system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

a convex approach to the absolute stability problem in this paper the problem of absolute stability is revisited it is show that it is possible to use convexity properties of the functions involved to develop an efficient algorithm for solving the absolute stability problem examples are presented to illustrate the results

weak stability of ricci expanders with positive curvature operator we prove the l infty stability of expanding gradient ricci solitons with positive curvature operator and quadratic curvature decay at infinity

stability analysis with popov multipliers and integral quadratic constraints it is shown that a general form of popov multipliers can be used in stability analysis based on integral quadratic constraints iqc the popov multiplier is nonproper and a condition that the nominal plant is strictly proper will be imposed in order to ensure boundedness of the iqc corresponding to the popov multiplier a consequence of our main result is that the classical popov criterion can be combined with a stability criterion for slope restricted nonlinearities developed by zames and falb an example shows that the combination of these two criteria is useful in applications

lmi searches for anticausal and noncausal rational zames falb multipliers abstract given a linear time invariant plant the search for a suitable multiplier over the class of zames falb multipliers is a challenging problem which has been studied for several decades recently a new linear matrix inequality search has been proposed over rational and causal zames falb multipliers this letter analyzes the conservatism of the restriction to causality on the multipliers and presents a complementary search for rational and anticausal multipliers the addition of a popov multiplier to the anticausal zames falb multiplier is implemented by analogy with the causal search as a result a search over a noncausal subset of zames falb multipliers is obtained a comparison between all the search methods proposed in the literature is given

reachability observability and realizability of continuous time positive systems this paper discusses reachability observability and realizability of single input single output linear time invariant systems in which state variables and or input output functions are restricted to be nonnegative to reflect physical constraints frequently encountered in real systems we define a set reachable from the origin with nonnegative inputs and also a set observable with nonnegative outputs we investigate geometrical structures of the sets through convex analysis and a duality relation between them is established next we consider positive realization of a given transfer function using the reachable set and the observable set we give a necessary and sufficient condition for positive realizability an example is given to demonstrate that a positive realizable transfer function does not in general have a jointly controllable and observable positive realization

stable and unstable einstein warped products in this article we systematically investigate the stability properties of certain warped product einstein manifolds we characterize stability of these metrics in terms of an eigenvalue condition of the einstein operator on the base manifold in particular we prove that all complete manifolds carrying imaginary killing spinors are strictly stable moreover we show that ricci flat and hyperbolic cones over kahler einstein fano manifolds and over nonnegatively curved einstein manifolds are stable if the cone has dimension n 10 on the other hand we construct unstable einstein metrics of negative scalar curvature

equivalence between classes of multipliers for slope restricted nonlinearities different classes of multipliers have been proposed in the literature for obtaining stability criteria using passivity theory integral quadratic constraint iqc theory or lyapunov theory some of these classes of multipliers can be applied with slope restricted nonlinearities in this paper the concept of phase containment is defined and it is shown that several classes are phase contained within the class of zames falb multipliers there are two main consequences firstly it follows that the class of zames falb multipliers remains to date the widest class of available multipliers for slope restricted nonlinearities secondly further restrictions may be avoided when exploiting the parametrization of the other classes of multipliers

multiobjective output feedback control via lmi optimization this paper presents an overview of a linear matrix inequality lmi approach to the multiobjective synthesis of linear output feedback controllers the design objectives can be a mix of h sub spl infin performance h sub 2 performance passivity asymptotic disturbance rejection time domain constraints and constraints on the closed loop pole location in addition these objectives can be specified on different channels of the closed loop system when all objectives are formulated in terms of a common lyapunov function controller design amounts to solving a system of linear matrix inequalities the validity of this approach is illustrated by a realistic design example

brief new results for analysis of systems with repeated nonlinearities this work presents new results for analysis of systems with repeated monotonic or slope restricted nonlinearities we first give new integral quadratic constraints iqcs that are satisfied by the inputs and outputs of diagonal operators with equal monotonic or slope restricted diagonal entries then we present new analysis results for systems with repeated nonlinearities obtained using the new iqcs for the type of nonlinearities considered in this work the new results are less conservative than previously known results the computation of the new conditions is discussed the results in this paper apply also to problems with nonrepeated nonlinearities whenever these are representable as the interconnection of multiple repeated nonlinearities and a constant matrix

dynamical stability and instability of ricci flat metrics in this short article we improve the dynamical stability and instability results for ricci flat metrics under ricci flow proved by sesum duke math j 133 1 26 2006 and haslhofer calc var partial differ equ 45 481 504 2012 getting rid of the integrability assumption

lyapunov functions and l 2 gain bounds for systems with slope restricted nonlinearities abstract the stability and l 2 performance analysis of systems consisting of an interconnection of a linear time invariant lti system and a static nonlinear element which is lipschitz slope restricted and sector bounded is revisited the main thrust of the paper is to improve and extend an existing result in the literature to enable i concise and correct conditions for asymptotic stability of the interconnection and ii reasonably tight bounds on the l 2 gain between an exogenous input and a given output to be obtained numerical examples indicate that the proposed algorithm performs well compared to competing results in the literature

a new approach to lipschitz continuity in state constrained optimal control for a linear quadratic state constrained optimal control problem it is proved in 11 that under an independence condition for the active constraints the optimal control is lipschitz continuous we now give a new proof of this result based on an analysis of the euler discretization given in 9 there we exploit the lipschitz continuity of the control to estimate the error in the euler discretization here we show that the theory developed for the euler discretization can be used to derive the lipschitz continuity of the optimal control

convergence analysis for approximations of optimal control problems subject to higher index differential algebraic equations and mixed control state constraints this paper establishes a convergence result for implicit euler discretizations of optimal control problems with differential algebraic equations of higher index and mixed control state constraints

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

stability analysis of optimal control problems with a second order state constraint this paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second order the strengthened legendre clebsch condition is assumed to hold and no assumption on the structure of the contact set is made under a weak second order sufficient condition taking into account the active constraints we show that the solutions are lipschitz continuous w r t the perturbation parameter in the l 2 norm and holder continuous in the l infty norm we use a generalized implicit function theorem in metric spaces by dontchev and hager siam j control optim 36 1998 pp 698 718 the difficulty is that multipliers associated with second order state constraints have a low regularity they are only bounded measures we obtain lipschitz stability of a primitive of the state constraint multiplier

stability analysis with dissipation inequalities and integral quadratic constraints this technical note considers the stability of a feedback connection of a known linear time invariant system and a perturbation the input output behavior of the perturbation is described by an integral quadratic constraint iqc iqc stability theorems can be formulated in the frequency domain or with a time domain dissipation inequality the two approaches are connected by a non unique factorization of the frequency domain iqc multiplier the factorization must satisfy two properties for the dissipation inequality to be valid first the factorization must ensure the time domain iqc holds for all finite times second the factorization must ensure that a related matrix inequality when feasible has a positive semidefinite solution this technical note shows that a class of frequency domain iqc multipliers has a factorization satisfying these two properties thus the dissipation inequality test with an appropriate factorization can be used with no additional conservatism

a survey of the maximum principles for optimal control problems with state constraints this paper gives a survey of the various forms of pontryagin s maximum principle for optimal control problems with state variable inequality constraints the relations between the different sets of optimality conditions arising in these forms are shown furthermore the application of these maximum principle conditions is demonstrated by solving some illustrative examples

a tutorial on the positive realization problem this paper is a tutorial on the positive realization problem that is the problem of finding a positive state space representation of a given transfer function and characterizing existence and minimality of such representation this problem goes back to the 1950s and was first related to the identifiability problem for hidden markov models then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory within this framework developing some ideas sprang in the 1960s during the 1980s the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem in this paper the reader is carried through the key ideas which have proved to be useful in order to tackle this problem in order to illustrate the main results contributions and open problems several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided

multiplier methods for nonlinear optimal control error estimates are derived for an augmented lagrangian approximation to an optimal control problem convex control constraints are treated explicitly while a lagrange multiplier is introduced for the nonlinear system dynamics the nonlinear optimal control problem does not fit the classical theory for estimating the error in multiplier approximations since the natural coercivity assumption is formulated in a hilbert space where the cost is not differentiable this discrepancy between the function space setting needed for coercivity and that needed for differentiability is compensated for by regularity results associated with the necessary conditions the paper concludes with an analysis of the optimal penalty parameter corresponding to a given finite element discretization

linear and dynamical stability of ricci flat metrics we can talk about two kinds of stability of the ricci flow at ricci flat metrics one of them is a linear stability defined with respect to perelman s functional f see 1 page 5 the other one is a dynamical stability and it refers to a convergence of a ricci flow starting at any metric in a neighborhood of a considered ricci flat metric we show that dynamical stability implies linear stability we also show that a linear stability together with the integrability assumption implies dynamical stability as a corollary we get a stability result for k3 surfaces part of which has been done in 11 corollary 4 15 theorem 4 16 our stability result applies to calabi yau manifolds as well

lipschitzian stability for state constrained nonlinear optimal control for a nonlinear optimal control problem with state constraints we give conditions under which the optimal control depends lipschitz continuously in the l2 norm on a parameter these conditions involve smoothness of the problem data uniform independence of active constraint gradients and a coercivity condition for the integral functional under these same conditions we obtain a new nonoptimal stability result for the optimal control in the l infty norm and under an additional assumption concerning the regularity of the state constraints a new tight l infty estimate is obtained our approach is based on an abstract implicit function theorem in nonlinear spaces

runge kutta methods in optimal control and the transformed adjoint system the convergence rate is determined for runge kutta discretizations of nonlinear control problems the analysis utilizes a connection between the kuhn tucker multipliers for the discrete problem and the adjoint variables associated with the continuous minimum principle this connection can also be exploited in numerical solution techniques that require the gradient of the discrete cost function

on uniqueness of tangent cones for einstein manifolds we show that for any ricci flat manifold with euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross section similarly for any noncollapsing limit of einstein manifolds with uniformly bounded einstein constants we show that local tangent cones are unique if one tangent cone has a smooth cross section

the euler approximation in state constrained optimal control we analyze the euler approximation to a state constrained control problem we show that if the active constraints satisfy an independence condition and the lagrangian satisfies a coercivity condition then locally there exists a solution to the euler discretization and the error is bounded by a constant times the mesh size the proof couples recent stability results for state constrained control problems with results established here on discrete time regularity the analysis utilizes mappings of the discrete variables into continuous spaces where classical finite element estimates can be invoked

technical communique lyapunov functions for the multivariable popov criterion with indefinite multipliers we demonstrate the existence of lur e postnikov type lyapunov functions for a reasonably general form of the multivariable popov criterion in particular the multipliers need not be positive semi definite we discuss the application to the special cases where the feedback element is either diagonal unstructured or linear

sufficient optimality conditions for optimal control subject to state constraints strong second order sufficient optimality conditions ssc are derived for optimal control problems of systems described by nonlinear odes subject to mixed control state and pure state constraints the obtained ssc are expressed in terms of a modified legendre clebsch condition and the associated riccati equation the role of ssc in stability analysis of solutions to parametric optimal control problems is briefly discussed

dual approximations in optimal control we analyze a dual approximation for the solution to an optimal control problem the differential equation is handled with a lagrange multiplier while other constraints are treated explicitly an algorithm for solving the dual problem is presented

error estimates for the euler discretization of an optimal control problem with first order state constraints we propose some error estimates for the discrete solution of an optimal control problem with first order state constraints where the trajectories are approximated with a classical euler scheme we obtain order 1 approximation results in the l infty norm as opposed to the order 2 3 results obtained in the literature we assume either a strong second order optimality condition or a weaker formulation in the case where the state constraint is scalar and satisfies some hypotheses for junction points and where the time step is constant our technique is based on some homotopy path of discrete optimal control problems that we study using perturbation analysis of nonlinear programming problems

metric regularity and stability of optimal control problems for linear systems this paper studies stability properties of the solutions of optimal control problems for linear systems the analysis is based on an adapted concept of metric regularity the strong bi metric regularity which is introduced and investigated in the paper it allows one to give a more precise description of the effect of perturbations on the optimal solutions in terms of a holder type estimate and to investigate the robustness of this estimate the holder exponent depends on a natural number k which is known as the controllability index of the reference solution an inverse function theorem for strongly bi metrically regular mappings is obtained which is used in the case k 1 for proving stability of the solution of the considered optimal control problem under small nonlinear perturbations moreover a new stability result with respect to perturbations in the matrices of the system is proved in the general case k geq 1

second order runge kutta approximations in control constrained optimal control in this paper we analyze second order runge kutta approximations to a nonlinear optimal control problem with control constraints if the optimal control has a derivative of bounded variation and a coercivity condition holds we show that for a special class of runge kutta schemes the error in the discrete approximating control is o h2 where h is the mesh spacing

delay integral quadratic constraints and stability multipliers for systems with mimo nonlinearities the formula for stability multipliers for systems with mimo nonlinearities is derived using the method of delay integral quadratic constraints the results applicable to both time invariant and time periodic systems are then used to derive a solution for a special case of a nonlinear regulation problem and a criterion for stability of forced oscillations a linear time periodic system is considered to illustrate the difference between the delay integral quadratic constraints that yield stability criteria in multiplier form and those that do not

stability criteria of sector and slope restricted lur e systems presents stability criteria of sector and slope restricted lur e systems in terms of linear matrix inequalities by fully exploiting inherent properties of sector and slope restrictions in the time domain interpreting the time domain criteria in the frequency domain furthermore supplies simpler expressions several examples show excellent performances of these criteria

analysis of systems with slope restricted nonlinearities using externally positive zames falb multipliers this paper proposes an approach for assessing the stability of feedback interconnections where one element is a static slope restricted nonlinearity and the other element is a linear system the approach is based on the use of zames falb multipliers where the dynamic portion of the multiplier is chosen as an externally positive noncausal transfer function by restricting attention to a subset of these multipliers a set of pure linear matrix inequality conditions is obtained which requires no initial parameterization by the user a useful byproduct of using externally positive systems is that the results are applicable to nonodd slope restricted nonlinearities which is not the case for all classes of zames falb multipliers

the riccati equation for optimal control problems with mixed state control constraints necessity and sufficiency the goal of this paper is to conduct a complete study of second order conditions for the optimal control problem with mixed state control constraints the conjugate point theory is presented and a necessary condition in terms of the corresponding riccati equation is obtained sufficiency criteria are developed in terms of strengthened necessary conditions including the riccati equation the results generalize the known ones for pure control constraints as well as for the mixed state control constraints

well posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control this paper deals with the shooting algorithm for optimal control problems with a scalar control and a regular scalar state constraint additional conditions are displayed under which the so called alternative formulation is equivalent to pontryagin s minimum principle the shooting algorithm appears to be well posed invertible jacobian iff i the no gap second order sufficient optimality condition holds and ii when the constraint is of order q geq 3 there is no boundary arc stability and sensitivity results without strict complementarity at touch points are derived using robinson s strong regularity theory under a minimal second order sufficient condition the directional derivatives of the control and state are obtained as solutions of a linear quadratic problem