rigidity for anosov actions of higher rank lattices this is the simplest example of a large class of analytic standard actions of lattices in semisimple lie groups on locally homogeneous spaces a basic problem is to understand the differentiable actions near to such a standard action in terms of their geometry and dynamics cf 13 50 51 a cr action so if x x x of a group f on a compact manifold x is said to be anosov if there exists at least one element yh e f such that p yh is an anosov diffeomorphism of x we begin in this paper to study the anosov differentiable actions of lattices including many standard algebraic examples and especially to study their stability properties our main theme is that either the cr rigidity or the cr deformation rigidity of an anosov action for 1 r cr or even for the real analytic case can be shown just by studying the behavior of the periodic orbits for the action there are two notions of structural stability that appear in this paper rigidity and deformation rigidity a cl perturbation of a cr action so is simply another cr action spl such that for a finite set of generators 81 ad of f the cr diffeomorphisms p 8i and pl 8i are cl close for all i an action so is said to be cr rigid or topologically rigid if r 0 if every sufficiently small c perturbation of so is cr conjugate to so for 0 r oo or r w in the case of real analytic actions a cl deformation of an action so is a continuous path of cr actions pt defined for some 0 t e with spo sd an action sd is said to be cr deform

on controllability of linear stochastic systems we discuss several concepts of controllability for partially observable stochastic systems complete controllability approximate controllability controllability and s controllability and show that complete and approximate controllability notions are equivalent and in turn are equivalent to the controllability for linear stochastic systems controlled by gaussian processes we derive necessary and sufficient conditions for these concepts of controllability these criteria reduce to the well known rank condition

a maximum principle for free endtime optimal control problems with data discontinuous in time recently a maximum principle has been proved for free endtime problems when the dynamics and the control constraint set are assumed to be merely measurable with respect to the time variable a new simple derivation of this optimality condition is given from standard fixed endtime results it involves a generalization of the customary boundary condition on the maximized hamiltonian function evaluated along the optimal state trajectory and costate function the hypotheses under which the maximum principle is proved permit unilateral state constraints data merely lipschitz continuous in the state variable and endpoint constraints expressed as general set inclusions the primary concern is with methodology and setting down a simple proof of the maximum principle for problems with data measurable in the time variable

infinitesimal rigidity for the action of sl n ℤ on n let r sl n z or any subgroup of finite index then the action of r on tn by automorphisms is infinitesimally rigid for n 7 i e the cohomology h i r vec tn 0 where vec t n denotes the module of c vector fields on t

smooth classification of cartan actions of higher rank semisimple lie groups and their lattices let g be a connected semisimple lie group without compact factors whose real rank is at least 2 and let g be an irreducible lattice we provide a c classification for volume preserving cartan actions of an d g also if g has real rank at least 3 we provide a c classification for volume preserving multiplicity free trellised anosov actions on compact manifolds

filtering for linear systems with shifted noises in this paper a method of approximation of a white noise by wide band noises is developed and based on this the complete set of equations of the kalman type optimal filter for the linear signal observation system with the signal noise delaying the observation noise is deduced the respective riccati equation is found as a system of equations containing partial differential equations a possible application of the results to navigation and guidance of spacecrafts is discussed

lie brackets and local controllability a sufficient condition for scalar input systems we develop a general formalism based on exponential lie series which can be used to study control variations we apply the formalism to prove a result conjectured by h hermes that gives a sufficient condition for small time local controllability for single input systems

on concepts of controllability for deterministic and stochastic systems the new necessary and sufficient conditions which are formulated in terms of convergence of a certain sequence of operators involving the resolvent of the negative of the controllability operator are found for deterministic linear stationary control systems to be completely and approximately controllable respectively these conditions are applied to study the s controllability a property of attaining an arbitrarily small neighborhood of each point in the state space with a probability arbitrarily close to one and c controllability the s controllability fortified with some uniformity of stochastic systems it is shown that the s controllability the c controllability of a partially observable linear stationary control system with an additive gaussian white noise disturbance on all the intervals 0 t for t 0 is equivalent to the approximate complete controllability of its deterministic part on all the intervals 0 t for t 0

controllability of semilinear stochastic systems the picard type approximation is used to derive sufficient conditions for approximate and complete controllability of semilinear stochastic systems with non lipschitz coefficients under the assumption that the corresponding linear system is appropriately controllable

controllability of linear stochastic systems abstract we discuss several concepts of controllability for stochastic systems and derive sufficient and in most cases also necessary algebraic conditions for the controllability of the state and the output of white and colored noise linear systems these criteria reduce to the well known deterministic ones in the absence of noise

applications of optimal multiprocesses in a previous paper a theory of necessary conditions for optimal multiprocess problems was supplied that is problems in dynamic optimization involving a collection of control systems coupled through constraints on the endpoints of the trajectories and the cost functions here the scope of the new theory is illustrated by application to optimization problems in investment planning impulse control optics robotics and renewable resources

partial controllability concepts the main results in theory of controllability are formulated for deterministic or stochastic control systems given in a standard form i e given as a first order differential equation driven by an infinitesimal generator of strongly continuous semigroup in an abstract hilbert or banach space and disturbed by a deterministic function or by a white noise process at the same time some deterministic or stochastic linear systems can be written in a standard form if the state space is enlarged respectively the ordinary controllability conditions for them are too strong since they assume extended state space it is reasonable to introduce partial controllability concepts which assume original state space in this paper we study necessary and sufficient conditions of partial controllability for deterministic and stochastic linear control systems given in a standard form and their implications to particular cases

brief paper multi modal control using adaptive motion description languages abstract in this paper we address the problem of adaptively enriching motion description languages for multi modal control by systematically enlarging the set of available modes this problem is formulated as an optimal control problem where new modes are designed by combining recurring mode string fragments into smooth new meta modes and variational arguments are employed to derive optimality conditions for this construction

stochastic maximum principle in the pontryagin s form for wide band noise driven systems the control problems for wide band noise driven systems have the following specific feature a wide band noise is given by its autocovariance function at the same time different wide band noises may have the same autocovariance function therefore it is important to obtain the results which are invariant under the change of wide band noises corresponding to the same autocovariance function in this paper one such result is obtained a stochastic maximum principle in the pontryagin s form is proved for a control system driven by a state dependent wide band noise the result is demonstrated on an example of controlled stochastic differential equation of risky asset price

representations of fundamental groups of manifolds with a semisimple transformation group group 1 m and the lie groups that can act on m more precisely let g be a connected semisimple lie group of higher real rank and suppose g acts continuously on a topological manifold m preserving a finite measure the main theme of this paper is that the representation theory of 7r i m in low dimensions is to a large extent controlled by that of g the latter of course being well understood in particular under natural hypotheses e g that the action of g on m is engaging i e there is no loss of ergodicity in passing to finite covers see definition 3 1 below we prove that if g has no nontrivial

livsic theorem for matrix cocycles we prove the liv sic theorem for arbitrary gl m r cocycles we con sider a hyperbolic dynamical system f x x and a holder continuous function a x gl m r we show that if a has trivial periodic data i e a f n 1 p a fp a p id for each periodic point p f n p then there exists a holder continuous function c x gl m r satisfying a x c fx c x 1 for all x2 x the main new ingredients in the proof are results of independent interest on relations between the periodic data lyapunov exponents and uniform estimates on growth of products along orbits for an arbitrary holder function a

on livsic s theorem superrigidity and anosov actions of semisimple lie groups we prove a generalization of livsic s theorem on the vanishing of the cohomology of certain types of dynamical systems as a consequence we strengthen a result due to zimmer concerning algebraic hulls of anosov actions of semisimple lie groups combining this with topological superrigidity we find a holder geometric structure for multiplicity free anosov actions

finding interpolating curves minimizing l infty acceleration in the euclidean space via optimal control theory we study the problem of finding an interpolating curve passing through prescribed points in the euclidean space the interpolating curve minimizes the pointwise maximum length i e l infty norm of its acceleration we reformulate the problem as an optimal control problem and employ simple but effective tools of optimal control theory we characterize solutions associated with singular and nonsingular controls some of the results we obtain are new even for the scalar interpolating function case we reduce the infinite dimensional interpolation problem to an ensuing finite dimensional one and derive closed form expressions for interpolating curves consequently we devise efficient numerical techniques and illustrate them with examples

on partial s controllability of semilinear partially observable systems in this paper the partial s controllability of semilinear partially observable control systems is investigated the essence of study is the presence of partial observations we define a weakened partial s controllability concept for partially observable systems a sufficient condition for this controllability concept is proved the method of proof differs from the traditional proofs by fixed point theorems the obtained result is demonstrated with examples

practical optimal state feedback control law for continuous time switched affine systems with cyclic steady state in this article a method for computing an optimal state feedback control law for continuous time switched affine systems exhibiting cyclic behaviour in steady state is presented the hybrid solutions are deduced from the fillipov solutions it is shown that the optimal trajectory synthesis implies to determine singular arcs algebraic conditions are given to obtain these particular arcs of the trajectory a numerical procedure is then proposed to generate optimal trajectories on a given state space area avoiding the classical two point boundary value problem occurring in optimal control synthesis the interpolation of the solutions set through a neural network yields a state feedback control law several examples in the power converters field show the feasibility and the efficiency of the method

partial controllability of stochastic linear systems the controllability concepts for linear stochastic differential equations driven by different kinds of noise processes can be reduced to the partial controllability concepts for the same systems driven by correlated white noises based on this fact in this article we study the conditions of exact and approximate controllability for linear stochastic control systems under various kinds of noise processes including correlated white noises as well as coloured wide band and shifted white noises it is proved that such systems are never exactly controllable while their approximate controllability is equivalent to the approximate controllability of the associated linear deterministic systems at all past time moments

on groups of polynomial subgroup growth let be a finitely generated group andan the number of its subgroups of indexn we prove that assuming is residually nilpotent e g linear thenan grows polynomially if and only if is solvable of finite rank this answers a question of segal the proof uses a new characterization ofp adic analytic groups the theory of algebraic groups and the prime number theorem the method can be applied also to groups of polynomial word growth

on controllability conception for stochastic systems the controllability notions for partially observed stochastic systems are defined their relation with complete and approximate controllabilities is shown in particular it is proven that the approximate controllability condition is necessary and the complete controllability condition is sufficient for the partially observed linear gaussian control system to attain the arbitrarily small neighborhood of each point in the state space with probability arbitrarily closely to one

approximate controllability of nonlinear stochastic impulsive systems with control acting on the nonlinear terms this paper is concerned with the approximate controllability of the stochastic impulsive system with control acting on the nonlinear terms in the case that the nonlinear terms are dependent on the control the control cannot be expressed explicitly and analysed in this situation we generate the control sequence by the approximate equations and give the properties of the control sequence and the driven solution sets some discussions on the assumptions are given to impose on the system the hausdorff measure is adopted to relax the requirement of the compactness condition it is also shown that under some sufficient conditions the stochastic impulsive system is approximately controllable without the requirement of the controllability of the associated linear system the results of this paper can be degraded into special cases and coincide with some existing ones

controllability of stochastic linear systems in the paper necessary and sufficient conditions for various types of stochastic controllability of the linear stochastic system of the form d x ax d t bu d t c d w r x 0 x 0 are given some possible extensions are indicated as well

an effective pseudospectral method for constraint dynamic optimisation problems with characteristic times dynamic optimisation problem with characteristic times widely existing in many areas is one of the frontiers and hotspots of dynamic optimisation researches this paper considers a class of dynam

bisimilar linear systems the notion of bisimulation in theoretical computer science is one of the main complexity reduction methods for the analysis and synthesis of labeled transition systems bisimulations are special quotients of the state space that preserve many important properties expressible in temporal logics and in particular reachability in this paper the framework of bisimilar transition systems is applied to various transition systems that are generated by linear control systems given a discrete time or continuous time linear system and a finite observation map we characterize linear quotient maps that result in quotient transition systems that are bisimilar to the original system interestingly the characterizations for discrete time systems are more restrictive than for continuous time systems due to the existence of an atomic time step we show that computing the coarsest bisimulation which results in maximum complexity reduction corresponds to computing the maximal controlled or reachability invariant subspace inside the kernel of the observations map these results establish strong connections between complexity reduction concepts in control theory and computer science

feedback can reduce the specification complexity of motor programs we show that when it is possible to use feedback in the specification of motor programs the length of the descriptions of the instruction sequences for carrying out a given task can be reduced by a factor that reflects the richness of the available feedback signals the model on which this work is based is that of a finite automaton modified in such a way that instruction processing is akin to the way in which difference or differential equations process piecewise constant inputs in terms of such free running automata we show that when feedback is available the length of the shortest description can be reduced by a factor depending on the ratio of the size of the entire state space to the size of the set of states for which feedback is locally effective

on the hybrid optimal control problem theory and algorithms a class of hybrid optimal control problems hocp for systems with controlled and autonomous location transitions is formulated and a set of necessary conditions for hybrid system trajectory optimality is presented which together constitute generalizations of the standard maximum principle these are given for the cases of open bounded control value sets and compact control value sets the derivations in the paper employ i classical variational and needle variation techniques and ii a local controllability condition which is used to establish the adjoint and hamiltonian jump conditions in the autonomous switching case employing the hybrid minimum principle hmp necessary conditions a class of general hmp based algorithms for hybrid systems optimization are presented and analyzed for the autonomous switchings case and the controlled switchings case using results from the theory of penalty function methods and ekeland s variational principle the convergence of these algorithms is established under reasonable assumptions the efficacy of the proposed algorithms is illustrated via computational examples

an optimal control approach to mode generation in hybrid systems abstract this paper presents a solution to the problem of constructing control programs i e sequences of control modes from a given motion alphabet in particular techniques are developed that enable reinforcement learning to act directly at the mode level and hence make learning applicable to continuous time control systems in a straightforward manner moreover given such a control program the issue of improving the system performance through the addition of new control laws is addressed as an optimal control problem in fact this is achieved through an optimal combination of recurring mode strings a number of examples are provided that illustrate the viability of the proposed methods