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

the beginning of the fucik spectrum for the p laplacian has a nontrivial solution u here 1 p 2p u div u p 2 u u max u 0 u uu in 1 1 denoting by 1 2 the first two eigenvalues of 2p on w 1 p 0 0 cf the end of this introduction 7p clearly contains in particular 1 1 2 2 and the two lines 1 r and r 1 this generalized notion of spectrum was introduced for p 2 i e in the linear case in the 1970s by fuc ik 21 and dancer 15 in connection article id jdeq 1999 3645 available online at http www idealibrary com on

boundedness in asymmetric oscillations abstract in this paper we are concerned with the boundedness of all the solutions of the equation x ax bx f t wheref t is a smooth 2 periodic function aandbare positive constants a b

periodic solutions of a second order ordinary differential equation a necessary and sufficient condition for nonresonance the nonlinearity g in 1 1 is a continuous function from r to r and the forcing term h is taken in l o 27r nonresonance means that 1 1 admits at least one solution x for any given h integrating eq 1 1 over a period one immediately sees that a necessary condition for nonresonance is that the function g be unbounded from above and from below on r it will appear later cf 1 6 that this unboundedness can be looked at as a condition relating the behaviour at infinity of the nonlinearity g with respect to the first eigenvalue 1 0 of the associated linear problem xf ax in 0 27r x 0 x 27r x 0 x1 271 1 2

weak maximum principle for optimal control problems with mixed constraints optimality conditions for control problems with mixed state control constraints have been the focus of attention for a long time in particular the subject of necessary conditions in the form of maximum principles have been addressed by a number of authors see for example 1 4 to name but a few weak maximum principles which apply to weak local solutions covering problems with possibly nonsmooth data have been considered in 5 and in a more general setting in 3 for nonsmooth problems strong maximum principles which in turn apply to strong local solutions have also received some attention recently see 6 7 various recent results including those of the present paper can be captured as special cases of the following optimal control problem with mixed constraints also known as

nonresonance conditions for asymptotically positively homogeneous differential systems the fucik spectrum and its generalization in this article we propose how to define the fucik spectra for differential systems in rnwith jumping nonlinearities the propertyp a generalization of the non fucik spectra to positively homogeneous nonautonomous differential systems is also defined in the most general setting a quantitative estimate result concerning the parameterized completely continuous positively homogeneous perturbations of the identity in banach spaces is given in this article such a result is a generalization of one property in the spectrum theory for completely continuous linear operators using this estimate result we will give some significantly improved nonresonance results from the fucik spectra and propertypfor asymptotically positively homogeneous differential systems

mixed constrained control problems necessary optimality conditions are derived in the form of a weak maximum principle for optimal control problems with mixed state control equality and inequality constraints in contrast to previous work these conditions hold when the jacobian of the active constraints with respect to the unconstrained control variable has full rank a feature of these conditions is that they are stated in terms of a joint clarke subdifferential furthermore the use of the joint subdifferential gives sufficiency for nonsmooth normal linear convex problems the main point of interest is not only the full rank condition assumption but also the nature of the analysis employed in this paper a key element is the removal of the constraints and application of ekeland s variational principle

periodic solutions of second order equations via rotation numbers abstract we consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems we show how a rotation number approach together with the poincare birkhoff theorem and the phase plane analysis of the spiral properties allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity these systems may also be resonant in particular our approach can be used to deal with the problems without both global existence of the cauchy problems associated to the equation and the sign assumption on f the typical example is a partially superlinear second order equation our main result generalizes some classical results of jacobowitz and hartman among others

periodic solutions of a second order nonlinear system abstract we discuss the existence of periodic solutions of the system x t d dt grad f x t g t x t e t under various asymptotic behaviours of g

optimal control problems with mixed constraints we develop necessary conditions of broad applicability for optimal control problems in which the state and control are subject to mixed constraints we unify subsume and significantly extend most of the results on this subject notably in the three special cases that comprise the bulk of the literature calculus of variations differential algebraic systems and mixed constraints specified by equalities and inequalities our approach also provides a new and unified calibrated formulation of the appropriate constraint qualifications and shows how to extend them to nonsmooth data other features include a very weak hypothesis concerning the type of local minimum nonrestrictive hypotheses on the data and stronger conclusions notably as regards the maximum or weierstrass condition the necessary conditions are stratified in the sense that they are asserted on precisely the domain upon which the hypotheses and the optimality are assumed to hold this leads to local intermediate and global versions of the necessary conditions according to how the hypotheses are formulated

quadratic forms weighted eigenfunctions and boundary value problems for non linear second order ordinary differential equations some known results for different kinds of boundary value problems for second order ordinary differential equations are generalised different approaches are compared with one another using topological and variational methods and the theory of weighted eigenvalue problems

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

the rotation number of the linear schrodinger equation with discontinuous almost periodic potentials abstract in this paper we study a linear schrodinger equation with a stepanov almost periodic potential where the potential is discontinuous from r to r we first show the existence of rotation numbers for the equation we then give the optimal estimate of rotation numbers

on a singular periodic ambrosetti prodi problem abstract we investigate the possibility of extending a classical multiplicity result by fabry mawhin and nkashama to a periodic problem of ambrosetti prodi type having a nonlinearity with possibly one or two singularities in the second part of the paper we study the existence of periodic rotating solutions for radially symmetric systems with nonlinearities of the same type

necessary conditions for constrained problems under mangasarian fromowitz conditions the focus of this paper is on first order necessary conditions for optimal control problems with mixed state control equality and inequality constraints we consider the case when the cost and dynamics are nonsmooth and the constraints satisfy mangasarian fromowitz type assumptions that weaken the commonly used hypothesis that the jacobian of the active constraints with respect to the free variable is of full rank the results are formulated as unmaximized hamiltonian inclusion type conditions involving not the customary product of partial subdifferentials but the joint subdifferential with respect to the state and control variables

necessary optimality conditions for optimal control problems with equilibrium constraints this paper introduces and studies the optimal control problem with equilibrium constraints ocpec the ocpec is an optimal control problem with a mixed state and control equilibrium constraint formulated as a complementarity constraint and it can be seen as a dynamic mathematical program with equilibrium constraints it provides a powerful modeling paradigm for many practical problems such as bilevel optimal control problems and dynamic principal agent problems in this paper we propose weak clarke mordukhovich and strong stationarities for the ocpec moreover we give some sufficient conditions to ensure that the local minimizers of the ocpec are fritz john type weakly stationary mordukhovich stationary and strongly stationary unlike pontryagain s maximum principle for the classical optimal control problem with equality and inequality constraints a counterexample shows that for general ocpecs there may exist two sets of multipliers for complementarity constraints a condition under which these

on the periodic fucik spectrum and a superlinear sturm liouville equation in the first part of the paper a variational characterisation of the periodic eigenvalues the so called fucik spectrum of a semilinear positive homogeneous sturm liouville equation is given the proof relies on the s 1 invariance of the equation in the second part a nonlinear sturm liouville equation with typically an exponential nonlinearity is considered it is proved that under certain conditions this equation is solvable for arbitrary forcing terms the proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values in the fucik spectrum

resonance and rotation numbers for planar hamiltonian systems multiplicity results via the poincare birkhoff theorem abstract in the general setting of a planar first order system 0 1 u g t u u r 2 with g 0 t r 2 r 2 we study the relationships between some classical nonresonance conditions including the landesman lazer one at infinity and in the unforced case i e g t 0 0 at zero and the rotation numbers of large and small solutions of 0 1 respectively such estimates are then used to establish via the poincare birkhoff fixed point theorem new multiplicity results for t periodic solutions of unforced planar hamiltonian systems j u u h t u and unforced undamped scalar second order equations x g t x 0 in particular by means of the landesman lazer condition we obtain sharp conclusions when the system is resonant at infinity

necessary and sufficient conditions for viability for semilinear differential inclusions given a set a in a banach space x we define the tangent set and the quasi tangent set to k atc g k concepts more general than the one of tangent vector introduced by bouligand 1930 and severi 1931 both notions prove very suitable in the study of viability problems referring to differential inclusions namely we establish several new necessary and even necessary and sufficient conditions for viability referring to both differential inclusions and semilinear evolution inclusions conditions expressed in terms of the tangency concepts introduced

boundedness in a piecewise linear oscillator and a variant of the small twist theorem consider the differential equation ddot x n 2 x h l x p t where n 1 2 dots is an integer p is a 2 pi periodic function and h l is the piecewise linear function h l x begin cases l text if x geq 1 lx text if x leq 1 l text if x leq 1 end cases a classical result of lazer and leach implies that this equation has a 2 pi periodic solution if and only if begin equation label ll hat p n 2l over pi end equation where hat p n 1 over 2 pi int 0 2 pi p t e int dt in this paper i prove that if p is of class c 5 then the condition ref ll is also necessary and sufficient for the boundedness of all the solutions of the equation the proof of this theorem motivates a new variant of moser s small twist theorem this variant guarantees the existence of invariant curves for certain mappings of the cylinder which have a twist that may depend on the angle 1991 mathematics subject classification 34c11 58f35

a viability theorem for set valued states in a hilbert space abstract applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact possibly non convex sets rather than vectors this leads to evolutions in a basic set which can be supplied with a metric like the well established pompeiu hausdorff distance but it does not have an obvious linear structure this article extends differential inclusions with state constraints to compact valued states in a separable hilbert space h the focus is on sufficient conditions such that a given constraint set of compact subsets is viable a k a weakly invariant our main result extends the tangential criterion in the well known viability theorem usually for differential inclusions in a vector space to the metric space of non empty compact subsets of h

periodic solutions of semilinear duffing equations with impulsive effects abstract in this paper we are concerned with the existence of periodic solutions for semilinear duffing equations with impulsive effects firstly for its autonomous equation any motion of the solution is same as the motion of the corresponding equation without impulses until it meets the first impulse time under the influence of impulses these two motions are likely to be quite different we introduce some reasonable assumptions on the impulsive functions to control these differences such that the information valid for the equation without impulses can always be used for the impulsive one basing on poincare birkhoff twist theorem we prove the existence of infinitely many periodic solutions secondly as for the nonautonomous equation where the autonomous case is taken as an auxiliary one we find the relation between the solutions of these two equations and then obtain the existence of infinitely many periodic solutions also by poincare birkhoff twist theorem lastly an example with special impulses satisfying the above assumptions is given

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

large amplitude subharmonic oscillations for scalar second order differential equations with asymmetric nonlinearities abstract we provide sufficient conditions for the existence of subharmonic solutions for equations whose nonlinearity may have a different qualitative behaviour to the right and to the left of the origin the proofs are based on variational methods

periodic solutions of duffing s equations with superquadratic potential abstract this paper is devoted to the study of harmonic and subharmonic solutions for the second order scalar nonlinear duffing s equation x g x p t x x where g and p are continuous functions with p bounded and periodic in the first variable and g satisfying the assumption g x sign x as x among other results we prove the existence of infinitely many harmonic and subharmonic solutions of any order p p t and if the potential g x of g x satisfies certain conditions of superquadratic growth at the new existence results can be applied to situations in which the more classical superlinear growth condition g x x as x is not satisfied in this manner various preceding theorems are improved and sharpened see the introduction for more details proofs are based on a generalized version of the poincare birkhoff twist theorem due to w ding

impulse capture basins of sets under impulse control systems given a target contained in a constrained set and an impulse control system governing the evolutions of runs or executions that are hybrids of continuous and discrete evolutions this paper studies and provides several characterizations of the capture basin of the target viable in the constrained set it is the subset of initial runs from which start at least one run viable in the constrained set until it reaches the target in finite time it also provides algorithms and regulation rules governing the runs that reach the targets while obeying state constraints

periodic solutions of some resonant hamiltonian systems abstract based on the galerkin approximation and the morse inequality an existence theorem of periodic solutions of the hamiltonian system j z h t z is given where h t z is asymptotically linear and resonant at infinity existence of periodic solutions of the resonant asymmetric p laplacian equation is also discussed

multiple solutions for a dirichlet problem with jumping nonlinearities ii on considere l existence des solutions pour le probleme x t g x t p t c t 0 1 x 0 x 1 0 ou p t est une fonction continue et c est une constante on suppose que g est de classe c 1 et que g est positive et strictement croissante sur 0

viability for nonautonomous semilinear differential equations abstract let x be a reflexive banach space a d a x x the infinitesimal generator of a compact c0 semigroup s t x x t 0 d a locally closed subset in x and t x f t x a function on a b d which is locally integrably bounded measurable with respect to t and continuous with respect to x the main result of the paper is t heorem under the general assumptions above a necessary and sufficient condition in order that for each a b dthere exists at least one mild solutionu t dof du dt t au t f t u t satisfyingu is the tangency condition below there is a negligible subset z of a b such that for eacht a b zand eachx d lim inf h 0 1 h d s h x hf t x d 0

optimality conditions for a constrained control problem this paper is devoted to the study of necessary or sufficient second order conditions for a weak local minimum in an optimal control problem the problem is stated in the mayer form and includes equality constraints both on the endpoints and on the state control trajectory the second order conditions are stated through an associated linear quadratic problem

half eigenvalues of periodic sturm liouville problems we consider the nonlinear sturm liouville problem 1 pu qu au bu u in 0 2 2 u 0 u 2 pu 0 pu 2 where 1 p q l1 0 2 with p 0 a e on 0 2 a b l1 0 2 is a real parameter and u t max u t 0 for t 0 2 values of for which 1 2 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions the set of half eigenvalues will be denoted by h we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having certain nodal properties and we obtain certain spectral and degree theoretic properties associated with h these properties yield results on the existence and non existence of solutions of the problem 3 pu qu f t u h in 0 2 together with 2 where h l1 0 2 f 0 2 r r is a caratheodory function and the limits a t lim f t b t lim f t exist for a e t 0 2 when the functions a and b are constant the set h is closely related to the fucik spectrum of the problem and equivalent solvability results are obtained using the two approaches however when a and b are not constant the half eigenvalue approach yields stronger results

periodic problems with asymmetric nonlinearities and nonsmooth potentials abstract we consider a nonlinear periodic boundary value problem driven by the ordinary scalar p laplacian and with a nonsmooth locally lipschitz potential which exhibits an asymmetric behaviour at and at using the fucik spectrum and a variational approach based on the nonsmooth critical point theory we obtain a solution for the problem using a general minimax principle concerning linking sets

set separation approximating multicones and the lipschitz maximum principle abstract we present a general necessary condition for separation of the reachable set of a lipschitz control system from another given set of states expressed in terms of an approximating multicone to the set in a sense that contains as special cases the clarke and mordukhovich cones we then show how this separation result implies a strengthened form of the usual sufficient condition for local controllability along the reference curve and the necessary condition for optimality

a new fixed point theorem and periodic solutions of nonconservative weakly coupled systems abstract in this paper we prove a new fixed point theorem for the coupling of twist conditions and poincare bohl type conditions as its applications we prove the existence of periodic solutions for various nonconservative mixed type weakly coupled systems

a multiplicity result for periodic solutions of second order differential equations with a singularity abstract by the use of the poincare birkhoff fixed point theorem we prove a multiplicity result for periodic solutions of a second order differential equation where the nonlinearity exhibits a singularity of repulsive type at the origin and has linear growth at infinity our main theorem is related to previous results by rebelo 1996 1997 4 5 and rebelo and zanolin 1996 6 7 in connection with a problem raised by del pino et al 1992 1

discrete analogue of fucik spectrum of the laplacian the discrete analog of the fucik spectrum for elliptic equations namely m matrices is shown to have properties analogous to the continuum in particular the fucik spectrum of a m matrix contains a continuous and decreasing curve which is symmetric with respect to the diagonal

feedback spreading control laws for semilinear distributed parameter systems the paper shows for the first time that a feedback spreading control law for semilinear distributed parameter systems with compact linear operator can be provided by a selection procedure which involves both the dynamics of the system and the property to be spread in the case of affine dependence upon the control a minimum energy feedback spreading control law will be derived by using a parametrized constrained optimization technique along with some facts of set valued analysis

viability in a non local population model structured by size and spatial position abstract motivated conceptually by multiscale models of cancer cell migration the focus is on a broad class of set valued population models structured both physiologically and spatially for time dependent sets of constraints given sufficient conditions are provided such that every permitted density function initializes at least one admissible weak solution this result about viability a k a weak invariance concerns non local hyperbolic differential inclusions of first order with memory and with one boundary condition for the minimal value of the scalar structure parameter

subharmonic solutions with prescribed minimal period for nonautonomous hamiltonian systems on etablit un resultat de multiplicite pour une certaine classe de systemes hamiltoniens periodiques en temps

multiple periodic solutions of scalar second order differential equations abstract we prove multiplicity of periodic solutions for a scalar second order differential equation with an asymmetric nonlinearity thus generalizing previous results by lazer and mckenna 1987 1 and del pino manasevich and murua 1992 2 the main improvement lies in the fact that we do not require any differentiability condition on the nonlinearity the proof is based on the use of the poincare birkhoff fixed point theorem

periodic solutions of superlinear impulsive differential equations a geometric approach abstract a geometric method is introduced to study superlinear second order differential equations with impulsive effects basing on a reference continuous polar lifting of a planar orientation preserving homeomorphism we prove via the poincare birkhoff twist theorem the existence of infinitely many periodic solutions of conservative superlinear second order equations with the finite twist impulsive terms we also prove by developing a new twist fixed point theorem the existence of periodic solutions for non conservative superlinear second order equations with degenerate impulsive terms

nagumo type condition for partial differential inclusions on etudie un probleme de la forme x t ax t g xc h x o x 0 k t o t x t k ou a est un operateur elliptique et k n est pas necessairement compact on trouve des conditions tangentielles de type nagumo

the rotation number approach to the periodic fuc caron ik spectrum abstract in this paper we study the fucik spectrum of the problem q t x q t x 0 with the 2 periodic boundary condition where q t are 2 periodic after introducing a rotation number function for we prove using the hamiltonian structure and the positive homogeneity of that for any positive integer n the two boundary curves of the domain 1 n 2 in the plane are fucik curves of the result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional fucik spectrum with the dirichlet condition in particular it remains open if the fucik spectrum of is composed of only these curves

tangent sets viability for differential inclusions and applications abstract let x be a real banach space let a d a x x be a given operator k a nonempty and possible non open subset in d a f k x a given multi function in this lecture we consider the differential inclusion u t a u t f u t with a a 0 b a linear and c a nonlinear and in each one of these cases we give a short survey of the most important and very recent necessary and sufficient conditions for viability expressed in terms of tangent sets and a quasi tangent sets to k at a given point k concepts recently introduced by the authors from a rather long list of applications we confined ourselves only to solutions in moving sets a comparison result for a reaction diffusion system a comparison result for a nonlinear diffusion inclusion and a sufficient condition for null controllability

necessary conditions for optimal control problems involving nonlinear differential algebraic equations abstract dynamic models which take the form of a coupled set of differential and algebraic equations daes are widely used in process systems engineering necessary conditions of optimality for optimal control problems involving such models are derived a strong maximum principle is obtained under a convexity hypothesis on the velocity set an example illustrates that the strong maximal principle may be violated when this hypothesis is dropped for problems involving nonconvex velocity sets however a weak maximum principle is valid

the exact number of solutions for a class of ordinary differential equations through morse index computation we consider a class of second order ordinary differential equations with jumping nonlinearities that cross the first k eigenvalues under zero boundary conditions and by means of a morse index computation show the existence of exactly 2 k solutions

viability theorems for a class of differential operator inclusions abstract this paper discusses the following viability problem of a differential inclusion x t ax t f x t x 0 x0 k t 0 x t k where k is a compact subset of a banach space x a is the infinitesimal generator of a compact differentiable semigroup of bounded linear operators on x and f k x is upper semicontinuous with compact convex values we show that a natural tanegential condition is necessary and sufficient for the existence of a global solution to this problem

feedback spreading control under speed constraints this paper is concerned with the control of spread in semilinear parabolic systems it first introduces a formula in order to measure the speed of a spread then feedback spreading control laws under speed constraints are studied by a set valued approach the optimality of such laws is examined in the case in which the system is affine dependent upon the control

the fucik spectrum of general sturm liouville problems abstract consider the boundary value problem pu qu u u in 0 c 00 u 0 c 01 u 0 0 c 10 u c 11 u 0 where u max u 0 the set of points r 2 for which this problem has a non trivial solution is called the fucik spectrum when p 1 q 0 and either dirichlet or periodic boundary conditions are imposed the fucik spectrum is known explicitly and consists of a countable collection of curves with certain geometric properties in this paper we show that similar properties hold for the general problem above and also for a further generalization of the fucik spectrum we also discuss some spectral type properties of a positively homogeneous half linear problem and use these results to consider the solvability of a nonlinear problem with jumping nonlinearities

multiplicity results for an ode problem with even nonlinearity on considere le probleme u g u h dans u 0 sur ou r n est un ouvert borne a frontiere lisse en se restreignant au cas u u 2 t sin x dans 0 u 0 u 0 etant donne n 0 n il existe t nd 0 tel que pour tout t t nd 0 l equation consideree a au moins n 0 solutions

feedback spreading controls for semilinear parabolic systems abstract distributed biophysical processes usually involve spreads in space e g pollution desertification in vegetation dynamics etc in this paper based on the fact that spreadability of a distributed system reduces to a monotonicity problem we show how feedback spreading control laws can be determined in semilinear parabolic systems in case of affine dependence upon the control the use of a technique which combines saddle points and contraction mapping theorem yields a minimum energy feedback spreading control law a mathematical example is examined in order to illustrate the derived results

an euler lagrange inclusion for optimal control problems a new first order necessary condition is proved for nonsmooth nonlinear optimal control problems with general endpoint constraints and for which the velocity set may be possibly nonconvex it is in the nature of a generalization of the euler lagrange equation of the calculus of variations to optimal control it resembles the weak form of the maximum principle but it is distinct from it because it employs a total generalized gradient instead of the customary product of partial generalized gradients the optimality condition is shown to be sufficient for optimality when it is specialized to apply to normal convex problems a counterexample illustrates that for such problems the maximum principle is not a sufficient condition

existence uniqueness and stability of oscillations in differential equations with asymmetric nonlinearities we give conditions for the existence uniqueness and asymptotic stability of periodic solutions of a second order differential equation with piecewise linear restoring and 27r periodic forcing where the range of the derivative of the restoring term possibly contains the square of an integer with suitable restrictions on the restoring and forcing in the undamped case we give a necessary and sufficient condition

optimal control problems with mixed and pure state constraints this paper provides necessary conditions of optimality for optimal control problems in which the pathwise constraints comprise both pure constraints on the state variable and mixed constraints on control and state variables the proofs are along the lines of earlier analysis for mixed constraint problems according to which clarke s theory of stratified necessary conditions is applied to a modified optimal control problem resulting from absorbing the mixed constraint into the dynamics the difference here is that necessary conditions which now take into account the presence of pure state constraints are applied to the modified problem necessary conditions are given for a rather general formulation of the problem containing both forms of the constraints and then these are specialized to problems having special structure while combined pure state and mixed control state problems have been previously treated in the literature the necessary conditions in this paper are proved under less restrictive

existence and multiplicity results for periodic solutions of semilinear duffing equations we obtain existence an multiplicity of harmonic and subharmonic solutions to the periodically forced nonlinear duffing s equation x g x p t in the semi linear case 0 a g x x b for x large we introduce new conditions of nonresonance with respect to the spectrum of the associated linear differential operator x x and show by some nonlinear counterexamples that the results are sharp

an extension of the schwarzkopf multiplier rule in optimal control the search for multiplier rules in dynamic optimization has been an important theme in the subject for over a century it was central in the classical calculus of variations and the pontryagin maximum principle of optimal control theory is part of this quest a more recent thread has involved problems with so called mixed constraints involving the control and state variables jointly a subject which now boasts a considerable literature recently clarke and de pinho proved a general multiplier rule for such problems that extends and subsumes rather directly most of the available results namely those which postulate some kind of rank condition or more generally a constraint qualification or generalized mangasarian fromowitz condition an exception to this approach is due to schwarzkopf whose well known theorem replaces the rank hypothesis for relaxed problems by one of covering the purpose of this article is to show how to obtain this type of theorem from the general multiplier rule of clarke and de pinho in so doing we subsume extend and correct the currently available versions of schwarzkopf s result