optimal control of evolution inclusions abstract we develop the method of discrete approximations for studying optimal control problems of the bolza type for evolution systems governed by generally nonconvex differential inclusions in infinite dimensional spaces the results obtained involve well posedness stability of discrete approximations necessary optimality conditions for general discrete time systems and necessary optimality conditions for continuous time inclusions established in the extended euler lagrange form our approach is based on the generalized differential constructions of variational analysis introduced earlier by the author

regularity of the hamiltonian along optimal trajectories this paper concerns state constrained optimal control problems in which the dynamic constraint takes the form of a differential inclusion if the differential inclusion does not depend on time then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory is independent of time if the differential inclusion is lipschitz continuous then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory is lipschitz continuous these two well known results are examples of the following principle the hamiltonian evaluated along the optimal state trajectory and the co state trajectory inherits the regularity properties of the differential inclusion regarding its time dependence we show that this principle also applies to another kind of regularity if the differential inclusion has bounded variation w r t time then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory has bounded variation two application

necessary conditions for optimal control problems with state constraints necessary conditions of optimality are derived for optimal control problems with pathwise state constraints in which the dynamic constraint is modelled as a differential inclusion the novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid an euler lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded nonconvex sets and satisfies a mild one sided lipschitz continuity hypothesis we recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by ioffe for problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one sided lipschitz hypothesis is replaced by a milder local hypothesis a recent dualization theorem permits us to infer a strengthened form of the hamiltonian inclusion from the euler lagrange condition the necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by loewen and rockafellar to achieve related necessary conditions for state constrained problems and improve on available results in certain respects even when specialized to the state constraint free case proofs make use of recent decoupling ideas of the authors which reduce the optimization problem to one to which pontryagin s maximum principle is applicable and a refined penalization technique to deal with the dynamic constraint

scale free animal movement patterns levy walks outperform fractional brownian motions and fractional levy motions in random search scenarios the movement patterns of a diverse range of animals have scale free characteristics these characteristics provide necessary but not sufficient conditions for the presence of movement patterns that can be approximated by levy walks nevertheless it has been widely assumed that the occurrence of scale free animal movements can indeed be attributed to the presence of levy walks this is in part because it is known that the super diffusive properties of levy walks can be advantageous in random search scenarios when searchers have little or no prior knowledge of target locations however fractional brownian motions fbms and fractional levy motions flms are both scale free and super diffusive and so it is possible that these motions rather than levy walks underlie some or all occurrences of scale free animal movement patterns here this possibility is examined in numerical simulations through a determination of the searching efficiencies of fbm and flm searches it is shown that these searches are less efficient than levy walk searches this finding does not rule out the possibility that some animals with scale free movement patterns are executing fbm and flm searches but it does make levy walk searches the more likely possibility

dynamic programming for nonlinear systems driven by ordinaryand impulsive controls a dynamic programming approach is considered for a class of minimum problems with impulses the minimization domain consists of trajectories satisfying an ordinary differential equation whose right hand side depends not only on a measurable control v but also on a second control u and on its time derivative dot u for this reason the control u and the differential equation are called impulsive the value function of the considered minimum problem turns out to depend on the time the state the u variable and the variation allowed to the impulsive control it is shown that the value function satisfies in a generalized sense a dynamic programming equation dpe which is obtained from a dynamic programming principle involving space time trajectories moreover the value function is the unique map solving equation dpe satisfying either an inequality condition or a supersolution condition at each point of the boundary incidentally this extends a result by barron jensen and menaldi nonlinear anal 21 1993 pp 241 268 where the impulsive control is scalar monotone and the corresponding vector field is independent of the state variable next a maximum principle is proved and the well known relationship between adjoint variables and value function is suitably extended to impulsive control systems a fully elaborated example concludes the paper

solutions to hybrid inclusions via set and graphical convergence with stability theory applications motivated by questions in stability theory for hybrid dynamical systems we establish some fundamental properties of the set of solutions to such systems using the notion of a hybrid time domain and general results on set and graphical convergence we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and upper semicontinuous with respect to initial conditions and system perturbations the general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets these results parallel what is already known for differential inclusions and difference inclusions for example the basin of attraction for a compact attractor is relatively open the attractivity is uniform from compact subsets of the basin of attraction and asymptotic stability is robust with respect to small perturbations

optimality of affine control system of several species in competition on a sequential batch reactor in this paper we analyze the optimalty of affine control system of several species in competition for a single substrate on a sequential batch reactors sbr with the objective being to reach a given low level of the substrate we allow controls to be bounded measurable functions of time plus possible impulses a suitable modification of the dynamics leads to a slightly different optimal control problem without impulsive controls for which we apply different optimality conditions de rived from pontryagin principle and the hamilton jacobi bellman equation we thus characterize the singular trajectories of our problem as the extremal trajecto ries keeping the substrate at a constant level we also establish conditions for which a immediate one impulse ioi strategy is optimal some numerical experiences are then included in order to illustrate our study and show that those conditions are also necessary to ensure the optimality of the ioi strategy

higher order runge kutta methods for impulsive differential systems abstract this paper studies higher order approximations of solutions of differential equations with non fixed times of impulses we assume that the right hand side is sufficiently smooth using a runge kutta method of higher order and natural assumptions on the impulsive surfaces and the impulses we calculate good approximations of the jump times which enables us to extend the classical results for higher order of convergence of runge kutta methods to more complicated systems

discrete approximations and refined euler lagrange conditions fornonconvex differential inclusions this paper deals with the bolza problem p for differential inclusions subject to general endpoint constraints we pursue a twofold goal first we develop a finite difference method for studying p and construct a discrete approximation to p that ensures a strong convergence of optimal solutions second we use this direct method to obtain necessary optimality conditions in a refined euler lagrange form without standard convexity assumptions in general we prove necessary conditions for the so called intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima in the case of a mayer cost functional or boundary solutions to differential inclusions this euler lagrange form holds without any relaxation the results obtained are expressed in terms of nonconvex valued generalized differentiation constructions for nonsmooth mappings and sets

necessary conditions and non existence results for autonomous nonconvex variational problems abstract classical one dimensional autonomous lagrange problems are considered in absence of any smoothness convexity or coercivity condition on the energy density we prove a dubois reymond type necessary condition expressed as a differential inclusion involving the subdifferential of convex analysis as a consequence a non existence result is obtained

necessary conditions for optimal impulsive control problems necessary conditions of optimality in the form of a maximum principle are derived for a class of optimal control problems certain of whose controls are represented by measures and whose state trajectories are functions of bounded variation state trajectories are interpreted as robust solutions of the dynamic equations a concept of solutions which takes account of the interaction between the measure control and the state variables during the jumps the maximum principle which is derived improves on earlier optimality conditions for problems of this nature by allowing nonsmooth data measurable time dependence and a possibly time varying constraint set for the conventional controls

levy flights in random searches we review the general search problem of how to find randomly located objects that can only be detected in the limited vicinity of a forager and discuss its quantitative description using the theory of random walks we illustrate levy flight foraging by comparison to brownian random walks and discuss experimental observations of levy flights in biological foraging we review recent findings suggesting that an inverse square probability density distribution p l l 2 of step lengths l can lead to optimal searches finally we survey the explanations put forth to account for these unexpected findings

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

on the extreme flights of one sided levy processes abstract we explore the statistical behavior of the order statistics of the flights of one sided levy processes olps we begin with the study of the extreme flights of general olps and then focus on the class of selfsimilar processes investigating the following issues i the inner hierarchy of the extreme flights for example how big is the 7th largest flight relative to the 2nd largest one and ii the relative contribution of the extreme flights to the entire flight aggregate for example how big is the 3rd largest flight relative to the olp s value furthermore we show that all hierarchical results obtained but not the aggregate results are explicitly extendable to the class of olps with arbitrary power law flight tails which is far larger than the selfsimilar class

discrete approximations and fixed set iterations in banach spaces we study autonomous differential inclusions with right hand sides satisfying a one sided lipschitz osl condition in banach spaces with uniformly convex duals we first show that the solution set is closed and obtain estimates for euler type discrete approximations we then use these results to derive an analogue of the exponential formula for the reachable set as well as results regarding the existence and approximation of a strongly invariant attractor in the case of a negative osl constant as a by product conditions for controllability of the reverse time system are obtained

existence stability and optimality for optimal control problems governed by maximal monotone operators abstract we study optimal control problems governed by maximal monotone differential inclusions with mixed control state constraints in infinite dimensional spaces we obtain some existence results for this kind of dynamics and construct the discrete approximations that allows us to strongly approximate optimal solutions of the continuous type optimal control problems by their discrete counterparts our approach allows us to apply our results for a wide class of mappings that are applicable in mechanics and material sciences

levy flights search patterns of biological organisms we discuss recent findings suggesting that an inverse square probability density distribution p l l 2 of step lengths l leads to an optimal random search strategy for organisms that can search efficiently for randomly located objects that can only be detected in the limited vicinity of the searcher and can be revisited any number of times we explore the extent to which these findings may be dependent on the dimensionality of the search space and the presence of short range correlations in the step lengths and directions

lack of bv bounds for impulsive control systems abstract for a nonlinear impulsive control system we extend the so called graph completion approach and introduce a notion of generalized solution x associated to a control u whose total variation is bounded on 0 t for every t t but possibly unbounded on 0 t we prove existence consistency with classical solutions and well posedness of this solution in particular we characterize it as a pointwise limit of certain regular solutions the notion that we consider provides the natural setting for controllability questions and for some non coercive optimal control problems where chattering phenomena at the final time are expected more in general it is well suited to describe the evolution of control systems subject to a train of impulses where no a priori bounds on the number and the amplitude of the impulses are imposed

a rigorous solution concept for geodesic and geodesic deviation equations in impulsive gravitational waves the geodesic as well as the geodesic deviation equation for impulsive gravitational waves involve highly singular products of distributions 2 2 a solution concept for these equations based on embedding the distributional metric into the colombeau algebra of generalized functions is presented using a universal regularization procedure we prove existence and uniqueness results and calculate the distributional limits of these solutions explicitly the obtained limits are regularization independent and display the physically expected behavior

l1 limit solutions for control systems abstract for a control cauchy problem x f t x u v 1 m g x u x a x on an interval a b we propose a notion of limit solution x verifying the following properties i x is defined for l 1 impulsive inputs u and for standard bounded measurable controls v ii in the commutative case i e when g g 0 for all 1 m x coincides with the solution one can obtain via a change of coordinates that makes the g simultaneously constant iii x subsumes former concepts of solution valid for the generic noncommutative case in particular when u has bounded variation we investigate the relation between limit solutions and single valued graph completion solutions furthermore we prove consistency with the classical caratheodory solution when u and x are absolutely continuous even though some specific problems are better addressed by means of special representations of the solutions we believe that various theoretical issues call for a unified notion of trajectory for instance this is the case of optimal control problems possibly with state and endpoint constraints for which no extra assumptions like e g coercivity bounded variation commutativity are made in advance

kernel independent component analysis we present a class of algorithms for independent component analysis ica which use contrast functions based on canonical correlations in a reproducing kernel hilbert space on the one hand we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence on the other hand building on recent developments in kernel methods we show that these criteria and their derivatives can be computed efficiently minimizing these criteria leads to flexible and robust algorithms for ica we illustrate with simulations involving a wide variety of source distributions showing that our algorithms outperform many of the presently known algorithms

necessary conditions for multiobjective optimal control problems with free end time necessary conditions of optimality are derived for multiobjective optimal control problems with free end time in which the dynamics constraint is modeled as a nonconvex differential inclusion the obtained results cover some previous results on necessary conditions for multiojective and single objective optimal control problems

optimal control of the sweeping process over polyhedral controlled sets abstract the paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given bolza type functional which depends on control and state variables as well as their velocities besides the highly non lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types all of this creates serious challenges for deriving necessary optimality conditions we develop here the method of discrete approximations and combine it with advanced tools of first order and second order variational analysis and generalized differentiation this approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions as a by product of the developed approach we prove the strong w 1 2 convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous time system and derive necessary optimality conditions for the discrete counterparts the established necessary optimality conditions for the sweeping process are illustrated by several examples

structured kernel dictionary learning with correlation constraint for object recognition in this paper we propose a new discriminative non linear dictionary learning approach called correlation constrained structured kernel ksvd for object recognition the objective function for dictionary learning contains a reconstructive term and a discriminative term in the reconstructive term signals are implicitly non linearly mapped into a space where a structured kernel dictionary each sub dictionary of which lies in the span of the mapped signals from the corresponding class is established in the discriminative term by analyzing the classification mechanism the correlation constraint is proposed in kernel form constraining the correlations between different discriminative codes and restricting the coefficient vectors to be transformed into a feature space where the features are highly correlated inner class and nearly independent between classes the objective function is optimized by the proposed structured kernel ksvd during the classification stage the specific form of the discriminative feature is needless to be known while the inner product of the discriminative feature with kernel matrix embedded is available and is suitable for a linear svm classifier experimental results demonstrate that the proposed approach outperforms many state of the art dictionary learning approaches for face scene and synthetic aperture radar vehicle target recognition

a maximum principle for optimal processes with discontinuous trajectories a maximum principle is proved which governs solutions to dynamic optimization problems in which the controls driving the system may be impulsive and give rise to discontinuous trajectories the approach which involves approximating the problem by a conventional one and using ekeland s theorem is new it permits us to weaken very considerably the hypotheses under which maximum principles for such problems have previously been proved

geodesics and geodesic deviation for impulsive gravitational waves the geometry of impulsive pp waves is explored via the analysis of the geodesic and geodesic deviation equation using the distributional form of the metric the geodesic equation involves formally ill defined products of distributions due to the nonlinearity of the equations and the presence of the dirac distribution in the space time metric thus strictly speaking it cannot be treated within schwartz s linear theory of distributions to cope with this difficulty we proceed by first regularizing the singularity then solving the regularized equation within classical smooth functions and finally obtaining a distributional limit as solution to the original problem furthermore it is shown that this limit is independent of the regularization without requiring any additional condition thereby confirming earlier results in a mathematically rigorous fashion we also treat the jacobi equation which despite being linear in the deviation vector field involves even more delicate singular expressions like the square of the dirac distribution again the same regularization procedure provides us with a perfectly well behaved smooth regularization and a regularization independent distributional limit hence it is concluded that the geometry of impulsive pp waves can be described consistently using distributions as long as careful regularization procedures are used to handle the ill defined products

d dimensional levy flights exact and asymptotic the analytic and asymptotic properties of the spherically symmetric d dimensional levy stable probability density function p d r are discussed in detail these isotropic stable probability density functions pdfs are analogous to the one dimensional symmetric levy stable pdfs previously studied by the present authors j math phys 43 2670 2002 we construct a hypergeometric representation of p d r when is rational and find a number of new representations of p d r in terms of special functions for various values of d and a recursion relation is found between p d r and p d 2 r which in particular implies there exists a simple map between p 1 r and p 3 r as in our previous paper we discuss the properties of p d r for both the cases 2 and 2 we demonstrate the existence of intricate exponentially small series in the large r asymptotics of p d r when is an integer which are dominant when is even we explicitly construct this beyond all orders expansion of p d r for arbitr

random strategies of contact tracking abstract one of several critical issues in the development of optimal disease containment and eradication strategies is the knowledge of underlying contacts between individuals here we employ random search strategies to identify all possible links representing direct or indirect interactions between individuals building up the system in order to recognize all contacts the searcher performs symmetric levy flights onto the accessible area we investigate the influence of local and non local information the exponent characterizing asymptotic behavior of levy flights boundary conditions density of links and type of a search strategy on the efficiency of the search process monte carlo examination of the suggested model reveals that the efficiency of the search process is sensitive to the type of boundary conditions depending on the assumed type of boundary conditions efficiency of the search process can be a monotonic or non monotonic function of the exponents characterizing asymptotic behavior of levy flights consequently among the whole spectrum of exponents characterizing the power law behavior of jumps length there exist distinguished values of stability index representing the most efficient search processes these exponents correspond to extreme minimal or maximal or intermediate values of stability index associated with gaussian maximally heavy tailed or cauchy like strategies respectively

minimal time sequential batch reactors with bounded and impulse controls for one or more species we consider the optimal control problem of feeding in minimal time a tank where several species compete for a single resource with the objective being to reach a given level of the resource we allow controls to be bounded measurable functions of time plus possible impulses for the one species case we show that the immediate one impulse strategy filling the whole reactor with one single impulse at the initial time is optimal when the growth function is monotonic for nonmonotonic growth functions with one maximum we show that a particular singular arc strategy precisely defined in section 3 is optimal these results extend and improve former ones obtained for the class of measurable controls only for the two species case with monotonic growth functions we give conditions under which the immediate one impulse strategy is optimal we also give optimality conditions for the singular arc strategy at a level that depends on the initial condition to be optimal the possibility for the immediate one impulse strategy to be nonoptimal while both growth functions are monotonic is a surprising result and is illustrated with the help of numerical simulations

existence theorems for optimal control and calculus of variations problems where the states can jump we examine optimal control and calculus of variations problems where the states can be discontinuous in the control theory setting this is caused by allowing impulse controls conditions under which optimal solutions will exist for these problems are determined after we derive suitable objective functionals that can handle the jump terms

a sampling method and approximation results for impulsive systems this paper studies an impulsive dynamical system that is in the form of a measure driven differential inclusion the employed solution concept depends upon a graph completion of the measure the first main result shows that a subsequence of discrete time trajectories graph converges to a solution and under lipschitz hypotheses that every solution can be obtained in this manner the second main result pursues a similar development with classical systems that approximate the given one

measure driven differential inclusions measure driven differential inclusions arise when we attempt to derive necessary conditions of optimality for optimal impulsive control problems with nonsmooth data we introduce the concept of a robust solution to a measure driven inclusion which extends to a multifunction setting interpretations of solutions to measure driven differential equations provided by dal maso and rampazzo and others closure properties of sets of robust solutions are established and notions of relaxation investigated implications for optimality conditions for impulsive control problems are pursued in a companion paper

hamilton jacobi bellman equation and feedback synthesis for impulsive control there is an increasing number of applications whose trajectories are better modeled by discontinuous or impulsive trajectories thus we explore optimality conditions for impulsive control system expressed in terms of an hamilton jacobi bellman equation we use a measure driven differential inclusion to model the impulsive behavior since it provides a formal framework in which the control space is complete additionally we use the impulsive euler solution and invariance results to derive feedback optimal control synthesis