controllability and strong controllability of differential inclusions abstract in this paper we prove sufficient conditions for controllability and strong controllability in terms of the mordukhovich subdifferential for two classes of differential inclusions the first one is the class of sub lipschitz multivalued functions introduced by loewen rockafellar 1994 10 the second one introduced recently by clarke 2005 18 is the class of multivalued functions which are pseudo lipschitz and satisfy the so called tempered growth condition to do this we establish an error bound result in terms of the mordukhovich subdifferential outside asplund spaces

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

the extended euler lagrange condition for nonconvex variational problems this paper provides necessary conditions of optimality for a general variational problem for which the dynamic constraint is a differential inclusion with a possibly nonconvex right side they take the form of an euler lagrange inclusion involving convexification in only one coordinate supplemented by the transversality and weierstrass conditions it is also shown that for time invariant free time problems the adjoint arc can be chosen so that the hamiltonian function is constant along the minimizing state arc the methods used here based on simple finite dimensional nonsmooth calculus clarke decoupling and a rudimentary version of the maximum principle offer an alternative and somewhat simpler derivation of such results to those used by ioffe and rockafellar in concurrent research

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

the hamiltonian inclusion for nonconvex velocity sets since clarke s 1973 proof of the hamiltonian inclusion for optimal control problems with convex velocity sets there has been speculation and more recently speculation relating to a stronger partially convexified version of the hamiltonian inclusion as to whether these necessary conditions are valid in the absence of the convexity hypothesis the issue was in part resolved by clarke himself when in 2005 he showed that l infty local minimizers satisfy the hamiltonian inclusion in this paper it is shown by counterexample that the hamiltonian inclusion and so also the stronger partially convexified hamiltonian inclusion are not in general valid for nonconvex velocity sets when the local minimizer in question is merely a w 1 1 local minimizer not an l infty local minimizer the counterexample demonstrates that the need to consider l infty local minimizers not w 1 1 local minimizers in the proof of the hamiltonian inclusion for nonconvex velocity sets is fundamental not j

periodic solutions for p laplacian neutral rayleigh equation with a deviating argument abstract by using topological degree theory and some analytical skill some criteria to guarantee the existence of periodic solutions are derived for p laplacian neutral rayleigh equation with a deviating argument of the following form p x t c x t f x t g x t t e t

some new results on the existence of periodic solutions to a kind of rayleigh equation with a deviating argument abstract by employing the continuation theorem of coincidence degree theory we study a kind of rayleigh equation with a deviating argument as follows x t f x t g x t t p t some new results on the existence of periodic solutions are obtained

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

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

discrete approximations relaxation and optimization of one sided lipschitzian differential inclusions in hilbert spaces we study discrete approximations of nonconvex differential inclusions in hilbert spaces and dynamic optimization optimal control problems involving such differential inclusions and their discrete approximations the underlying feature of the problems under consideration is a modified one sided lipschitz condition imposed on the right hand side i e on the velocity sets of the differential inclusion which is a significant improvement of the conventional lipschitz continuity our main attention is paid to establishing efficient conditions that ensure the strong approximation in the w1 p norm as p 1 of feasible trajectories for the one sided lipschitzian differential inclusions under consideration by those for their discrete approximations and also the strong convergence of optimal solutions to the corresponding dynamic optimization problems under discrete approximations to proceed with the latter issue we derive a new extension of the bogolyubov type relaxation density theorem to the case of differential inclusions satisfying the modified one sided lipschitzian condition all the results obtained are new not only in the infinite dimensional hilbert space framework but also in finite dimensional spaces

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

remarks on the paper controllability of second order differential inclusion in banach spaces j math anal appl 285 2003 537 550 abstract an error is pointed out in the paper j r kang y c kwun j y park controllability of second order differential inclusion in banach spaces j math anal appl 285 2003 537 550 by an additional condition the considered system is controllable

controllability of the second order differential inclusion in banach spaces abstract the purpose of this paper is to study the controllability for the second order differential inclusion in banach spaces we rely on a fixed point theorem for condensing maps due to martelli we consider the damping term x and find a control u such that the solution satisfies x t x1 and x t y1

periodic solutions for a kind of rayleigh equation with a deviating argument existence of periodic solutions for a kind of rayleigh equation with a deviating argument x t f x t g x email protected t p t is studied and some new results are obtained our work generalizes and improves the known results in the literature

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

dualization of the euler and hamiltonian inclusions optimal control emerged as a distinct eld of research in the late 1950s with the derivation of the pontryagin maximum principle a fundamental set of necessary conditions on minimizing arcs which takes account of di erential constraints the pontryagin maximum principle directly generates the euler equation and other classical necessary conditions from the calculus of variations when specialised to problems without dynamic constraints however deeper connections between necessary conditions in optimal control and the calculus of variations were only subsequently revealed when clarke developed new analytic machinery now referred to as nonsmooth analysis a term introduced by clarke himself for extending classical necessary conditions to apply to the generalized

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

periodic solutions to neutral differential equation with deviating arguments by employing the continuation theorem of coincidence degree theory and some analysis techniques we study a kind of neutral differential equation with deviating arguments as follows x t cx t r f x t g x t t p t some new results on the existence of periodic solutions are obtained our result generalizes main work of wang app math lett 12 1999 41 44

on existence of periodic solutions of a kind of rayleigh equation with a deviating argument abstract existence of periodic solutions for a kind of rayleigh equation with a deviating argument x t f x t g t x t t p t is studied and some new results are obtained our work generalizes and improves the known results in the literature

variational analysis of evolution inclusions the paper is devoted to optimization problems of the bolza and mayer types for evolution systems governed by nonconvex lipschitzian differential inclusions in banach spaces under endpoint constraints described by finitely many equalities and inequalities with generally nonsmooth functions we develop a variational analysis of such problems mainly based on their discrete approximations and the usage of advanced tools of generalized differentiation satisfying comprehensive calculus rules in the framework of asplund and hence any reflexive banach spaces in this way we establish extended results on stability of discrete approximations with the strong w 1 2 convergence of optimal solutions under consistent perturbations of endpoint constraints and derive necessary optimality conditions for nonconvex discrete time and continuous time systems in the refined euler lagrange and weierstrass pontryagin forms accompanied by the appropriate transversality inclusions in contrast to the case of geometric endpoint constraints in infinite dimensions the necessary optimality conditions obtained in this paper do not impose any nonempty interiority finite codimension normal compactness assumptions the approach and results developed in the paper make a bridge between optimal control dynamic optimization and constrained mathematical programming problems in infinite dimensional spaces

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

equivalent subgradient versions of hamiltonian and euler llagrange equations in variational analysis much effort in recent years has gone into generalizing the classical hamiltonian and euler lagrange equations of the calculus of variations so as to encompass problems in optimal control and a greater variety of integrands and constraints these generalizations in which nonsmoothness abounds and gradients are systematically replaced by subgradients have succeeded in furnishing necessary conditions for optimality that reduce to the classical ones in the classical setting but important issues have remained unsettled especially concerning the exact relationship of the subgradient versions of the hamiltonian equations versus those of the euler lagrange equations here it is shown that new tighter subgradient versions of these equations are actually equivalent to each other the theory of epi convergence of convex functions provides the technical basis for this development

on non exact controllable systems in this paper we are concerned with linear control systems of first or second order defined on banach spaces we established some negative facts for these systems fundamentally related with linear compact operators in particular we show several sufficient conditions of a general character for the non exact controllability or these systems we also show that it is not possible to uniformly stabilize a first order control system with states in a banach space using a compact control feedback this generalizes gibson s 1980 result for systems on hilbert spaces

periodic solutions for a kind of rayleigh equation with a deviating argument abstract in this paper we use the coincidence degree theory to establish new results on the existence of t periodic solutions for the rayleigh equation with a deviating argument of the form x f x t g t x t t p t