reachability and minimal times for state constrained nonlinear problems without any controllability assumption we consider a target problem for a nonlinear system under state constraints we give a new continuous level set approach for characterizing the optimal times and the backward reachability sets this approach leads to a characterization via a hamilton jacobi equation without assuming any controllability assumption we also treat the case of time dependent state constraints as well as a target problem for a two player game with state constraints our method gives a good framework for numerical approximations and some numerical illustrations are included in the paper

sonic hyperbolic phase transitions and chapman jouguet detonations abstract we prove that the cauchy problem for an n n system of strictly hyperbolic conservation laws in one space dimension admits a weak global solution also in presence of sonic phase boundaries applications to chapman jouguet detonations liquid vapor transitions and elastodynamics are considered

infinite horizon problems on stratifiable state constraints sets this paper deals with a state constrained control problem it is well known that unless some compatibility condition between constraints and dynamics holds the value function has not enough regularity or can fail to be the unique constrained viscosity solution of a hamilton jacobi bellman hjb equation here we consider the case of a set of constraints having a stratified structure under this circumstance the interior of this set may be empty or disconnected and the admissible trajectories may have the only option to stay on the boundary without possible approximation in the interior of the constraints in such situations the classical pointing qualification hypothesis are not relevant the discontinuous value function is then characterized by means of a system of hjb equations on each stratum that composes the state constraints this result is obtained under a local controllability assumption which is required only on the strata where some chattering phenomena could occur

regularity and large time behaviour of solutions of a conservation law without convexity a l aide de la methode des caracteristiques generalisees on etudie la regularite et le comportement a grand des solutions faibles admissibles d une seule loi de conservation a 1 variable d espace avec un point d inflexion

on the maximal smoothing effect for multidimensional scalar conservation laws abstract in 1994 lions perthame and tadmor conjectured an optimal smoothing effect for entropy solutions of multidimensional scalar conservation laws this effect estimated in fractional sobolev spaces is linked to the flux nonlinearity we use a new definition of a nonlinear smooth flux in order to show that the conjectured smoothing effect cannot be exceeded first one dimensional solutions are considered in fractional b v spaces then the multidimensional case is handled with a monophase solution

formation of diffusion waves in a scalar conservation law with convection we study the scalar conservation law ut c x u x b u2 x uxx c 0 which is a model equation for the behavior of weak transverse waves near a viscous transitional shock solutions are shown to decay in l to a pair of diffusion waves moving apart at speeds c oo and c oo behavior that has been observed numerically in solutions of the full equations the interesting aspect of the analysis is that the asymptotic state of the solution is not known a priori in contrast to cases treated previously

how parabolic free boundaries approximate hyperbolic fronts a rather complete study of the existence and qualitative behaviour of the boundaries of the support of solutions of the cauchy problem for nonlinear first order and second order scalar conservation laws is presented among other properties it is shown that under appropriate assumptions parabolic interfaces converge to hyperbolic ones in the vanishing viscosity limit

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

large time behavior for conservation laws with source in a bounded domain here u u x t u r precise assumptions will be given later we are in particular interested in the asymptotic behavior of solutions to problem 1 3 equation of the form 1 is called reaction convection equation or balance law the correct setting to deal with for such equations is the one given by entropy solutions in the sense of kruz kov in 15 in such class it is well known that the cauchy problem has a unique solution continuously depending on initial data in l norm in this article we consider boundary conditions and we analyze the influence of such conditions on the whole solution as time goes on such conditions for equations of the form 1 were considered in 1 it turns out that in order to have existence uniqueness and continuous dependence conditions 3 have to be interpreted in a non classical fashion following 1 we look for entropy solutions of 1 such that their boundary values belong to appropriate sets details will be given later on article id jdeq 1999 3669 available online at http www idealibrary com on

asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to n waves abstract the goal of this article is to develop a new technique to obtain better asymptotic estimates for scalar conservation laws general convex flux f u 0 is considered with an assumption lim u 0 uf u f u 1 we show that under suitable conditions on the initial value its solution converges to an n wave in l1 norm with the optimal convergence order of o 1 t the technique we use in this article is to enclose the solution with two rarefaction waves we also show a uniform convergence order in the sense of graphs a numerical example of this phenomenon is included

an oleinik type estimate for a convection diffusion equation and convergence to n waves in this article we propose an oleinik type estimate for sign changing solutions to a convection diffusion equation ut u x uxx u x 0 u0 x u x r 1 0 since the oleinik entropy inequality holds for nonnegative solutions or inviscid case 0 only the theoretical progress for the case was limited in this paper we show that its solution satisfies an oleinik type estimate t2 ux c 1 0 where c c u0 0 using this estimate the convergence to an n wave is proved for sign changing solutions and the theoretical gap in asymptotic convergence of the corresponding problem is filled

optimal control problems on manifolds a dynamic programming approach abstract dynamic programming identifies the value function of continuous time optimal control with a solution to the hamilton jacobi equation appropriately defined this relationship in turn leads to sufficient conditions of global optimality which have been widely used to confirm the optimality of putative minimisers in continuous time optimal control the dynamic programming methodology has been used for problems with state space a vector space however there are many problems of interest in which it is necessary to regard the state space as a manifold this paper extends dynamic programming to cover problems in which the state space is a general finite dimensional c manifold it shows that also in a manifold setting we can characterise the value function of a free time optimal control problem as a unique lower semicontinuous lower bounded generalised solution of the hamilton jacobi equation the application of these results is illustrated by the investigation of minimum time controllers for a rigid pendulum

filippov s and filippov wazewski s theorems on closed domains abstract the celebrated filippov s theorem implies that given a trajectory x1 0 r n of a differential inclusion x f t x with the set valued map f measurable in t and k lipschitz in x for any initial condition x2 0 r n there exists a trajectory x2 starting from x2 0 such that x1 t x2 t ekt x1 0 x2 0 filippov wazewski s theorem establishes the possibility of approximating any trajectory of the convexified differential inclusion x co f t x by a trajectory of the original inclusion x f t x starting from the same initial condition in the present paper we extend both theorems to the case when the state variable x is constrained to the closure of an open subset r n the latter is allowed to be non smooth we impose a generalized soner type condition on f and yielding extensions of the above classical results to infinite horizon constrained problems applications to the study of regularity of value functions of optimal control problems with state constraints are discussed as well

hamilton jacobi equations with state constraints in the present paper we consider hamilton jacobi equations of the form h x u vu 0 x e q where q is a bounded open subset of rn h is a given continuous real valued function of x s p e q x r x rn and vu is the gradient of the unknown function u we are interested in particular solutions of the above equation which are required to be supersolutions in a suitable weak sense of the same equation up to the boundary of q this requirement plays the role of a boundary condition the main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system as well from related questions in constrained geodesics

non classical riemann solvers with nucleation we introduce a new nonclassical riemann solver for scalar conservation laws with concave convex flux function this solver is based on both a kinetic relation which determines the propagation speed of undercompressive nonclassical shock waves and a nucleation crite rion which makes a choice between a classical riemann solution and a nonclassical one we establish the existence of nonclassical entropy solutions of the cauchy problem and discuss several examples of wave interactions we also show the existence of a class of solutions called splitting merging solutions which are made of two large shocks and small bv bounded variation perturbations the nucleation solvers as we call them are applied to and actually motivated by the theory of thin film flows they help explain numerical results observed for such flows

mayer problem for quantum stochastic control this paper is concerned with the value function for optimal quantum stochastic control we established the lipschitz continuity of the value function which is inherent from the lipschitz property of the set valued map describing the quantum stochastic control by using a feedback multivalued map we also prove a characterization of optimal solutions relating to a noncommutative generalization of mayer problem

on diagnosability of large fault sets in regular topology based computer systems the classical diagnosability approach has its limitation when dealing with large fault sets in large multiprocessor systems this is due to limited diagnosability of large multiprocessor systems connected using regular interconnection structures we propose an alternative approach to system diagnosis by allowing a few upper bounded number of units to be diagnosed incorrectly this measure is called t k diagnosability using this new measure it is possible to increase the degree of diagnosability of large system considerably the t k diagnosis guarantees that all the faulty units processors in a system are detected provided the number of faulty units does not exceed t while at most k units are incorrectly diagnosed we provide necessary and sufficient conditions for t k diagnosability and discuss their implication to demonstrate the power of this approach we analyze the diagnosability of large systems connected as hypercube star graph and meshes it is shown that a substantial increase in the degree of diagnosability of these structures is achieved compared with the degree of diagnosability achieved using the classic diagnosability approach at the cost of a comparably small number of incorrectly diagnosed units

a new formulation of state constraint problems for first order pdes the first order hamilton jacobi bellman equation associated with the state constraint problem for optimal control is studied instead of the boundary condition which soner introduced a new and appropriate boundary condition for the pde is proposed the uniqueness and lipschitz continuity of viscosity solutions for the boundary value problem are obtained

lower semicontinuous solutions of hamilton jacobi bellman equations the value function of mayer s problem arising in optimal control is investigated and lower semicontinuous solutions of the associated hamilton jacobi bellman equation are defined in three equivalent ways under quite weak assumptions about the control system the value function is the unique solution moreover it is stable with respect to perturbations of the control system and the cost it coincides with the viscosity solution whenever it is continuous

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

asymptotic behavior of solutions of a conservation law without convexity conditions in this paper we study the asymptotic behavior for the hyperbolic conservation law u i 0 t 0 oo x co 1 1 u x 0 uo x al x co 1 2 with riemann like data for x large the function f is a smooth nonlinear function of u in general eq 1 1 does not have a continuous solution for all time shock curves appear after finite time we will consider a piecewise continuous weak solution of 1 1 19 lo it is well known that across a discontinuity line x x t the solution satisfies the rankine hugoniot condifion r h and the entropy condition e 111

source solutions and asymptotic behavior in conservation laws abstract we study the uniqueness of the solutions to the scalar conservation law u t u x 0 when the initial datum is a finite measure the case of a dirac mass is particularly emphasized it is shown how it provides a description of the asymptotic behavior of the solutions initiated by an arbitrary integrable function this behavior is proved to depend on one parameter in the case when is odd while it depends on two when is convex

a discontinuous galerkin scheme for front propagation with obstacles we are interested in front propagation problems in the presence of obstacles we extend a previous work bokanowski et al siam j sci comput 33 2 923 938 2011 to propose a simple and direct discontinuous galerkin dg method adapted to such front propagation problems we follow the formulation of bokanowski et al siam j control optim 48 7 4292 4316 2010 leading to a level set formulation driven by min u t h x nabla u u g x 0 min u t h x u u g x 0 where g x g x is an obstacle function the dg scheme is motivated by the variational formulation when the hamiltonian h h is a linear function of nabla u u corresponding to linear convection problems in the presence of obstacles the scheme is then generalized to nonlinear equations written in an explicit form stability analysis is performed for the linear case with euler forward a heun scheme and a runge kutta third order time discretization using the technique proposed in zhang and shu siam j numer anal 48 1038 1063 2010 several numerical examples are provided to demonstrate the robustness of the method finally a narrow band approach is considered in order to reduce the computational cost

semicontinuous solutions of hamilton jacobi bellman equations with degenerate state constraints abstract in this paper the value function of a bolza optimal control problem with state constraints is characterized as the unique lower semicontinuous solution of a hamilton jacobi equation the state constraints are given by an arbitrary closed set with possibly empty interior in particular soner s inward pointing condition is extended here to the case of degenerate state constraints

dynamic programming principle of control systems on manifolds and its relations to maximum principle abstract we study the dynamic programming principle dpp for short on manifolds obtain the hamilton jacobi bellman hjb for short equation and prove that the value function is the only viscosity solution to the hjb equation then we investigate the relation between dpp and pontryagin s maximum principle pmp for short from which we obtain pmp on manifolds

adjoint and compensated compactness methods for hamilton jacobi pde we investigate the vanishing viscosity limit for hamilton jacobi pde with nonconvex hamiltonians and present a new method to augment the standard viscosity solution approach the main idea is to introduce a solution e of the adjoint of the formal linearization and then to integrate by parts with respect to the density e this procedure leads to a natural phase space kinetic formulation and also to a new compensated compactness technique