the local structure of a liouville vector field 1 introduction a liouville vector field on a symplectic manifold is a smooth vector field whose flow consists of conformally flat transformations nagano 7 characterized the cotangent bundle of the singular set of a 1 form whose exterior derivative is symplectic by means of a liouville field in this work we investigate the local structure of a liouville vector field of a 2n dimensional symplectic manifold p q which vanishes on a k dimensional isotropic submanifold q of p some of the eigenvalues of its linear part are assumed to be in resonance a vector field with this properties appears in a natural geometric problem it arises as the gradient field with respect to some kaihler metric of the square of the distance function to an isotropic submanifold in nagano s theorem the singular set is a lagrangian submanifold and the eigenvalues of the linear part of the liouville field at the singular points are 0 and 1 here we assume that at every point q of q the vector field vanishes and the eigenvalues of its 1 jet are 1 1 2 and 0 this means precisely that in some coordinates the 1 jet of at q is

the unique limit of the glimm scheme we introduce the definitions of a standard riemann semigroup and of a viscosity solution for a nonlinear hyperbolic system of conservation laws for a class including general 2 2 systems it is proved that the solutions obtained by a wavefront tracking algorithm or by the glimm scheme are precisely the semigroup trajectories in particular these solutions are unique and depend lipschitz continuously on the initial data in the l1 norm

dissipative boundary conditions for 2 2 hyperbolic systems of conservation laws for entropy solutions in bv in this article we investigate the bv stability of 2 2 hyperbolic systems of conservation laws with strictly positive velocities under dissipative boundary conditions more precisely we derive sufficient conditions guaranteeing the exponential stability of the system under consideration for entropy solutions in bv our proof is based on a front tracking algorithm used to construct approximate piecewise constants solutions whose bv norms are controlled through a lyapunov functional this lyapunov functional is inspired by the one proposed in j glimm s seminal work j glimm comm pure appl math 18 697 715 1965 modified with some suitable weights in the spirit of the previous works j m coron g bastin and b d andrea novel siam j control optim 47 3 1460 1498 2008 and j m coron b d andrea novel and g bastin ieee trans automat control 52 1 2 11 2007

on the controllability of the non isentropic 1 d euler equation in this paper we examine the question of the boundary controllability of the one dimensional non isentropic euler equation for compressible polytropic gas in the context of weak entropy solutions we consider the system in eulerian coordinates and the one in lagrangian coordinates we obtain for both systems a result of controllability toward constant states with the limitation gamma frac 5 3 on the adiabatic constant for the lagrangian system the solutions that we obtain remain of small total variation in space if the initial condition is itself of sufficiently small total variation

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

existence of solutions to some singular initial value problems on etudie l existence des solutions et leurs domaines de definition pour des problemes aux valeurs initiales pour des equations differentielles d ordre 1 et 2 qui sont singulieres soit dans les variables dependantes soit dans les variables independantes

minimal positive solutions to some singular second order differential equations abstract the existence of a minimal c1 0 1 positive solution is established for some second order singular boundary value and initial value problems by new schemes which are related to x our nonlinearity may be singular at t 0 1 x 0 or x 0

optimal control of cell mass and maturity in a model of follicular ovulation in this paper we study optimal control problems associated with a scalar hyperbolic conservation law modeling the development of ovarian follicles changes in the age and maturity of follicular cells are described by a two dimensional conservation law where the control terms act on the velocities the control problem consists in optimizing the follicular cell resources so that the follicular maturity reaches a maximal value in fixed time formulating the optimal control problem within a hybrid framework we prove necessary optimality conditions in the form of the hybrid maximum principle then we derive the optimal strategy and show that there exists at least one optimal bang bang control with one single switching time

decay estimate for a new measure of rarefaction waves in general systems of conservation laws abstract the sharp decay estimate of rarefaction waves in terms of a partial ordering among positive measures is not only interesting in itself but also crucial in the study of convergence rate of vanishing viscosity approximations cf bressan and yang 2004 10 such an estimate is well established for genuinely nonlinear system of conservation laws cf bressan and yang 2004 9 but similar result is not available for non genuinely nonlinear system in this paper we give a new measure about the rarefaction waves in addition a sharp decay estimate of the new measure is given for the cubic nonlinear system of conservation laws

existence of positive solution to a singular second order initial value problem with impulses this paper deals with the existence of positive solutions to singular initial value problem with impulse effects the right hand side of the differential equation can be singular in its dependent variable

on the boundary controllability of first order hyperbolic systems abstract we discuss some results concerning the boundary controllability and stabilizability of a hyperbolic system of conservation laws u t f u x 0 t 0 x 0 1 where we regard the boundary data or a partial number of their components as boundary input controls in particular we consider the problem of the global exact boundary controllability of a first order linear hyperbolic system with constant coefficients relative to the linear boundary conditions under generic orthogonality assumptions on the boundary and control matrices and assuming a non resonance condition of the characteristic speeds we show that one can steer in finite time the solution from any initial condition l 1 to any terminal state l 1 even in the case where only a partial control of the boundary values is available

well posedness theory for hyperbolic conservation laws the paper presents a well posedness theory for the initial value problem for a general system of hyperbolic conservation laws we will start with the refinement of glimm s existence theory and discuss the principle of nonlinear through wave tracing our main goal is to introduce a nonlinear functional for two solutions with the property that it is equivalent to the l1 x distance between the two solutions and is time decreasing moreover the functional is constructed explicitly in terms of the wave patterns of the solutions through the nonlinear superposition it consists of a linear term measuring the l1 x distance a quadratic term measuring the coupling of waves and distance and a generalized entropy functional 1999 john wiley sons inc

some existence results for differential equations of second order existence results for initial value problem and dirichlet boundary value problem for nonlinear differential equations of second order are obtained

the boundary riemann solver coming from the real vanishing viscosity approximation we study the limit of the hyperbolic parabolic approximation left begin array l quad l quad l v varepsilon t tilde a left v varepsilon varepsilon v varepsilon x right v varepsilon x varepsilon tilde b v varepsilon v varepsilon xx quad v varepsilon in mathbb r n cr tilde rm ss v varepsilon t 0 equiv bar g cr v varepsilon 0 x equiv bar v 0 end array right the function tilde ss is defined in such a way as to guarantee that the initial boundary value problem is well posed even if tilde b is not invertible the data bar g and bar v 0 are constant when tilde b is invertible the previous problem takes the simpler form left begin array l quad l quad l v varepsilon t tilde a left v varepsilon varepsilon v varepsilon x right v varepsilon x varepsilon tilde b v varepsilon v varepsilon xx quad v varepsilon in mathbb r n cr v varepsilon t 0 equiv bar v b cr v varepsilon 0 x equiv bar v 0 end array right again the data bar v b and bar v 0 are constant the conservative case is included in the previous formulations convergence of the v varepsilon smallness of the total variation and other technical hypotheses are assumed and a complete characterization of the limit is provided the most interesting points are the following first the boundary characteristic case is considered that is one eigenvalue of tilde a can be 0 second as pointed out before we take into account the possibility that tilde b is not invertible to deal with this case we take as hypotheses conditions that were introduced by kawashima and shizuta relying on physically meaningful examples we also introduce a new condition of block linear degeneracy we prove that if this condition is not satisfied then pathological behaviors may occur

continuous dependence for 2 2 conservation laws with boundary on the domain 0 t x r t 0 and x 9 t for some boundary profile 9 r r as usual 1 1 is assumed to be strictly hyperbolic and with each characteristic field either linearly degenerate or genuinely non linear an initial data u 0 x u x having sufficiently small total variation is given we consider two different kinds of boundary conditions along x 9 t and in both cases we construct a lipschitzean flow whose trajectories are solutions of an initial boundary value problem for 1 1 thus we prove the continuous dependence of the solution upon the initial data upon the boundary condition and upon the boundary profile the existence theory for global bv solutions to the cauchy problem for 1 1 goes back to the fundamental paper 15 by glimm more recently in 6 a new approach has been introduced it relies on the construction of a lipschitzean semigroup the standard riemann semigroup srs whose trajectories extend the local standard lax 17 solutions of riemann problems at present such a srs has been constructed in the 2 2 case in 8 9 while for the general n n case see 7 10 and 3 in article no de973274

a contractive metric for systems of conservation laws with coinciding shock and rarefaction curves abstract we introduce two algorithms for the construction of weak entropy admissible solutions to a class of systems of conservation laws with coinciding shock and rarefaction curves global existence uniqueness and continuous dependence are proved for all solutions obtained by our constructive procedure the generated semigroup is contractive with respect to a riemann type metric defined in terms of glimm s wave interaction functional equivalent to the usual l 1 distance

decay estimation in nonlinear hyperbolic system of conservation laws abstract in this paper we study some decay estimates in nonlinear hyperbolic system of conservation laws this research is not only interesting in itself but also crucial in studying the large time behavior problem by introducing a proper glimm functional we obtain some useful decay estimates which are proved helpful in obtaining decay rates of the admissible solutions to nonlinear hyperbolic conservation laws as t

no contractive metrics for systems of conservation laws soit u t f u x 0 un systeme 2 2 de lois de conservation satisfaisant certaines hypotheses generiques sur f dans un voisinage 7b n de l espace u on montre que pour chaque metrique non degeneree d sur l u espace il existe des etats u 1 et u 2 dans 7b n tels que d u x t u 1 dx est une fonction strictement croissante de t dans un voisinage de t 0 ou u est la solution admissible du systeme avec u x 0 u 1 x 0 u 2 0 x 1 u 1 x 1

on the convergence of finite difference methods for pde under temporal refinement abstract in this note we present an issue in the verification of order of convergence for finite difference approximations to partial differential equations discretized in time and space when one refines both space and time together the convergence rate for a correct implementation will match that expected from an analysis of the asymptotic convergence rate or local truncation error of the method sometimes however only the time step size is reduced while the spatial mesh is held fixed the observed rate of convergence in this case may then differ from the formal order of accuracy of the method in space time in particular one class of methods second order lax wendroff time differencing produces only first order when the time step is refined at fixed spatial mesh resolution because the time and space differencing are intrinsically linked our method to arrive at this result is through analyzing the error at a defined point in time associated with many time steps we demonstrate our results computationally

a generalized upper and lower solutions method and multiplicity results for nonlinear first order ordinary differential equations abstract we introduce a generalized upper and lower solutions method for the solvability of first order ordinary differential equations u t t u t u 0 u 1 in order to cover the case when the function satisfies caratheodory conditions this method is then applied to get multiplicity results when the nonlinearity interacts with the real eigenvalue of the linearized problem our proofs are based on differential inequalities and classical leray schauder degree

boundary layers in weak solutions of hyperbolic conservation laws this paper is concerned with the initial boundary value problem for a nonlinear hyperbolic system of conservation laws we study the boundary layers that may arise in approximations of entropy discontinuous solutions we consider both the vanishing viscosity method and finite difference schemes lax friedrichs type schemes and the godunov scheme we demonstrate that different regularization methods generate different boundary layers hence the boundary condition can be formulated only if an approximation scheme is selected first assuming solely uniform l infty bounds on the approximate solutions and so dealing with l infty solutions we derive several entropy inequalities satisfied by the boundary layer in each case under consideration a young measure is introduced to describe the boundary trace when a uniform bound on the total variation is available the boundary young measure reduces to a dirac mass from the above analysis we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not each formulation is based on a set of admissible boundary values following the terminology of dubois lefloch 15 the local structure of these sets and the well posedness of the corresponding initial boundary value problem are investigated the results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics

exact boundary observability for a kind of second order quasilinear hyperbolic system abstract in this paper by means of a constructive method based on the existence and uniqueness of the semi global classical solution we establish the local exact boundary observability for a kind of second order quasilinear hyperbolic system in which the number of positive eigenvalues and the number of negative ones are not equal as an application we obtain the one sided local exact boundary observability and two sided local exact boundary observability with fewer observed values for first order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative ones are decoupled

backstepping boundary control for first order hyperbolic pdes and application to systems with actuator and sensor delays abstract we consider a problem of boundary feedback stabilization of first order hyperbolic partial differential equations pdes these equations serve as a model for physical phenomena such as traffic flows chemical reactors and heat exchangers we design controllers using a backstepping method which has been recently developed for parabolic pdes with the integral transformation and boundary feedback the unstable pde is converted into a delay line system which converges to zero in finite time we then apply this procedure to finite dimensional systems with actuator and sensor delays to recover a well known infinite dimensional controller analog of the smith predictor for unstable plants we also show that the proposed method can be used for the boundary control of a korteweg de vries like third order pde the designs are illustrated with simulations

global well posedness of traffic flow models with phase transitions abstract this note is devoted to the study of traffic flow models that develop phase transitions from the analytical point of view this is a first example of a well posedness result for conservation laws developing phase transitions which is independent from the number of phase boundaries in the initial data or in the solutions we consider below the cauchy problem as well as the problem with boundaries

the symplectic structure of kahler manifolds of nonpositive curvature on montre que la forme de kahler sur une variete de kahler complete simplement connexe w de courbure non positive est diffeomorphe a la forme symplectique standard sur r n ceci signifie que la structure symplectique sur un espace symetrique hermitien de type non compact est standard

systems of conservation laws with invariant submanifolds systems of conservation laws with coinciding shock and rarefaction curves arise in the study of oil reservoir simulation multicomponent chromatogra phy as well as in the study of nonlinear motion in elastic strings here we characterize this phenomenon by deriving necessary and sufficient conditions on the geometry of a wave curve in order that the shock wave curve coincide with its associated rarefaction wave curve for a system of conservation laws this coinci dence is the one dimensional case of a submanifold of the state variables being invariant for the system of equations and the necessary and sufficient conditions are derived for invariant submanifolds of arbitrary dimension in the case of 2 x 2 systems we derive explicit formulas for the class of flux functions that give rise to the coupled nonlinear conservation laws for which the shock and rarefaction wave curves coincide introduction systems of conservation laws which have coinciding shock and rarefaction curves arise in the study of oil reservoir simulation nonlinear wave motion in elastic strings as well as in multicomponent chromatography 1 4 5 6 9 11 12 these systems have many interesting features the riemann problem for these equations can be explicitly solved in the large and wave interactions have a simplified structure even in the presence of a nonconvex flux function for this reason these systems represent some of the few examples for which the cauchy problem has been solved for arbitrary data of bounded variation also hyperbolic degeneracies appear in each of these systems in the present paper we are concerned with characterizing the phenomenon of coinciding shock and rarefaction curves this phenomenon turns out to be a special case of the general phenomenon of a submanifold of r being invariant for weak solutions of a given system of n conservation laws invariant manifolds of higher dimension also appear in physical conservation equations in which the conserved quantities represent concentrations i e when the concentrations c of any of the species vanish the system must reduce to a lower order system that expresses the conservation of the remaining species thus c 0 defines a manifold which is invariant in the sense that solutions that start on the manifold remain there for all time moreover such an invariant manifold of dimension n 1 is the boundary of an dimensional invariant region in the sense of chueh conley and smoller 14 in the case of

a new glimm functional and convergence rate of glimm scheme for general systems of hyperbolic conservation laws in this paper we introduce a new glimm functional for general systems of hyperbolic conservation laws this new functional is consistent with the classical glimm functional for the case when each characteristic field is either genuinely nonlinear or linearly degenerate so that it can be viewed as optimal in some sense with this new functional the consistency of the glimm scheme is proved clearly for general systems moreover the convergence rate of the glimm scheme is shown to be the same as the one obtained in bressan marson arch ration mech anal 142 2 155 176 1998 for systems with each characteristic field being genuinely nonlinear or linearly degenerate

scalar non linear conservation laws with integrable boundary data where u u t x is the state variable u are integrable possibly unbounded initial and boundary data and f is assumed to be a superlinear strictly convex function for problems of this type since classical solutions develop discontinuities in nite time no matter how smooth their initial and boundary data it is natural to consider weak solutions satisfying the usual entropy conditions 13 15 u t x u t x t x 0 1 4

symplectic duality of symmetric spaces let be a complex n dimensional hermitian symmetric space endowed with the hyperbolic form hyp denote by m fs the compact dual of m hyp where fs is the fubini study form on m our first result is theorem 1 1 where with the aid of the theory of jordan triple systems we construct an explicit symplectic duality namely a diffeomorphism satisfying and for the pull back of m where 0 is the restriction to m of the flat kahler form of the hermitian positive jordan triple system associated to m amongst other properties of the map m we also show that it takes complete complex and totally geodesic submanifolds of m through the origin to complex linear subspaces of as a byproduct of the proof of theorem 1 1 we get an interesting characterization theorem 5 3 of the bergman form of a hermitian symmetric space in terms of its restriction to classical complex and totally geodesic submanifolds passing through the origin

a complex analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit abstract we revisit a result by coron and guerrero stating that the one dimensional transport diffusion equation u t m u x e u x x 0 in 0 t 0 l controlled by the left dirichlet boundary value is zero controllable at a bounded cost as e 0 when t 4 3 l m if m 0 and when t 57 2 l m if m 0 by a completely different method relying on complex analysis we prove that this still holds when t 4 2 l m if m 0 and when t 6 1 l m if m 0

optimal boundary control of nonlinear hyperbolic conservation laws with switched boundary data we consider the optimal control of initial boundary value problems for entropy solutions of scalar hyperbolic conservation laws in particular we consider initial boundary value problems where the initial and boundary data switch between different c 1 functions at certain switching points and both the functions and the switching points are controlled we show that the control to state mapping is differentiable in a certain generalized sense which implies frechet differentiability with respect to the control functions and the switching points for the composition with a tracking type functional even in the presence of shocks we also present an adjoint based formula for the gradient of the reduced objective functional

on the attainable set for temple class systems with boundary controls consider the initial boundary value problem for a strictly hyperbolic genuinely nonlinear temple class system of conservation laws begin equation label eq 1 u t f u x 0 qquad u 0 x ov u x qquad left begin array ll u t a widetilde u a t 2pt u t b widetilde u b t end array right end equation on the domain omega t x in r 2 t geq 0 a le x leq b we study the mixed problem ref eq 1 from the point of view of control theory taking the initial data overline u fixed and regarding the boundary data widetilde u a widetilde u b as control functions that vary in prescribed sets u a u b of li boundary controls in particular we consider the family of configurations a t doteq u t cdot u mbox is a sol to ref eq 1 widetilde u a in u a widetilde u b in u b that can be attained by the system at a given time t 0 and we give a description of the attainable set a t in terms of suitable oleinik type conditions we also establish closure and compactness of the set a t in the lu topology

the semigroup generated by 2 2 conservation laws consider the cauchy problem for a strictly hyperbolic 2 2 system of conservation laws in one space dimension ie1 01 assuming that each characteristic field is either linearly degenerate or genuinely nonlinear this paper develops a new algorithm based on wave front tracking which yields a cauchy sequence of approximate solutions converging to a unique limit depending continuously on the initial data the solutions that we obtain constitute a semigroup s defined on a set ie1 02 of integrable functions with small total variation for some lipschitz constant l we have the estimate ie1 03

smooth linearization near a fixed point in this paper we extend a theorem of sternberg and bi leckii we study a vector field or a diffeomorphism in the vicinity of a hyperbolic fixed point we assume that the eigenvalues of the linear part a at the fixed point satisfy qth order algebraic inequalities where q 2 2 then there is ck linearization in the vicinity of the fixed point where k is the q smoothness of a we give an explicit and simple algorithm for com puting k an illustrative example from celestial mechanics is included which shows that our main theorem is an improvement over previously known theories we also show that if the vector field or diffeomorphism depends smoothly on a parameter then the linearizing conjugation de pends smoothly on the parameter

penalty techniques for state constrained optimal control problems with the wave equation we consider a system that is governed by a pde of hyperbolic type namely the wave equation and is controlled by a dirichlet boundary control for the penalization of terminal constraints we compare the convergence of three penalization techniques a differentiable penalty method an exact penalization and a smoothed exact penalization the error of the solution of the differentiably penalized problem is at most of the order 1 sqrt gamma where gamma is the penalty parameter the exact penalization yields the exact solution if gamma is sufficiently large if gamma is sufficiently large for the smoothed exact penalization we obtain an error bound of the order 1 sqrt beta where beta is the additional smoothing parameter this method yields differentiable objective functions that remain uniformly strongly convex since gamma need not tend to infinity to obtain convergence we also consider the penalization of distributed inequality state constraints that prescribe an upper bound for the l infty norm of the state we prove the convergence with respect to the l 2 norm for a penalization of the constraint violation measured in the l 1 norm we use this penalization technique to prove the existence of optimal controls for the inequality state constrained problem

twin solutions to singular semipositone problems abstract in this paper by a specially constructed cone and the fixed point index theory we investigate the existence of multiple positive solutions for the following singular semipositone problem y f t y 0 t 0 1 y 0 y 1 0 the nonlinear term f t y may be singular at t 0 t 1 and y 0 also may be negative for some values of t and y and is a positive parameter

exact controllability of scalar conservation laws with an additional control in the context of entropy solutions in this paper we study the exact controllability problem for nonlinear scalar conservation laws on a compact interval with a regular convex flux and in the framework of entropy solutions with the boundary data and a source term depending only on the time as controls we provide sufficient conditions for a state to be reachable in arbitrary small time to do so we introduce a slightly modified wave front tracking algorithm

oleinik type estimates and uniqueness for n n conservation laws abstract let ut f u x 0 be a strictly hyperbolic n n system of conservation laws in one space dimension relying on the existence of a semigroup of solutions we first establish the uniqueness of entropy admissible weak solutions to the cauchy problem under a mild assumption on the local oscillation of u in a forward neighborhood of each point in the t x plane in turn this yields the uniqueness of weak solutions which satisfy a decay estimate on positive waves of genuinely nonlinear families thus extending a classical result proved by oleinik in the scalar case

positive solutions of some second order nonlinear singular differential equations abstract with new schemes the existence of the positive solutions is proved to some second order nonlinear singular boundary value problems and initial value problems which have higher order singularities

exact boundary controllability for quasi linear hyperbolic systems using a result on the existence and uniqueness of the semiglobal c1 solution to the mixed initial boundary value problem for first order quasi linear hyperbolic systems with general nonlinear boundary conditions we establish the exact boundary controllability for quasi linear hyperbolic systems if the c1 norm of initial and final states is small enough

time varying riemann solvers for conservation laws on networks abstract we consider a conservation law on a network and generic riemann solvers at nodes depending on parameters which can be seen as control functions assuming that the parameters have bounded variation as functions of time we prove existence of solutions to cauchy problems on the whole network

multivalued equations for granular avalanches we study existence and uniqueness for the scalar equation of first order hyperbolic type the rhs is in the form of a multifunction which is physically motivated by an avalanche flow model therefore some kind of monotonicity to the rhs has to be assumed

on the convergence rate of vanishing viscosity approximations given a strictly hyperbolic genuinely nonlinear system of conservation laws we prove the a priori boundu t u e t l1 o 1 1 t e ln e on the distance between an exact bv solution u and a viscous approximation u e letting the viscosity coefficient e 0 in the proof starting from u we construct an approximation of the viscous solution u e by taking a mollification u e and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed e error estimates are then obtained by introducing new lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families c 2004 wiley periodicals inc

well posedness of balance laws with boundary consider the initial boundary value problem for a temple system of balance laws aim of this paper is to prove the well posedness of this problem for large times and without requiring the total variation of the initial data be small

on the boundary control of systems of conservation laws this paper is concerned with the boundary controllability of entropy weak solutions to hyperbolic systems of conservation laws we prove a general result on the asymptotic stabilization of a system near a constant state on the other hand we give an example showing that exact controllability in finite time cannot be achieved in general

on the attainable set for scalar nonlinear conservation laws with boundary control we consider the initial value problem with boundary control for a scalar nonlinear conservation law begin equation u t f u x 0 qquad qquad u 0 x 0 qquad u cdot 0 tilde u in cal u tag ast end equation on the domain omega t x in real 2 t geq 0 x geq 0 here u u t x is the state variable cal u is a set of bounded boundary data regarded as controls and f is assumed to be strictly convex we give a characterization of the set of attainable profiles at a fixed time t 0 and at a fixed point bar x 0 begin equation begin aligned rag u t cdot u hbox is a solution of ast ragx u cdot bar x u hbox is a solution of ast end aligned qquad quad cal u ellein bbb r end equation moreover we prove that rag and ragx are compact subsets of elleuno and elleuno loc respectively whenever cal u is a set of controls which pointwise satisfy closed convex constraints together with some additional integral inequalities

l1 stability of spatially periodic solutions in relativistic gas dynamics this paper proves the well posedness of spatially periodic solutions of the relativistic isentropic gas dynamics equations the pressure is given by a law with initial data of large amplitude provided 1 is sufficiently small as a byproduct of our techniques we obtain the same results for the classical case at the limit c the solutions of the relativistic system converge to the solutions of the classical one the convergence rate being 1 c2 we also construct the semigroup of solutions of the cauchy problem for initial data with bounded total variation which can be large as long as 1 is small

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented

nonlinear superlinear singular and nonsingular second order boundary value problems abstract many new existence theorems are presented for singular and nonsingular second order boundary value problems singularities at y 0 t 0 and t 1 will be discussed

regularity and lyapunov stabilization of weak entropy solutions to scalar conservation laws we consider the problem of lyapunov boundary stabilization of the weak entropy solution to a scalar conservation law with strictly convex flux in one dimension of space around a uniform equilibrium we show that for a specific class of boundary conditions the solution to the initial boundary value problem for an initial condition with bounded variations can be approximated arbitrarily closely in the l 1 norm by a piecewise smooth solution with finitely many discontinuities the constructive method we present designs explicit boundary conditions in this class which guarantee lyapunov stability of the weak entropy solution to the initial boundary value problem we show how the greedy control obtained by maximizing the decrease of the natural lyapunov function may fail to asymptotically stabilize and a brute force control generates unbounded variation of traces we then design a stabilizing control which avoid oscillations and propose a nonlocal technique depending on time and the whole initial datum which optimizes the convergence time controllers performance is illustrated on numerical benchmarks using the godunov scheme