bounded along the calabi flow on x when the dimension is 2 partially confirming chen s conjecture moreover we show that the calabi flow exponentially converges to the flat kahler metric for arbitrary dimension if the curvature is uniformly bounded partially confirming donaldson s conjecture

observability analysis of conewise linear systems via directional derivative and positive invariance techniques belonging to the broad framework of hybrid systems conewise linear systems clss form a class of lipschitz piecewise linear systems subject to state triggered mode switchings motivated by state estimation of nonsmooth switched systems this paper exploits directional derivative and positive invariance techniques to characterize finite time and long time local observability of a general cls for the former observability notion directional derivative results are developed from the simple switching property and they yield improved observability conditions for the latter notion we focus on the case where a nominal trajectory has finitely many switchings in order to characterize long time behaviors of the cls necessary and sufficient conditions are obtained for the interior of a positively invariant cone by employing these conditions we establish connections between finite time and long time local observability underlying positive invariance properties are unveiled

on the calabi flow in this paper we study the calabi flow on a polarized kahler manifold and some related problems we first give a precise statement on the short time existence of the calabi flow for any c 3 a m initial kahler potential as an application we prove a stability result any metric near a constant scalar curvature kahler csck metric will flow to a nearby csck metric exponentially fast secondly we prove that a compactness theorem in the space of the kahler metrics given uniform ricci bound and potential bound as an application we prove the calabi flow can be extended once ricci curvature stays uniformly bounded lastly we prove a removing singularity result about a weak constant scalar curvature metric in a punctured disc

lyapunov stability of complementarity and extended systems a linear complementarity system lcs is a piecewise linear dynamical system consisting of a linear time invariant ordinary differential equation ode parameterized by an algebraic variable that is required to be a solution to a finite dimensional linear complementarity problem lcp whose constant vector is a linear function of the differential variable continuing the authors recent investigation of the lcs from the combined point of view of system theory and mathematical programming this paper addresses the important system theoretic properties of exponential and asymptotic stability for an lcs with a c 1 state trajectory the novelty of our approach lies in our employment of a quadratic lyapunov function that involves the auxiliary algebraic variable of the lcs when expressed in the state variable alone the lyapunov function is piecewise quadratic and thus nonsmooth the nonsmoothness feature invalidates standard stability analysis that is based on smooth lyapunov functions in addition to providing sufficient conditions for exponential stability we establish a generalization of the well known lasalle invariance theorem for the asymptotic stability of a smooth dynamical system to the lcs which is intrinsically a nonsmooth system sufficient matrix theoretic copositivity conditions are introduced to facilitate the verification of the stability properties properly specialized the latter conditions are satisfied by a passive like lcs and certain hybrid linear systems having common quadratic lyapunov functions we provide numerical examples to illustrate the stability results we also develop an extended local exponential stability theory for nonlinear complementarity systems and differential variational inequalities based on a new converse theorem for odes with b differentiable right hand sides the latter theorem asserts that the existence of a b differentiable lyapunov function is a necessary and sufficient condition for the exponential stability of an equilibrium of such a differential system

complementarity modeling of hybrid systems a complementarity framework is described for the modeling of certain classes of mixed continuous discrete dynamical systems the use of such a framework is well known for mechanical systems with inequality constraints but we give a more general formulation which also applies for instance to switching control systems the main theoretical results in the paper are concerned with uniqueness of smooth continuations the solution of this problem requires the construction of a map from the continuous state to the discrete state a crucial technical tool is the so called linear complementarity problem from mathematical programming and we introduce various generalizations of this problem

the calabi flow on kahler surfaces with bounded sobolev constant i we consider the formation of singularities along the calabi flow by assuming the uniformly bounded sobolev constants on kahler surfaces we prove that if curvature tensor is not uniformly bounded then one can form a singular model called deepest bubble such deepest bubble has to be a scalar flat ale kahler metric in certain kahler classes on toric fano surfaces the sobolev constants are a priori bounded along the calabi flow with small calabi energy we can also show in certain cases no deepest bubble can form along the flow it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently to illustrate our results more clearly we focus on an example on mathbb cp 2 blown up three points at generic position our result also implies existence of constant scalar curvature metrics on mathbb cp 2 blown up three points at generic position in the kahler classes where the exceptional divisors have the same area

local observability of nonlinear systems a necessary and sufficient condition for local observability of analytic systems is presented for smooth systems this condition is only sufficient but it is still weaker than the hermann krener rank condition it is expressed with the language of ideals of germs of analytic or smooth functions and real radicals of such ideals the condition may be practically checked in many cases

linear complementarity systems zeno states a linear complementarity system lcs is a hybrid dynamical system defined by a linear time invariant ordinary differential equation coupled with a finite dimensional linear complementarity problem lcp the present paper is the first of several papers whose goal is to study some fundamental issues associated with an lcs specifically this paper addresses the issue of zeno states and the related issue of finite number of mode switches in such a system the cornerstone of our study is an expansion of a solution trajectory to the lcs near a given state in terms of an observability degree of the state on the basis of this expansion and an inductive argument we establish that an lcs satisfying the p property has no strongly zeno states we next extend the analysis for such an lcs to a broader class of problems and provide sufficient conditions for a given state to be weakly non zeno while related mode switch results have been proved by brunovsky and sussmann for more general hybrid systems our analysis exploits the special structure of the lcs and yields new results for the latter that are of independent interest and complement those by these two and other authors

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

linear complementarity systems we introduce a new class of dynamical systems called linear complementarity systems the time evolution of these systems consists of a series of continuous phases separated by events which cause a change in dynamics and possibly a jump in the state vector the occurrence of events is governed by certain inequalities similar to those appearing in the linear complementarity problem of mathematical programming the framework we describe is suitable for certain situations in which both differential equations and inequalities play a role for instance in mechanics electrical networks piecewise linear systems and dynamic optimization we present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions

reorientation of linear switched systems using state feedback a characterisation is obtained for the controlled and conditioned invariants of a linear switched system under feedback the existence of a maximal consistent subspace with feedback is shown and is computed a few sufficient conditions under which the feedback has no effect on the controlled and conditioned invariants are obtained sufficient conditions for achieving the minimal jump subspace with feedback when it exists were also derived the impossibility of forcing the jump subspace to a trivial subspace with state feedback is shown some sufficient conditions on the feedback for the consistent and jump subspaces to intersect trivially have also been stated

computation of nonautonomous invariant and inertial manifolds we derive a numerical scheme to compute invariant manifolds for time variant discrete dynamical systems i e nonautonomous difference equations our universally applicable method is based on a truncated lyapunov perron operator and computes invariant manifolds using a system of nonlinear algebraic equations which can be solved both locally using nonsmooth inexact newton and globally using continuation algorithms compared to other algorithms our approach is quite flexible since it captures time dependent nonsmooth noninvertible or implicit equations and enables us to tackle the full hierarchy of strongly stable stable and center stable manifolds as well as their unstable counterparts our results are illustrated using a test example and are applied to a population dynamical model and the henon map finally we discuss a linearly implicit euler bubnov galerkin discretization of a reaction diffusion equation in order to approximate its inertial manifold

on the existence and nonexistence of extremal metrics on toric kahler surfaces abstract in this paper we study the existence of extremal metrics on toric kahler surfaces we show that on every toric kahler surface there exists a kahler class in which the surface admits an extremal metric of calabi we found a toric kahler surface of 9 t c 2 fixed points which admits an unstable kahler class and there is no extremal metric of calabi in it moreover we prove a characterization of the k stability of toric surfaces by simple piecewise linear functions as an application we show that among all toric kahler surfaces with 5 or 6 t c 2 fixed points cp 2 3 cp 2 is the only one which allows vanishing futaki invariant and admits extremal metrics of constant scalar curvature

three modeling paradigms in mathematical programming celebrating the sixtieth anniversary since the zeroth international symposium on mathematical programming was held in 1949 this paper discusses several promising paradigms in mathematical programming that have gained momentum in recent years but have yet to reach the main stream of the field these are competition dynamics and hierarchy the discussion emphasizes the interplay between these paradigms and their connections with existing subfields including disjunctive equilibrium and nonlinear programming and variational inequalities we will describe the modeling approaches mathematical formulations and recent results of these paradigms and sketch some open mathematical and computational challenges arising from the resulting optimization and equilibrium problems our goal is to elucidate the need for a systematic study of these problems and to inspire new research in the field

bounds on the number of switchings for trajectories of piecewise analytic vector fields abstract we study the trajectories of systems x x x where x is a continuous extendably piecewise analytic vector field i e a continuous vector field x such that the domain of admits a locally finite partition i into sets such that for each a i there is a vector field xa which is analytic on a neighborhood of the closure of a and whose restriction to a coincides with that of x we prove that the trajectories are piecewise analytic with a priori bounds on the number of switchings for all trajectories that stay in a fixed compact set and whose duration does not exceed a fixed number t this result implies the existence of a regular synthesis for optimal control problems with a strictly convex lagrangian and a linear dynamics with polyhedral constraints on the controls

filtrations and test configurations we introduce a strengthening of k stability based on filtrations of the homogeneous coordinate ring this allows for considering certain limits of families of test configurations which arise naturally in several settings we prove that if a manifold with no automorphisms admits a csck metric then it satisfies this stronger stability notion we also discuss the relation with the birational transformations in the definition of b stability