the fully compressible semi geostrophic system from meteorology the semi geostrophic equations are an approximation to the 3 d euler equations for an atmosphere where the effects of rotation dominate they are used by meteorologists to model the formation of fronts mathematically rigorous results were obtained by benamou and brenier and by cullen and gangbo in the incompressible case in this paper we extend the results of benamou and brenier to the fully compressible case and show the existence of weak solutions to a reformulation of these equations in so called dual variables the monge kantorovich theory appears as a material tool in our study

on the existence and weak stability of solutions to the compressible semigeostrophic equations abstract in this article existence and weak stability of solutions to the compressible semigeostrophic equations are studied both in physical and in dual spaces we present a proof of the existence of a weak solution to these equations in dual space assuming initial data in l 1 r 3 extending previous results cullen and maroofi 2003 8 an extension to the concept of lagrangian solutions due to cullen and feldman 2006 6 is presented to the compressible semigeostrophic system and it is shown the existence of lagrangian solutions in physical space with initial potential vorticities in l 1 as well a weak stability result with respect to perturbations of the potential vorticity in l 1

a variational approach for the 2 dimensional semi geostrophic shallow water equations existence of weak solutions to the 3 d semi geostrophic equations with rigid boundaries was proved by benamou and brenier 3 using monge transport theory this paper extends the results to a free surface boundary condition which is more physically appropriate this extension is at present for the 2 d shallow water case only in addition we establish stronger time regularity than was possible in 3

holder regularity of the 2d dual semigeostrophic equations via analysis of linearized monge ampere equations we obtain the holder regularity of time derivative of solutions to the dual semigeostrophic equations in two dimensions when the initial potential density is bounded away from zero and infinity our main tool is an interior holder estimate in two dimensions for an inhomogeneous linearized monge ampere equation with right hand side being the divergence of a bounded vector field as a further application of our holder estimate we prove the holder regularity of the polar factorization for time dependent maps in two dimensions with densities bounded away from zero and infinity our applications improve previous work by g loeper who considered the cases of densities sufficiently close to a positive constant

euler poisson systems as action minimizing paths in the wasserstein space this paper uses a variational approach to establish existence of solutions t vt for the 1 d euler poisson system by minimizing an action we assume that the initial and terminal points 0 t are prescribed in mathcal p 2 mathbb r the set of borel probability measures on the real line of finite second order moments we show existence of a unique minimizer of the action when the time interval 0 t satisfies t these solutions conserve the hamiltonian and they yield a path t t in mathcal p 2 mathbb r when t y t is a dirac mass the euler poisson system reduces to ddot y y 0 the kinetic version of the euler poisson that is the vlasov poisson system was studied in ambrosio and gangbo comm pure appl math to appear as a hamiltonian system

the semigeostrophic equations discretized in reference and dual variables we study the evolution of a system of n particles x i v i i 1 n in mathbb r 2d that system is a conservative system with a hamiltonian of the form h mu w 2 2 mu nu n where w2 is the wasserstein distance and is a discrete measure concentrated on the set x i v i i 1 n typically 0 is a discrete measure approximating an initial l density and can be chosen randomly when d 1 our results prove convergence of the discrete system to a variant of the semigeostrophic equations we obtain that the limiting densities are absolutely continuous with respect to the lebesgue measure when nu n n 1 infty converges to a measure concentrated on a special d dimensional set we obtain the vlasov monge ampere vma system when d 1 the vma system coincides with the standard vlasov poisson system

the existence of periodic solutions of the equation x t ƒ x t x t τ abstract in this paper we study the existence of periodic solutions of the equation x t x t x t where gt 0 x y is continuous on r2 we obtain a sufficient condition in which 1 has at least k non constant oscillatory periodic solutions

hamiltonian odes in the wasserstein space of probability measures in this paper we consider a hamiltonian h on p 2 r 2d the set of probability measures with finite quadratic moments on the phase space r 2d r d r d which is a metric space when endowed with the wasserstein distance w 2 we study the initial value problem d t dt j d v t t 0 where j d is the canonical symplectic matrix 0 is prescribed and v t is a tangent vector to p 2 r 2d at t belonging to h t the subdifferential of h at t two methods for constructing solutions of the evolutive system are provided the first one concerns only the case where 0 is absolutely continuous it ensures that fit remains absolutely continuous and v t h t is the element of minimal norm in h t the second method handles any initial measure 0 if we further assume that h is convex proper and lower semicontinuous on p 2 r 2d we prove that the hamiltonian is preserved along any solution of our evolutive system h t h 0

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

existence of a weak solution for the semigeostrophic equation with integrable initial data in this article we present a proof of existence of a weak solution for the semigeostrophic system of equations formulated as an active scalar transport equation with monge ampere coupling with initial data in this is an extension of a 1998 result due to benamou and brenier who proved existence with initial data in

moreau yosida approximation and convergence of hamiltonian systems on wasserstein space abstract in this paper we study the stability property of hamiltonian systems on the wasserstein space let h be a given hamiltonian satisfying certain properties we regularize h using the moreau yosida approximation and denote it by h we show that solutions of the hamiltonian system for h converge to a solution of the hamiltonian system for h as converges to zero we provide sufficient conditions on h to carry out this process

weak existence for the semigeostrophic equations formulated as a coupled monge ampere transport problem hoskins s semigeostrophic equations are reformulated as a coupled monge ampere transport problem b j hoskins quart j royal met soc 97 1971 pp 139 153 existence of global weak solutions is obtained for this formulation

the stability principle and global weak solutions of the free surface semi geostrophic equations in geostrophic coordinates the semi geostrophic equations are used widely in the modelling of large scale atmospheric flows in this note we prove the global existence of weak solutions of the incompressible semi geostrophic equations in geostrophic coordinates in a three dimensional domain with a free upper boundary the proof based on an energy minimization argument originally inspired by the stability principle as studied by cullen purser and others uses optimal transport techniques as well as the analysis of hamiltonian odes in spaces of probability measures as studied by ambrosio and gangbo we also give a general formulation of the stability principle in a rigorous mathematical framework

nonlinear h control of nonsmooth time varying systems with application to friction mechanical manipulators nonlinear h controller synthesis is developed for nonsmooth time varying systems via measurement feedback a local h controller is derived by means of a certain perturbation of the differential riccati equations appearing in solving the h control problem for the linearized system when these unperturbed equations have bounded positive semidefinite solutions stabilizability and detectability properties of the control system are thus ensured by the existence of the proper solutions of the unperturbed differential riccati equations and hence the proposed synthesis procedure obviates an extra formidable in the nonlinear case work on verification of these properties theoretical results are applied to a position tracking control problem and particularly a regulation problem for mechanical systems with friction performance issues of the nonlinear h tracking controller are illustrated in a simulation study made for a two degrees of freedom robot manipulator

continuity equations and ode flows with non smooth velocity in this paper we review many aspects of the well posedness theory for the cauchy problem for the continuity and transport equations and for the ordinary differential equation ode in this framework we deal with velocity fields that are not smooth but enjoy suitable weak differentiability assumptions we first explore the connection between the partial differential equation pde and the ode in a very general non smooth setting then we address the renormalization property for the pde and prove that such a property holds for sobolev velocity fields and for bounded variation velocity fields finally we present an approach to the ode theory based on quantitative estimates

bifurcations of periodic solutions of delay differential equations abstract in this paper we develop kaplan yorke s method and consider the existence of periodic solutions for some delay differential equations we especially study hopf and saddle node bifurcations of periodic solutions with certain periods for these equations with parameters and give conditions under which the bifurcations occur we also give application examples and find that hopf and saddle node bifurcations often occur infinitely many times

w2 1 regularity for solutions of the monge ampere equation in this paper we prove that a strictly convex alexandrov solution u of the monge ampere equation with right hand side bounded away from zero and infinity is w 2 1 mathrm loc this is obtained by showing higher integrability a priori estimates for d 2 u namely d 2 u llog k l for any k

on the semi geostrophic system in physical space with general initial data in order to accommodate general initial data an appropriately relaxed notion of renormalized lagrangian solutions for the semi geostrophic system in physical space is introduced this is shown to be consistent with previous notions generalizing them a weak stability result is obtained first followed by a general existence result whose proof employs said stability and approximating solutions with regular initial data the renormalization property ensures the return from physical to dual space as consequences we get conservation of hamiltonian energy and some weak time regularity of solutions

periodic solutions of impulsive systems with a small delay for an impulsive system with delay it is proved that if the corresponding system without delay has an isolated omega periodic solution then in any neighbourhood of this orbit the system considered also has an omega periodic solution if the delay is small enough

an extension of the small gain theorem in ℒ we consider the class of control systems that consist of a linear time invariant forward path and a memoryless possibly time varying nonlinear feedback in this work we extend the small gain theorem in for these systems in two ways on the one hand we generalize the approach to nonlinearities with bounds of arbitrary shape on the other hand we add structure to the analysis dividing the vectors involved in the problem outputs nonlinearities disturbances into smaller dimension subvectors this leads to less conservative conditions in the cases in which some structure of the nonlinearity is known one important application is to the closed loop robust stability analysis of a nonlinear plant controlled by a feedback linearizing and stabilizing controller this is the case of an uncertain nonlinear model to be controlled affected by measurement noise and external disturbances

on l sub 2 and l sub infinity stability approaches for the robust control of robot manipulators robustness results were recently obtained for the linear and nonlinear control of robot manipulators using the l sub 2 and l sub infinity stability approaches respectively critical remarks on these approaches and further investigations are presented in order to investigate l sub 2 stability it is shown that additional assumptions have to be made and a reformulation of the control problem has to be carried out

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

duality theorems for marginal problems given topological spaces x1 xn with product space x probability measures i on xi together with a real function h on x define a marginal problem as well as a dual problem using an extended version of choquet s theorem on capacities an analogue of the classical duality theorem of linear programming is established imposing only weak conditions on the topology of the spaces xi and the measurability resp boundedness of the function h applications concern among others measures with given support stochastic order and general marginal problems

analysis of the semi geostrophic shallow water equations abstract the semi geostrophic shallow water equations are an accurate approximation to the rotating shallow water equations on scales larger than the rossby radius of deformation the global existence of weak solutions of the lagrangian form of the equations is known however uniqueness is not known except for short times for smooth initial data similarly a rigorous proof of existence of solutions has not been achieved for variable rotation rate which is important for atmospheric applications the obstacle is the lack of regularity of solutions to the monge ampere equation which underlies the solution procedure in this note it is shown formally that the modified form of the monge ampere equation that appears in the solution procedure for the semi geostrophic shallow water equations is better controlled on large scales than that of the standard equation that is used in the incompressible case this suggests that more progress should be possible with the rigorous theory and in particular that it should be possible to prove uniqueness

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159