the proximal normal formula in banach space approximation by proximal normals to the clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis we give simple general versions of such formulae in infinite dimensional banach spaces which satisfy different geometrical conditions our first class of spaces with uniformly g teaux differentiable norm includes the hilbert space case and the formulae is attained through dense subsets our second class of reflexive kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of ekeland and lebourg

weak and strong solutions for the incompressible navier stokes equations with damping in this paper we show that the cauchy problem of the navier stokes equations with damping u 1u 0 has global weak solutions for any 1 global strong solution for any 7 2 and that the strong solution is unique for any 7 2 5

lipschitz regularity for minima without strict convexity of the lagrangian abstract we give in a non smooth setting some conditions under which some of the minimizers of f u x d x g x u x d x among the functions in w 1 1 that lie between two lipschitz functions are lipschitz we weaken the usual strict convexity assumption in showing that if just the faces of the epigraph of a convex function f r n r are bounded and the boundary datum u 0 satisfies a generalization of the bounded slope condition introduced by a cellina then the minima of f u x d x on u 0 w 0 1 1 whenever they exist are lipschitz a relaxation result follows

the existence of solutions to variational problems of slow growth abstract we consider the existence of solutions in the space w 1 1 to the problem minimize l v x d x on w 0 1 1 where l is of slow linear or at most quadratic growth we present a necessary and sufficient condition in order that for any smooth boundary datum and for any bounded with smooth boundary the minimum problem be solvable

existence of global weak solutions for a 2d viscous shallow water equations and convergence to the quasi geostrophic model we consider a two dimensional viscous shallow water model with friction term existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi geostrophic equation with free surface term is proven in the well prepared case the ill prepared data case is also discussed

preservation of prox regularity of sets with applications to constrained optimization in this paper we first provide counterexamples showing that sublevels of prox regular functions and levels of differentiable mappings with lipschitz derivatives may fail to be prox regular then we prove the uniform prox regularity of such sets under usual verifiable qualification conditions the preservation of uniform prox regularity of intersection and inverse image under usual qualification conditions is also established applications to constrained optimization problems are given

integral representation of abstract functionals of autonomous type in this work we extend the results of amar et al to a family of abstract functionals of autonomous type satisfying suitable locality and additivity properties and general integral growth conditions of superlinear type we single out a condition which is necessary and sufficient in order for a functional of this class to admit an integral representation and sufficient to have an integral representation for its lower semicontinuous envelope we also show that the integrand f x q satisfies some nice regularity properties in the q variable in particular a convexity type property along lines by adapting the reparametrization techniques introduced in an earlier work by the author to the case at issue we then prove that the family of integral functionals associated to integrands of this kind do meet the condition mentioned above in particular it is closed by vargamma convergence

on the 3d kelvin voigt brinkman forchheimer equations in some unbounded domains abstract considered here is the first initial boundary value problem for the kelvin voigt brinkman forchheimer equations in three dimensional domains satisfying the poincare inequality the existence of a weak solution to the problem is proved by using the faedo galerkin method then we show the existence of a unique minimal pullback d attractor for the process associated to the problem with respect to a large class of non autonomous forcing terms finally the existence uniqueness and stability of a stationary solution is studied when the external force is time independent and small

the classical problem of the calculus of variations in the autonomous case relaxation and lipschitzianity of solutions we consider the problem of minimizing b a l x t x t dt x a a x b b under the assumption that the lagrangian l is continuous and satisfies a growth assumption that does not imply superlinear growth we provide a result on the relaxation of the functional and show that a solution to the minimum problem is lipschitzian

regularity properties of optimal controls a class of optimal control problems is considered involving integral cost functions and linear dynamics broad verifiable hypotheses he are known under which the problems have solutions these must be supplemented for standard optimality conditions in the form of a maximum principle to apply new regularity properties of optimal controls when merely hypotheses he are in force are established a byproduct is a maximum principle governing optimal controls under such circumstances where the ostate function is possibly unbounded criteria for minimizing controls to be essentially bounded are also deduced

variational approach to second order sufficient optimality conditions in optimal control we consider a bolza optimal control problem with convex pure state constraints and general control constraints and propose new second order sufficient optimality conditions for a weak local minimum that can be applied in a general framework in particular the reference control may be merely measurable and the dynamics may be discontinuous in the time variable our approach is based on a new decomposition result for control constraints involving second order tangents

integration of subdifferentials of lower semicontinuous functions on banach spaces abstract this paper studies the lower semicontinuous functions on banach spaces whose subdifferentials are not disassociated from several conditions are given to ensure that two lower semicontinuous functions with the same subdifferential are equal up to an additive constant

preservation of stability in delay equations under delay perturbations we consider a class of linear delay equations with perturbed time lags and present conditions which guarantee that the asymptotic stability of the trivial solution of the equation at hand is preserved under these perturbations as an example we show how our results can be used to obtain an estimate on the maximum allowable sampling interval in the stabilization of a hybrid system with feedback delays we also present applications of our perturbation theorem to obtain stability conditions for delay equations with multiple delays

existence and lipschitz regularity of solutions to bolza problems in optimal control in this paper we investigate the existence and lipschitz continuity of optimal trajectories for the autonomous bolza problem in control theory the main feature of our results is that they relax the usual fast growth condition for the lagrangian furthermore we show that optimal solutions do satisfy the maximum principle

integration of subdifferentials of nonconvex functions in nonsmooth analysis and optimization subgradients come in many different flavors e g approximate dini proximal clarke generalized see 2 5 6 12 14 191 these subgradients are important and valuable tools however many questions remain unsolved concerning the exact link between the function and its subgradients for instance can two functions not differing by an additive constant have the same subgradients in this paper we study the fundamental problem of determining functions that can be recovered up to an additive constant from the knowledge of their subgradients this integration problem is not very well understood and very few functions or classes of functions are known to be recoverable from their subgradients in section 4 we show that if the basic constraint qualification holds at r then the composition of a closed i e lowersemicontinuous proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients actually in this case all above mentioned subgradients are the same beside the obvious theoretical interest of this integration problem it is our hope or perhaps our long term goal that once this problem is better understood we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives an example of such an equation that well deserves study is the extended hamilton jacobi equation used in optimal control see clarke 2 let us also mention that a problem similar to the integration problem is the one of determining the set valued mappings that are in fact subgradient set valued mappings uniqueness is not mandatory for a contribution to this problem see janin 8 before we look at some of the known cases where the function can be recovered from its subgradients let us look at some negative examples it is clear that not every function can be recovered up to an additive constant from its subgradients we only need to look at the following two functions 0 x50 0 x10 f x 1 x o g x 2 x o

degenerate optimal control problems with state constraints standard necessary conditions for optimal control problems with pathwise state constraints supply no useful information about minimizers in a number of cases of interest e g when the left endpoint of state trajectories is fixed at x0 and x0 lies in the boundary of the state constraint set in these cases a nonzero but nevertheless trivial set of multipliers exists we give conditions for the existence of nontrivial multipliers a feature of these conditions is that they allow nonconvex velocity sets and measurably time dependent data the proof techniques are based on refined estimates of the distance of a given state trajectory from the set of state trajectories satisfying the state constraint originating in the dynamic programming literature

exponential dichotomies and homoclinic orbits in functional differential equations abstract suppose an autonomous functional differential equation has an orbit which is homoclinic to a hyperbolic equilibrium point the purpose of this paper is to give a procedure for determining the behavior of the solutions near of a functional differential equation which is a nonautonomous periodic perturbation of the original one the procedure uses exponential dichotomies and the fredholm alternative it is also shown that any smooth function p t defined on the reals which approaches zero monotonically as t is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero examples illustrating the results are also given

lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands the study of regularity properties for minimizers of problems of the form min u u dt u e iv q r u a a u b b 1 1 has been undertaken for a long time in calculus of variations and a number of results are by now available on this subject when f t s z is a smooth coercive function problem 1 1 was considered by tonelli see 18 19 who proved that every minimizer u t of the problem 1 1 is locally lipschitz on a relatively open set of full measure in a b under some additional assumptions on f it is also possible to prove global regularity theorems see for instance cesari 3 chap 2 sect 61 in recent years much attention has been devoted to problems of the form l l on the one hand in weakening the hypotheses of tonelli s regularity results and on the other in finding counterexamples of nonregular minimizers for coercive functionals with respect to this last subject we recall that it has been shown see ball and mizel l and clarke and vinter 6 that there exist polynomial functions f t s z such that for suitable c 0 and p

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

on the uniqueness of strong solution to the incompressible navier stokes equations with damping abstract in this paper we show that the cauchy problem of the incompressible navier stokes equations with damping u 1 u 0 has global strong solution for any 3 and the strong solution is unique when 3 5 this improves earlier results

discontinuous galerkin methods for the stokes equations with nonlinear damping term on general meshes abstract this paper presents the interior penalty discontinuous galerkin ipdg methods to solve the stationary stokes equations with nonlinear damping term the ipdg discrete schemes are established on general meshes the corresponding consistency and stabilization of those schemes are proved subsequently we analyze the existence boundedness and uniqueness of the discrete solutions then the optimal error estimates are derived in the l 2 norm and h 1 like dg norm for the velocity variable and l 2 norm for the pressure variable respectively finally some numerical experiments in two dimensional are reported to demonstrate the theoretical results and show the robustness of our discrete schemes

a connection between the maximum principle and dynamic programming for constrained control problems we consider the mayer optimal control problem with dynamics given by a nonconvex differential inclusion whose trajectories are constrained to a closed set and obtain necessary optimality conditions in the form of the maximum principle together with a relation between the costate and the value function this additional relation is applied in turn to show that the maximum principle is nondegenerate we also provide a sufficient condition for the normality of the maximum break principle to derive these results we use convex linearizations of differential inclusions and convex linearizations of constraints along optimal trajectories then duality theory of convex analysis is applied to derive necessary conditions for optimality in this way we extend the known relations between the maximum principle and dynamic programming from the unconstrained problems to the constrained case

exponential dichotomies in intermediate spaces with applications to a diffusively perturbed predator prey model abstract intermediate spaces are useful for obtaining maximal regularity when studying reaction diffusion equations we give a definition of exponential dichotomies in these spaces roughness of the exponential dichotomies and other basic properties are also proved we then use these notions to investigate the stability of a periodic orbit in a predator prey model studied by conway and smoller when diffusion terms are added

not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks we present two examples of hyperbolic partial differential equations which are stabilized by boundary feedback controls and then destabilized by small delays in these controls we show that in a general case when the controls are distributed stabilized hyperbolic systems possess nontrivial periodic solutions if small time delays are introduced into their feedbacks we also indicate by means of an example that the general case of this phenomenon is harder to demonstrate for boundary control problems

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

sweeping by a continuous prox regular set abstract the evolution problem known as sweeping process is considered for a class of nonconvex sets called prox regular or convex assuming essentially that such sets contain in the interior a suitable subset and move continuously w r t the hausdorff distance we prove local and global existence as well as uniqueness of solutions which are continuous functions with bounded variation some examples are presented

lipschitz continuity of optimal controls for state constrained problems this paper provides new conditions under which optimal controls are lipschitz continuous for dynamic optimization problems with functional inequality constraints a control constraint expressed in terms of a general closed convex set and a coercive cost function it is shown that the linear independence condition on active state constraints present in the earlier literature can be replaced by a less restrictive positive linear independence condition that requires linear independence merely with respect to nonnegative weighting parameters smoothness conditions on the data imposed in earlier work are also relaxed the new conditions for lipschitz continuity of optimal controls are obtained by a detailed analysis of the implications of first order optimality conditions in the form of a nonsmooth maximum principle

iterative algorithms for a new class of extended general nonconvex set valued variational inequalities abstract some new classes of extended general nonconvex set valued variational inequalities and the extended general wiener hopf inclusions are introduced by the projection technique equivalence between the extended general nonconvex set valued variational inequalities and the fixed point problems as well as the extended general nonconvex wiener hopf inclusions is proved then by using this equivalent formulation we discuss the existence of solutions of the extended general nonconvex set valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set valued variational inequalities we also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set valued variational inequalities the results presented in this paper extend and improve some known results from the literature

regularity properties of solutions to the basic problem in the calculus of variations this paper concerns the basic problem in the calculus of variations minimize a functional j defined by j x j l t x t x t dt a over a class of arcs x whose values at a and b have been specified existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs conditions which must be strengthened in order that the standard necessary conditions apply the question arises what necessary conditions hold merely under hypotheses of existence theory say the classical tonelli conditions it is shown that given a solution x there exists a relatively open subset q of a b of full measure on which x is locally lipschitz and satisfies a form of the euler lagrange equation the main theorem of which this is a corollary can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions three such classes are identified and results of bernstein tonelli and morrey are extended one of these classes is of a novel nature and its study implies the new result that when l is independent of t the solution has essentially bounded derivative

linear estimates for trajectories of state constrained differential inclusions and normality conditions in optimal control abstract we consider solutions to a differential inclusion x f x constrained to a compact convex set here f is a compact possibly non convex valued lipschitz continuous multifunction whose convex closure co f satisfies a strict inward pointing condition at every boundary point x given a reference trajectory x taking values in an e neighborhood of we prove the existence of a second approximating trajectory x 0 t which satisfies the linear estimate x x ac 0 t k e if one of the following two cases occurs i the initial datum x 0 x 0 is given but lies in a compact set containing only points where the boundary is smooth ii the initial point x 0 of the approximating trajectory x can be chosen arbitrarily subsequently we employ these linear ac estimates to establish conditions for normality of the generalized euler lagrange condition for optimal control problems with state constraints in which we have an integral term in the cost we finally provide an illustrative example which underlines the fact that if conditions i and ii above are not satisfied then we can find a degenerate minimizer

another theorem of classical solvability in small for one dimensional variational problems in this paper we suggest a direct method for studying local minimizers of one dimensional variational problems which naturally complements the classical local theory this method allows us both to recover facts of the classical local theory and to resolve a number of problems which were previously unreachable the basis of these results is a regularity theory a priori estimates and compactness in c1 for solutions of obstacle problems with sufficiently close obstacles in these problems we establish that solutions exist and inherit regularity of the obstacles even under assumptions on integrands that are much weaker than those required in the classical local theory

the turnpike property in finite dimensional nonlinear optimal control turnpike properties have been established long time ago in finite dimensional optimal control problems arising in econometry they refer to the fact that under quite general assumptions the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces the first and the last of which being transient short time arcs and the middle piece being a long time arc staying exponentially close to the optimal steady state solution of an associated static optimal control problem we provide in this paper a general version of a turnpike theorem valuable for nonlinear dynamics without any specific assumption and for very general terminal conditions not only the optimal trajectory is shown to remain exponentially close to a steady state but also the corresponding adjoint vector of the pontryagin maximum principle the exponential closedness is quantified with the use of appropriate normal forms of riccati equations we show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method or a shooting method in particular we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector not at the initial time but at the middle of the trajectory

existence of minimizers for nonconvex noncoercive simple integrals we consider the problem of minimizing autonomous simple integrals such as min left int 0 t f left x t x prime t right dt colon text x in ac 0 t x 0 x 0 x t x t right tag cal p where f mathbb r times mathbb r to 0 infty is a possibly nonconvex function with either superlinear or slow growth at infinity assuming that the relaxed problem cal p ast ast obtained from cal p by replacing f with its convex envelope f with respect to the derivative variable x prime admits a solution we prove attainment for cal p under mild regularity and growth assumptions on f and f we discuss various instances of growth conditions on f that yield solutions to the corresponding relaxed problem cal p ast ast and we present examples that show that the hypotheses on f and f considered here for attainment are essentially sharp

relaxation and regularity in the calculus of variations in this work we prove that if l t u is a continuous function in t and u borel measurable in with bounded non convex pieces in then any absolutely continuous solution u to the variational problem min abl t u t u t dt u w01 1 a b is quasi regular in the sense of tonelli i e u is locally lipschitz on an open set of full measure of a b under the further assumption that either l is lipschitz continuous in u locally uniformly in but not necessarily in t or l is invariant under a group of c1 transformations as in the noether s theorem without one of those further assumptions the solution could be not regular as shown by a recent example in gratwick and preiss 2010 13 our result is then optimal in this sense moreover we improve the standard hypothesis used so far in buttazzo et al 1998 1 clarke and vinter 1985 5 6 csornyei et al 2008 7 tonelli 1915 15 which have been the lipschitz continuity of l in u locally uniform in and t and some growth condition in we also show that the relaxed and the original problem have the same solutions without assuming any of the two further assumptions above this extends a result in mariconda and treu 2004 14 to the non autonomous case

reparametrizations and approximate values of integrals of the calculus of variations abstract we prove an approximation result that implies the non occurrence of the lavrentiev phenomenon

partial extensions of attouch s theorem with applications to proto derivatives of subgradient mappings attouch s theorem which gives on a reflexive banach space the equivalence between the mosco epi convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients has many important applications in convex optimization in particular generalized derivatives have been defined in terms of the epi convergence or graph convergence of certain difference quotient mappings and attouch s theorem has been used to relate these various generalized derivatives these relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems we prove in a hilbert space several partial extensions of attouch s theorem to functions more general than convex these functions are called primal lower nice furthermore we use our extensions to derive a relationship between the second order epi derivatives of primal lower nice functions and the proto derivative of their associated subgradient mappings

asymptotic behavior of strong solutions to the 3d navier stokes equations with a nonlinear damping term abstract this paper deals with the asymptotic behavior of strong solutions to the 3d navier stokes equations with a nonlinear damping term u 1 u 3 first we establish an upper bound for the difference between the solution of our equation and the heat equation in l 2 space then we optimize the upper bound of decay for the solutions and obtain their algebraic lower bound by using fourier splitting method

investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints in this paper we study the degeneracy phenomenon in optimal control problems with state constraints it is shown that this phenomenon occurs because of the incompleteness of the standard variants of pontryagin s maximum principle for problems with state constraints a new maximum principle containing additional information about the behavior of the hamiltonian at the endtimes is developed we also obtain some sufficient and necessary conditions for nondegeneracy and pointwise nontriviality of the maximum principle the results obtained pertain to optimal control problems with systems described by differential inclusions and ordinary differential equations

an integral type constraint qualification to guarantee nondegeneracy of the maximum principle for optimal control problems with state constraints abstract for optimal control problems involving ordinary differential equations and functional inequality state constraints the maximum principle may degenerate producing no useful information about minimizers this is known as the degeneracy phenomenon several non degenerate forms of the maximum principle valid under different constraint qualifications have been proposed in the literature in this paper we propose a new constraint qualification under which a nondegenerate maximum principle is validated in contrast with existing results our constraint qualification is of an integral type an advantage of the proposed constraint qualification is that it is verified on a larger class of problems with nonsmooth data and convex velocity sets

normal forms of necessary conditions for dynamic optimization problems with pathwise inequality constraints abstract there has been a longstanding interest in deriving conditions under which dynamic optimization problems are normal that is the necessary conditions of optimality nco can be written with a nonzero multiplier associated with the objective function this paper builds upon previous results on nondegenerate nco for trajectory constrained optimal control problems to provide even stronger normal forms of the conditions the nco developed may address problems with nonsmooth less regular data the particular case of calculus of variations problems is here explored to show a favorable comparison with existent results

variational approach to second order optimality conditions for control problems with pure state constraints for optimal control problems with set valued control constraints and pure state constraints we propose new second order necessary optimality conditions in addition to the usual second order derivative of the hamiltonian these conditions contain extra terms involving second order tangents to the set of feasible trajectory control pairs at the extremal process under consideration the second order necessary optimality conditions of the present work are obtained by using a variational approach in particular we present a new second order variational equation this approach allows us to make direct proofs as opposed to the classical way of obtaining second order necessary conditions by using an abstract infinite dimensional optimization problem no convexity assumptions on the constraints are imposed and optimal controls are required to be merely measurable

convergence rate for compressible euler equations with damping and vacuum we study the asymptotic behavior of l weak entropy solutions to the compressible euler equations with damping and vacuum previous works on this topic are mainly concerned with the case away from the vacuum and small initial data in the present paper we prove that the entropy weak solution strongly converges to the similarity solution of the porous media equations in lp r 2 p with decay rates the initial data can contain vacuum and can be arbitrary large a new approach is introduced to control the singularity near vacuum for the desired estimates

on the resolvent of linear nonautonomous partial functional differential equations abstract we give sufficient conditions for the existence of the resolvent operator for nonautonomous linear partial differential equations with delay where the highest order derivatives are undelayed furthermore we analyse the connection between the resolvent and the solution operator of the homogeneous equation

a characteristic equation for non autonomous partial functional differential equations abstract we characterize the exponential dichotomy of non autonomous partial functional differential equations by means of a spectral condition extending known characteristic equations for the autonomous or time periodic case from this we deduce robustness results we further study the almost periodicity of solutions to the inhomogeneous equation our approach is based on the spectral theory of evolution semigroups

on partial regularity results for the navier stokes equations looking at the regularity results of scheffer respectively caffarelli kohn and nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the navier stokes equations

parabolic evolution equations with asymptotically autonomous delay we study retarded parabolic non autonomous evolution equations whose coefficients converge as t such that the autonomous problem in the limit has an exponential dichotomy then the non autonomous problem inherits the exponential dichotomy and the solution of the inhomogeneous equation tends to the stationary solution at infinity we use a generalized characteristic equation to deduce the exponential dichotomy and new representation formulas for the solution of the inhomogeneous equation

existence and differentiability of metric projections in hilbert spaces this paper considers metric projections onto a closed subset s of a hilbert space if the set s is convex then it is well known that the corresponding metric projections always exist unique and directionally differentiable at boundary points of s these properties of metric projections are considered for possibly nonconvex sets s in particular existence and directional differentiability of metric projections for certain classes of sets are established and will be referred to as nearly convex sets

normality of the maximum principle for nonconvex constrained bolza problems we consider a bolza optimal control problem with state constraints it is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle in general the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls in the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate cf a v arutyanov s m aseev investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints siam j control optim 35 1997 930 952 f rampazzo r b vinter a theorem on existence of neighbouring trajectories satisfying a state constraint with applications to optimal control ima 16 4 1999 335 351 f rampazzo r b vinter degenerate optimal control problems with state constraints siam j control optim 39 4 2000 989 1007 our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle in particular we allow the initial condition to be fixed and the state constraints to be nonsmooth to prove normality we use j yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined in turn by linearization of the original set of constraints along an extremal trajectory

a mixed fem for navier stokes type variational inequality with nonlinear damping term abstract in this work we consider a mixed finite element method fem for incompressible navier stokes equations with nonlinear damping term and friction boundary conditions after establishing the variational formulation we show the well posedness of this problem subsequently we focus our attention on the mixed fem and analyze the galerkin approximation to this system then some optimal error estimates are deduced in the end some iterative algorithms are presented and numerical results are given to verify the theoretical analysis

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

an extension of attouch s theorem and its application to second order epi differentiation of convexly composite functions in 1977 hedy attouch established that a sequence of closed proper convex functions epi converges to a convex function if and only if the graphs of the subdierentials converge in the mosco sense to the subdifferential of the limiting function and roughly speaking there is a condition that fixes the constant of integration we show that the theorem is valid if instead one considers functions that are the composition of a closed proper convex function with a twice continuously differentiable mapping in addition a constraint qualification is imposed

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

a new variational inequality in the calculus of variations and lipschitz regularity of minimizers abstract we consider a problem of the calculus of variations of the form p minimize i x a b t x t x t d t x a x b subject to x w 1 m a b r n x t c a e x t t a b where a b r n r n r is borel measurable c is a cone is a nonempty subset of r n and is an arbitrary extended valued function we prove that if t t x satisfies a nonsmooth version of cesari s condition s then any local minimizer of p in the ac norm is lipschitz whenever is coercive the proof is obtained via a new variational inequality formulated here that holds under the extended condition s just borel measurability if is autonomous for a e t in a b w t x t x t v v t x t x t p t v 1 v 0 where p t is an absolutely continuous arc the derivative of which belongs to a suitable subdifferential of x t x t a e on a b the proof of the directional weierstrass type condition w is based on clarke s nonsmooth recent versions of the maximum principle

existence of vector minimizers for nonconvex 1 dim integrals with almost convex lagrangian this paper proves new results of existence of minimizers for the nonconvex integral abl x x dt among the ac functions x a b rn with x a a x b b our lagrangian l is e g lsc with superlinear growth assuming values freely we replace convexity by almost convexity a hypothesis which in the radial superlinear case l s f s is automatically satisfied provided f s is convex at zero

generalized hessian properties of regularized nonsmooth functions we take up the question of second order expansions for a class of functions of importance in optimization namely moreau envelope regularizations of nonsmooth functions f it is shown that when f is prox regular which includes convex functions and the extended real valued functions representing problems of nonlinear programming the many second order properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself this clarifies the circumstances conducive to developing computational methods based on envelope functions such as second order approximations in nonsmooth optimization and variants of the proximal point algorithm the results establish that generalized second order expansions of moreau envelopes at least can be counted on in most situations of interest in finite dimensional optimization

nondegenerate necessary conditions for nonconvex optimal control problems with state constraints standard versions of the maximum principle for optimal control problems with pathwise state inequality constraints are satisfied by a trivial set of multipliers in the case when the left endpoint is fixed and lies in the boundary of the state constraint set and so give no useful information about optimal controls recent papers have addressed the problem of overcoming this degenerate feature of the necessary conditions in these papers it is typically shown that if a constraint qualification is imposed requiring existence of inward pointing velocities then sets of multipliers exist in addition to the trivial ones a simple new approach for deriving nondegenerate necessary conditions is presented which permits relaxation of hypotheses previously imposed concerning data regularity and convexity of the velocity set

necessary and sufficient conditions for optimality of nonconvex noncoercive autonomous variational problems with constraints we consider the classical autonomous constrained variational problem of minimization of a b v t v t dt in the class v w 1 1 a b v a v b v t 0 a e in a b where a x 0 r is a lower semicontinuous nonnegative integrand which can be nonsmooth nonconvex and noncoercive we prove a necessary and sufficient condition for the optimality of a trajectory v 0 in the form of a dubois reymond inclusion involving the subdifferential of convex analysis moreover we also provide a relaxation result and necessary and sufficient conditions for the existence of the minimum expressed in terms of an upper limitation for the assigned mean slope 0 b a applications to various noncoercive variational problems are also included

optimal sampled data controls with running inequality state constraints pontryagin maximum principle and bouncing trajectory phenomenon sampled data control systems have steadily been gaining interest for their applications in automatic engineering where they are implemented as digital controllers and recently results have been obtained in optimal control theory for nonlinear sampled data control systems and certain generalizations in this paper we derive a pontryagin maximum principle for general nonlinear finite dimensional optimal sampled data control problems with running inequality state constraints in particular we obtain a nonpositive averaged hamiltonian gradient condition with the adjoint vector being a function of bounded variations our proof is based on the ekeland variational principle in general optimal control problems with running inequality state constraints are difficult to solve using numerical methods due to the discontinuities the jumps and the singular part of the adjoint vector however in our case we find that under certain general hypotheses the adjoint vector only experiences jumps at most at the sampling times and moreover the trajectory only contacts the running inequality state constraints at most at the sampling times we call this behavior a bouncing trajectory phenomenon and it constitutes the second major focus of this paper finally taking advantage of the bouncing trajectory phenomenon we numerically solve three examples with different kinds of constraints and in several dimensions