morphological shape decomposition a technique for decomposing a binary shape into a union of simple binary shapes is presented the decomposition is shown to be unique and invariant to translation rotation and scaling the techniques used in the decomposition are based on mathematical morphology the shape description produced can be used in object recognition and in binary image coding

morphological decomposition of 2 d binary shapes into modestly overlapped octagonal and disk components one problem with several leading morphological shape representation algorithms is the heavy overlapping among the representative disks of the same size a shape component formed by grouping connected disk centers may use many heavily overlapping disks to represent a simple shape part sometimes these representative disks form complicated structures a generalized skeleton transform was recently introduced which allows a shape to be represented as a collection of modestly overlapped octagonal shape parts however the generalized skeleton transform needs to be applied many times furthermore an octagonal component is not easily matched up with another octagonal component in this paper we describe a octagon fitting algorithm which identifies a special maximal octagon for each image point in a given shape this transform leads to the development of two new shape decomposition algorithms these algorithms are more efficient to implement the octagon fitting algorithm only needs to be applied once the components generated are better characterized mathematically the disk components used in the second decomposition algorithm are more primitive than octagons and easily matched up with other disk components from another shape the experiments show that the new decomposition algorithms produce as efficient representations as the old algorithm for both exact and approximate cases a simple shape matching algorithm using disk components is also demonstrated

optimal control problems with set valued control and state constraints in this paper a general optimal control problem with pure state and mixed control state constraints is considered these constraints are of the form of set inclusions second order necessary optimality conditions for weak local minimum are derived for this problem in terms of the original data in particular the nonemptiness of the set of critical directions and the evaluation of its support function are expressed in terms of the given functions and set valued maps in order that the lagrange multiplier corresponding to the mixed control state inclusion constraint be represented via an integrable function a strong normality condition involving the notion of the critical tangent cone is introduced

on the stabilization of bilinear systems via constant feedback we study the problem of stabilization of a bilinear system via a constant feedback the question reduces to an eigenvalue problem on the pencil a 0b of two matrices using the idea of simultaneous triangularization of the matrices involved some easily checkable conditions for the solvability of this question are obtained algorithms for checking these conditions are given and illustrated by a few examples

stabilizing controllers for bilinear systems the problem of finding stabilizing feedback controllers for continuous bilinear systems where the controls act additively and multiplicatively simultaneously is treated the applicability of the quadratic control law of jacobson 5 and others is extended to the case when the a matrix has arbitary eigenvalues under certain conditions a class of bilinear systems common among biochemical flow systems is defined the dyadic bilinear systems a control scheme the so called division controller for dyadic bilinear systems is suggested the practicality of the control schemes is demonstrated on the problem of controlling the neutron level in a fission reactor

morphological decomposition of 2 d binary shapes into conditionally maximal convex polygons mathematical morphology is an approach to image analysis based on the geometric concept of shape and size naturally it provides a very effective tool for shape analysis in this paper we present a morphological shape segmentation algorithm that decomposes a 2 d two dimensional binary shape into a collection of restricted convex polygons we use simple morphological operations to extract shape information from different skeleton subsets to form a group of conditionally maximal convex polygonal shape components then a set of more meaningful and natural convex polygonal components are constructed from these maximal components the resulting algorithm is very simple and the decomposition is always unique the components produced have well defined mathematical characterizations and they seem to be in good agreement with the nature structures of the given shapes the shape segments produced can be used to construct structural shape description and for other shape analysis purposes

first and second order necessary conditions for control problems with constraints second order necessary conditions are developed for an abstract nonsmooth control problem with mixed state control equality and inequality constraints as well as a constraint of the form g x u e if where f is a closed convex set of a banach space with nonempty interior the inequality constraints g s x u 0 depend on a parameter s belonging to a compact metric space s the equality constraints are split into two sets of equations k x u 0 and h x u 0 where the first equation is an abstract control equation and h is assumed to have a full rank property in u the objective function is maxtet f t x u where t is a compact metric space f is upper semicontinuous in t and lipschitz in x u the results are in terms of a function a that disappears when the parameter spaces t and s are discrete we apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints thus we obtain a generalization and extension of the existing results on this problem

a generalized discrete morphological skeleton transform with multiple structuring elements for the extraction of structural shape components a common problem shared by several leading morphological shape representation algorithms is that there is much overlapping among the representative disks of the same size a shape component represented by a group of connected disk centers sometimes uses many heavily overlapping representative disks to represent a relatively simple shape part a shape component may also contain a large number of representative disks that form a complicated structure we introduce a generalized discrete morphological skeleton transform that uses eight structuring elements to generate skeleton subsets so that no two skeletal points from the same skeleton subset are adjacent to each other each skeletal point represents a shape part that is in general an octagon with four pairs of parallel opposing sides the number of representative points needed to represent a given shape is significantly lower than that in the standard skeleton transform a collection of shape components needed to build a structural representation is easily derived from the generalized skeleton transform each shape component covers a significant area of the given shape and severe overlapping is avoided the given shape can also be accurately approximated using a small number of shape components

morphological reversible contour representation in this paper a novel morphological reversible contour representation of discrete binary images is proposed a binary image is represented by a set of nonoverlapping multilevel contours and a residual image in this proposed representation the total number of pixels representing an image is far less than the total number of pixels obtained by the seed based morphological contour skeleton lossless representation the proposed contour representation is simple unique and general without restrictions on the binary image to be represented the resulting multicontour image component is robust to noise an efficient differential chain contour coding scheme is employed to further compress the represented image the proposed method yields very low bit rates compared to the existing morphological techniques an automatic filling procedure which properly fills a proper multicontour image according to its topological structure without need of seed points is proposed

second order derivatives of extremal value functions and optimality conditions for semi infinite programs second order differential properties of an extremal value function v are studied it is shown that under certain assumptions in particular the strong second order sufficiency conditions v is locally expressible as the maximum of a finite number of c2 functions after removing the assumption of strong sufficiency lower and upper bounds on the second order local behaviour of v are proposed the obtained results are applied to give second order optimally conditions for a semi infinite program

conjugate points for variational problems with equality and inequality state constraints in this paper variational problems with equality and inequality state constraints are considered the theory of conjugate points for these problems is developed and necessary conditions for weak local optimality are derived in terms of this concept and the legendre condition for the case of inequality constraints the envelope like effect is taken into consideration in the accessory problem

on the decomposition of binary shapes into meaningful parts abstract a novel decomposition scheme for binary shapes is proposed based on a newly developed notion of maximal approximately convex subpart macs macss are first derived from concepts of integral geometry and it is argued that macss are the simplest possible constituents of a decomposition an algorithm is developed that utilizes the notion of macss to achieve a decomposition into possibly overlapping parts the developed algorithm is based on concepts of mathematical morphology which have been extended by an elongation and a fattening operation the validity of the chosen approach is shown by applying it to several examples synthetic as well as real objects thus corroborating as well the validity of the initial concept of macss

global feedback stabilization of new class of bilinear systems abstract in this paper we consider some new classes of bilinear systems we give necessary and sufficient conditions for the asymptotic stabilization by using homogeneous feedbacks

uniting local and global output feedback controllers we consider control systems for which we know two stabilizing output feedback controllers one is globally asymptotically stabilizing while the other one is only locally asymptotically stabilizing we look for a composite output feedback control law that is equal to the local feedback on a neighborhood of the origin and that is globally asymptotically stabilizing since we want some robustness with respect to measurement noise actuator errors and external disturbances we need to consider hybrid output feedback controllers under an input output to state stability assumption we exhibit a solution of this uniting problem by means of a dynamic hybrid output feedback controller then we particularize our study to linear control systems with saturation at the input for which we know two stabilizing output feedback controllers one is a nonlinear globally asymptotically stabilizing controller while the other one is a high performance linear only locally asymptotically stabilizing controller we specify numerically tractable conditions to solve this uniting problem finally we illustrate our main results by means of numerical examples

invariance principles for hybrid systems with connections to detectability and asymptotic stability this paper shows several versions of the lasalle s invariance principle for general hybrid systems the broad framework allows for nonuniqueness of solutions zeno behaviors and does not insist on continuous dependence of solutions on initial conditions instead only a mild structural property involving graphical convergence of solutions is posed the general invariance results are then specified to hybrid systems given by set valued data further results involving invariance as well as observability detectability and asymptotic stability are given

brief output feedback stabilization of bilinear systems using dead beat observers an output feedback stabilizing controller for bilinear systems is proposed in this paper key techniques in the proposed method are periodic switching of the controller and the use of a dead beat observer the controller is proved to globally stabilize the plant both in noiseless and noisy conditions the only condition on the plant for the proposed method to be applicable is that the linear part of it is observable no condition on nonlinear observability is required

no gap second order optimality conditions for optimal control problems with a single state constraint and control the paper deals with optimal control problems with only one control variable and one state constraint of arbitrary order we consider the case of finitely many boundary arcs and touch times we obtain a no gap theory of second order conditions allowing to characterize second order quadratic growth

a generalized morphological skeleton transform using both internal and external skeleton points the morphological skeleton transform mst is a leading morphological shape representation scheme in the mst a given shape is represented as the union of all the maximal disks contained in the shape the concept of external skeleton points and external maximal disks has been used for shape description and characterization purposes in this paper we develop a generalized morphological skeleton transform that combines the concepts of internal and external maximal disks into a unified framework in this framework a shape is described in terms of disk components that need to be added as well as disk components that need to be removed the procedures and formulae describing the extraction of the disk components and the reconstruction of the original shape from these components are developed the correctness of the procedures and formulae is established this new framework seems to provide a more powerful and more natural way of modeling the approximation and reconstruction of binary shapes using primitive shape components

efficient morphological shape representation mathematical morphology is well suited to capturing geometric information hence morphology based approaches have been popular for object shape representation the two primary morphology based approaches the morphological skeleton and the morphological shape decomposition msd each represent an object as a collection of disjoint sets a practical shape representation scheme though should give a representation that is computationally efficient to use unfortunately little work has been done for the morphological skeleton and the msd to address efficiency we propose a flexible search based shape representation scheme that typically gives more efficient representations than the morphological skeleton and msd our method decomposes an object into a number of simple components based on homothetics of a set of structuring elements to form the representation the components are combined using set union and set difference operations we use three constituent component types and a thorough cost based search strategy to find efficient representations we also consider allowing object representation error which may yield even more efficient representations

second order necessary optimality conditions for a class of semilinear elliptic optimal control problems with mixed pointwise constraints this paper deals with first and second order necessary optimality conditions for a class of optimal control problems governed by semilinear elliptic partial differential equations with mixed pointwise state control constraints

efficient morphological shape representation with overlapping disk components this paper proposes a new morphological shape representation algorithm in which a two dimensional 2 d binary shape is represented as a union of certain disks contained in the given shape the representative disks of different sizes may overlap but excessive overlapping between them is avoided the algorithm combines the advantages of the morphological skeleton transform mst and the morphological shape decomposition msd the representative disks have simple and well defined mathematical characterizations the algorithm is simple and efficient to implement the experimental results show that the number of representative disks used by our algorithm is significant lower than that used by the msd the overlapping level between the representative disks is much lower than that of the mst a simple procedure can be used to combine the representative disks into more meaningful shape components these shape components seem to correspond better to the natural shape parts than those generated by the msd it is also possible to build a good approximation for a given shape using only a small number of major components

brief paper state feedback design for input saturating quadratic systems this paper proposes a method to design stabilizing state feedback control laws for nonlinear quadratic systems subject to input saturation based on a quadratic lyapunov function a modified sector condition and a particular representation for the quadratic terms synthesis conditions in a quasi lmi form are stated in a regional local context an lmi based optimization problem is then derived for computing the state feedback gains maximizing the estimate of the stability region of the closed loop system

optimum problems with nonsmooth equality constraints abstract in this paper first and higher order necessary conditions for constrained extremum problems are obtained under weak regularity assumptions concerning the functions describing the equality as well as the inequality conditions of the problem

the upper and lower second order directional derivatives of a sup type function the purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup type functions x sup f x t t t in terms of the first and second derivatives off x t wheret is a compact set in a metric space and we assume thatf f x and 2f x2 are continuous on n t we will give a geometrical meaning of the formula we will moreover give a sufficient condition fors x to be directionally twice differentiable

optimum problems with certain lower semicontinuous set valued constraints in this paper we provide a complete analysis of second order admissible variations to inequality type constraints that are given in terms of lower semicontinuous set valued functions whose images are closed convex sets with nonempty interior as an application we consider optimization problems where such constraints are present and we deduce second order necessary conditions for optimality

legendre type optimality conditions for a variational problem with inequality state constraints they differ from the legendre clebsch condition they give information about the hesse matrix of the integrand at not only inactive points but also active points on the other hand since the inequality state constraints can be regarded as an infinite number of inequality constraints they sometimes form an envelope according to a general theory 9 one has to take the envelope into consideration when one consider second order necessary optimality conditions for an abstract optimization problem with a generalized inequality constraint however we show that we do not need to take it into account when we consider legendre type conditions finally we show that any inequality state constraint forms envelopes except two trivial cases we prove it by presenting an envelope in a visible form

variational analysis of a composite function a formula for the lower second order epi derivative an exact formula is established for the lower second order epi derivative of a function of the form g f x where f is a smooth map from one banach space into another and g is a convex function generally not everywhere finite unconstrained minimization of such functions typically arise as an equivalent in one or another sense reduction form for many important classes of constrained optimization problems the formula is further applied to study epi differentiability of the max function x max q x q q

stability analysis of bilinear systems under aperiodic sampled data control this note considers the problem of local stability of bilinear systems with aperiodic sampled data linear state feedback control the sampling intervals are time varying and upper bounded it is shown that the feasibility of some linear matrix inequalities lmis implies the local asymptotic stability of the sampled data system in an ellipsoidal region containing the equilibrium the method is based on the analysis of contractive invariant sets and it is inspired by the dissipativity theory the results are illustrated by means of numerical examples

global asymptotic stabilization for a class of bilinear systems by hybrid output feedback this technical note deals with the global asymptotic stabilization problem for a class of bilinear systems a state feedback controller solving this problem is obtained uniting a local controller having an interesting behavior in a neighborhood of the origin and a constant controller valid outside this neighborhood the approach developed is based on the use of a hybrid loop and more precisely a hybrid state feedback this result is extended to the case where the state of the plant is not fully available and only the measured output can be used for control purposes in this case a dynamical controller constituted by an observer and a state feedback is built by means of hybrid output feedback framework in both cases the conditions are expressed by a set of linear matrix inequalities

high order approximations and generalized necessary conditions for optimality in this paper we derive generalized necessary conditions for optimality for an optimization problem with equality and inequality constraints in a banach space the equality constraints are given in operator form as q x in x f x 0 where f x rightarrow y is an operator between banach spaces the inequality constraints are given by smooth functionals or by closed convex sets models of this type are common in the optimal control problem the paper addresses the case when the frechet derivative f x is not onto and hence the classical lyusternik theorem does not apply to describe the tangent space to q in this case the classical euler lagrange type necessary conditions are trivially satisfied generating abnormal cases a high order generalization of the lyusternik theorem derived earlier u ledzewicz and h schattler nonlinear anal 34 1998 pp 793 815 is used to calculate high order tangent cones to the equality constraint at points x in q where f x is not onto combining these with high order approximating cones related to the other constraints of the problem feasible cones respectively cones of decrease a high order generalization of the dubovitskii milyutin theorem is given and then applied to derive generalized necessary conditions for optimality these conditions reduce to classical conditions for normal cases but they give new and nontrivial conditions for abnormal cases

second order necessary optimality conditions for problems without a priori normality assumptions in this article we derive second order necessary conditions of optimality for an abstract optimization problem with equality and inequality constraints and constraints in the form of an inclusion into a given closed set an important feature is that our optimality conditions dispense with any a priori normality assumptions such as robinsons constraint qualification and remain informative even for abnormal points moreover our optimality conditions take into account the second order effect of the curvature of the set in the inclusion constraints

efficient morphological shape representation by varying overlapping levels among representative disks abstract the morphological skeleton transform the morphological shape decomposition and the overlapped morphological shape decomposition are three basic morphological shape representation schemes in this paper we propose a new way of generalizing these basic representation algorithms to improve representational efficiency in all three basic algorithms a fixed overlapping policy is used to control the overlapping relationships among representative disks of different sizes in our new algorithm different overlapping policies are used to generate shape components that have different overlapping relationships among themselves the overlapping policy is selected dynamically according to local shape features experiments show that compared to the three basic algorithms our algorithm produces more efficient representations with lower numbers of representative points

optimal control of a parabolic equation with time dependent state constraints in this paper we study the optimal control problem of the heat equation with a cubic nonlinearity by a distributed control over a subset of the domain in the presence of a state constraint the latter is integral over the space and has to be satisfied each time using for the first time the technique of alternative optimality systems in the context of optimal control of partial differential equations we show that both the control and multiplier are continuous in time under some natural geometric hypotheses we can prove that extended polyhedricity holds allowing us to obtain no gap second order optimality conditions that characterize quadratic growth an expansion of the value function and of approximate solutions can be computed for a directional perturbation of the right hand side of the state equation

morphological decomposition of 2 d binary shapes into convex polygons a heuristic algorithm in many morphological shape decomposition algorithms either a shape can only be decomposed into shape components of extremely simple forms or a time consuming search process is employed to determine a decomposition in this paper we present a morphological shape decomposition algorithm that decomposes a two dimensional 2 d binary shape into a collection of convex polygonal components a single convex polygonal approximation for a given image is first identified this first component is determined incrementally by selecting a sequence of basic shape primitives these shape primitives are chosen based on shape information extracted from the given shape at different scale levels additional shape components are identified recursively from the difference image between the given image and the first component simple operations are used to repair certain concavities caused by the set difference operation the resulting hierarchical structure provides descriptions for the given shape at different detail levels the experiments show that the decomposition results produced by the algorithm seem to be in good agreement with the natural structures of the given shapes the computational cost of the algorithm is significantly lower than that of an earlier search based convex decomposition algorithm compared to nonconvex decomposition algorithms our algorithm allows accurate approximations for the given shapes at low coding costs

optimum problems with measurable set valued constraints in this paper we provide a complete analysis of second order admissible variations to inequality type constraints which are given in terms of measurable set valued functions whose images are closed convex sets with nonempty interior as an application we consider optimization problems where such constraints are present and we deduce second order necessary conditions for optimality

exponentially stabilizing division controllers for dyadic bilinear systems it is difficult to asymptotically stabilize a dyadic bilinear system with only multiplicative control inputs when the open loop dynamics are unstable the previous approach of cascading a division controller with a constant size dead zone can only stabilize but not asymptotically stabilize the system this note proposes a new control design which cascades a division controller with a modified dead zone whose size is proportional to the modulus of the system state it is shown that this new division controller can globally and exponentially stabilize any open loop unstable dyadic bilinear system as long as it is controllable

comparison between the morphological skeleton and morphological shape decomposition the morphological skeleton and morphological shape decomposition msd are two popular approaches for morphological shape representation each method represents an object as an algebraic combination of a number of components where each component is given by a locus of points dilated by a specified structuring element homothetic this correspondence develops a theoretical comparison between the two methods combining the theoretical results with several representation cost measures we make a concrete comparison of the efficiency of the two methods the results indicate that for complex objects i e objects requiring a full range of homothetic sizes in the morphological skeleton representation the msd represents objects more efficiently than the morphological skeleton for three of four suggested cost measures

optimality conditions in mathematical programming and composite optimization new second order optimality conditions for mathematical programming problems and for the minimization of composite functions are presented they are derived from a general second order fermat s rule for the minimization of a function over an arbitrary subset of a banach space the necessary conditions are more accurate than the recent results of kawasaki 1988 and cominetti 1989 but more importantly in the finite dimensional case they are twinned with sufficient conditions which differ by the replacement of an inequality by a strict inequality we point out the equivalence of the mathematical programming problem with the problem of minimizing a composite function our conditions are especially important when one deals with functional constraints when the cone defining the constraints is polyhedral we recover the classical conditions of ben tal zowe 1982 and cominetti 1990

nonsmooth optimum problems with constraints this paper develops second order necessary conditions for nonsmooth infinite dimensional optimization problems with banach space valued equality and real valued inequality constraints another constraint in the form of a closed convex set is also present the objective function is the maximum over a parameter of functions f t z that are lipschitz in z and upper semicontinuous in t the inequality constraints g s z depend on a parameter s the technique we use is a generalization of that of dubovitskii and milyutin the second order conditions obtained here are in terms of a certain function sigma that disappears when the parameters and a certain set that derives from the given convex set are absent the presence of the function sigma and that set is due to the envelope like effect discovered by kawasaki

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

second order necessary conditions in semismooth optimization first and second order conditions are given which are necessary for a functionf to have a local minimal value atx inrn it is assumed thatf is locally lipschitzian nearx and semismooth atx the necessary conditions are expressed in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives the method of proof bears no resemblance to standard methods three special cases are discussed here but applications to constrained problems are made elsewhere

second order necessary optimality conditions for minimizing a sup type function the second derivative of an envelope cannot be expressed only by second derivatives of the constituent functions by taking account of this fact we derive new second order necessary optimality conditions for minimization of a sup type function the conditions involve an extra term besides the second derivative of the lagrange function furthermore we will comment on the relationship between the extra term and a kind of second order directional derivative of the sup type function

morphological shape representation abstract a new approach for shape representation is described which provides a general scheme for object description and unifies some of the existing representation techniques the proposed method has the advantage that it is based on the use of simple geometric objects which are intuitively used by humans in their perception of shapes furthermore its unifying framework can be used both in computer graphics and computer vision thus providing a tool to close the existing gap between object modeling and object recognition in certain applications