new singular solutions of the nonlinear schrodinger equation we present numerical simulations of a new type of singular solutions of the critical nonlinear schrodinger equation nls that collapse with a quasi self similar ring profile at a square root blowup rate we find and analyze the equation of the ring profile we observe that the self similar ring profile is an attractor for a large class of radially symmetric initial conditions but is unstable under symmetry breaking perturbations the equation for the ring profile admits also multi ring solutions that give rise to collapsing self similar multi ring solutions but these solutions are unstable even in the radially symmetric case and eventually collapse with a single ring profile collapsing ring solutions are also observed in the supercritical nls

a mathematical analysis of the influence of adverse and favourable winds on sprinting abstract a mathematical analysis of running performance based on the first law of thermodynamics and originally derived for still air conditions is extended to account for the effects of favourable and adverse winds solutions to the full theory have been obtained by numerical integration of the governing equations simplifications to the full calculation procedure have also been investigated calculations for races over a distance of 100 m show that the advantage to an athlete of a following wind increases progressively with increasing wind speed a favourable wind of 2 m s 1 provides a benefit of about 0 18 s on running time

technical communique optimally switched linear systems in this paper we address the problem of optimal switching for switched linear systems the uniqueness of our approach lies in describing the switching action by multiple control inputs this allows us to embed the switched system in a larger family of systems and apply pontryagin s minimum principle for solving the optimal control problem this approach imposes no restriction on the switching sequence or the number of switchings this is in contrast to search based algorithms where a fixed number of switchings is set a priori in our approach the optimal solution can be determined by solving the ensuing two point boundary value problem results of numerical simulations are provided to support the proposed method

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

uniqueness of positive radial solutions of δu f u 0 in ℝ n ii we prove a uniqueness result for the positive solution of u f u 0 in r n which goes to 0 at the result applies to a wide class of nonlinear functions f including the important model case f u u u p 1 p n 2 n 2 the result is proved by reducing to an initial boundary problem for the ode u n 1 r f u 0 and using a shooting method

optimal control and scheduling of switched systems this technical note addresses optimal control and scheduling controlled switching of discrete time switched linear systems a receding horizon control and scheduling rhcs problem is introduced and solved by dynamic programming leading to a combinatorial optimization problem with exponential complexity by relaxed dynamic programming complexity is reduced while relaxing optimality within prespecified bounds the resulting rhcs strategy is expressed explicitly as a piecewise linear state feedback control law defined over regions implied by quadratic forms closed loop stability is not guaranteed inherently for the rhcs strategy therefore a posteriori stability criteria based on piecewise quadratic lyapunov functions are proposed finally a region reachability criterion is presented

blow up of h1 solution for the nonlinear schrodinger equation we consider the blow up of the solution in h1 for the following nonlinear schrodinger equation i tu u u p 1u x rn t 0 u 0 x u0 x x rn t 0 where n 2 and 1 4 n p min n 2 n 2 5 we prove that if the initial data u0 in h1 are radially symmetric and have negative energy then the solution of in h1 blows up in finite time we do not assume that xu0 l2 and therefore our result is the generalization of the results of glassey 4 and m tsutsumi 18 for the radially symmetric case

asymptotic and limiting profiles of blowup solutions of the nonlinear schrodinger equation with critical power this paper is a sequel to previous ones 38 39 41 we continue the study of the blowup problem for the nonlinear schrodinger equation with critical power nonlinearity nsc we introduce a new idea to prove the existence of a blowup solution in h1 n without any weight condition and reduce the problem to a kind of variational problem our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions for a certain class of initial data the blowup solution behaves like a finite superposition of zero energy h1 bounded global in time solutions of nsc these singularities stay in a bounded region in n and one can see that the so called shoulder emerges outside these singularities as suggested by some numerical computations see e g 26 we investigate the asymptotic behavior of zero energy global in time solutions of nsc and find that such a solution behaves like a multisoliton however it is not an assemblage of free particles the solitons interact with each other 1999 john wiley sons inc

on an elliptic equation related to the blow up phenomenon in the nonlinear schrodinger equation this paper is devoted to the study of the asymptotic behaviour of radial solutions to an elliptic equation in n the equation is derived from the blow up problem in the non linear schrodinger equation

supersonic limit for the zakharov rubenchik system abstract we study the asymptotic behavior of the solutions of zakharov rubenchik system when appropriate parameters go to zero namely we state weak and strong convergence results of these solutions to solutions of zakharov system the proof of the weak limit is a classical argument in the theory of compactness whose main ingredient is the aubin lions theorem and the ascoli theorem strong limits are conveniently treated by decomposing the nonlinearities and using the strichartz estimates associated with the group of the schrodinger equation and the wave group

infinite horizon switched lqr problems in discrete time a suboptimal algorithm with performance analysis this paper studies the quadratic regulation problem for discrete time switched linear systems dslqr problem on an infinite time horizon a general relaxation framework is developed to simplify the computation of the value iterations based on this framework an efficient algorithm is developed to solve the infinite horizon dslqr problem with guaranteed closed loop stability and suboptimal performance due to these guarantees the proposed algorithm can be used as a general controller synthesis tool for switched linear systems

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

optimal control of switched systems based on parameterization of the switching instants this paper presents a new approach for solving optimal control problems for switched systems we focus on problems in which a prespecified sequence of active subsystems is given for such problems we need to seek both the optimal switching instants and the optimal continuous inputs in order to search for the optimal switching instants the derivatives of the optimal cost with respect to the switching instants need to be known the most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state costate stationarity equations the boundary and continuity conditions along with their differentiations this method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed an extension of the method is also applied to problems with internally forced switching examples are shown to illustrate the results in the paper

profiles and quantization of the blow up mass for critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation open image in new window for a suitable class of initial data in the energy space h1 we prove that the solution splits in two parts the first part corresponds to the singular part and accumulates a quantized amount of l2 mass at the blow up point the second part corresponds to the regular part and has a strong l2 limit at blow up time

comment on a model for the calculation of mechanical power during distance running abstract previous studies have found widely varying estimates of mechanical power for constant velocity running and corresponding widely varying efficiency ratios these results are interpreted as pointing to a need for further research before confidence in measured mechanical power could be established however newtonian mechanics applied to external power together with data on running athletes show that almost as wide a variation is within normal limits for athletes concern should therefore rather be focussed on the measurement of metabolic power and in the procedure of efficiency calculation by division

optimal control of switching systems this paper considers an optimal control problem for a switching system for solving this problem we do not make any assumptions about the number of switches nor about the mode sequence they are determined by the solution of the problem the switching system is embedded into a larger family of systems and the optimization problem is formulated for the latter it is shown that the set of trajectories of the switching system is dense in the set of trajectories of the embedded system the relationship between the two sets of trajectories 1 motivates the shift of focus from the original problem to the more general one and 2 underlies the engineering relevance of the study of the second problem sufficient and necessary conditions for optimality are formulated for the second optimization problem if they exist bang bang type solutions of the embedded optimal control problem are solutions of the original problem otherwise suboptimal solutions are obtained via the chattering lemma

a switched lq regulator design in continuous time in this technical note the design of a lq regulator for linear switched systems in continuous time is investigated from a relaxation of the optimal control problem a lyapunov based switching law is provided even if the subsystems are all unstable the state feedback switching law can be applied subject to a positiveness condition in any cases the real cost is always upper bounded by the lyapunov function value the optimality of the switching law is also discussed and we prove that the switching conditions are optimal in some generic cases this point explains why the obtained results over examples approach finely the optimal solutions finally a design strategy is also given that extends the results to the cases where the subsystems are controlled linear systems

the instability of bourgain wang solutions for the l2 critical nls we consider the two dimensional l 2 critical nonlinear schr odinger equation i partial tu delta u uu 2 0 in their pioneering 1997 work bourgain and wang have constructed smooth solutions which blow up in finite time t nabla u t l 2 sim 1 over t t and which display some decoupling between the regular and the singular part of the solution at blow up time we prove that this dynamic is unstable more precisely we show that any such solution with small super critical l 2 mass lies on the boundary of both h 1 open sets of global solutions that scatter forward and backwards in time and solutions that blow up in finite time on the right in the log log regime exhibited in work by f merle and p rapha e l we moreover exhibit some continuation properties of the scattering solution after blow up time and recover the chaotic phase behavior first exhibited in f merle s 1992 it comm pure appl math paper in the critical mass case

virial type blow up solutions for the zakharov system with magnetic field in a cold plasma abstract in this paper we study blow up solutions of virial type to the zakharov system with magnetic field in a cold plasma in r n n 2 3 after obtaining some a priori estimates on those terms generated by the magnetic field we obtain a virial type blow up result to the system under consideration the result suggests that the magnetic field in a cold plasma doesn t affect the virial type blow up character of the zakharov system

successive method for general multiple linear quadratic control problem in discrete time a successive method is proposed for the general multiple linear quadratic control problem in discrete time a family of auxiliary parametric linear quadratic control problems is constructed such that its solution sequence converges to the optimal solution of the original problem theoretical analysis for the computational procedure and global convergence of the successive method is provided the successive method utilizes the special structure of the problem thus being computationally efficient and being able to furnish a closed loop optimal control law

a mathematical theory of running based on the first law of thermodynamics and its application to the performance of world class athletes following a survey of existing mathematical models of running a new analysis is developed based on the first law of thermodynamics the method properly accounts for each term in the energy balance and avoids the use of mechanical efficiency factors a relationship is derived between race distance and the time taken to run that distance an excellent correlation of results from recent olympic games is established for events over distances from 100 m to 10 000 m the velocity time relationship for a sprinter running 100 m at maximum available power is obtained by numerical integration of the power equation it is shown that the peak velocity is achieved in the middle stages of the race a result which is consistent with practice but which previous calculations based on newton s laws have failed to predict further applications of the analysis are indicated

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

optimal lower bound estimates for the blow up rate for the zakharov system in a nonhomogeneous medium abstract in this paper we obtain lower bound estimates for the blow up rate of finite time blow up solutions to the cauchy problem for the zakharov system in a nonhomogeneous medium in two space dimensions by introducing suitable scale transformations of space and time and the use of compactness arguments we derive an optimal lower bound estimate in the energy space h 2 r 2 l 2 r 2 h 1 r 2 for the blow up rate for t near the finite blow up time t also we give an application to the virial identity for the zakharov system under study

existence of self similar blow up solutions for zakhrov equation in dimension two part i we consider the zakharov equation in space dimension two left begin array 20 c iu t delta u nu frac 1 c 0 2 n tt delta n delta left u right 2 end array right we prove the existence of blow up solutions stable self similar blow up solutions for this problem and we study various properties of these solutions

existence of blow up solutions in the energy space for the critical generalized kdv equation from these conservation laws h appears as an energy space so that it is a natural space in which to study the solutions note that p 2 is a special case for equation 2 indeed from the integrability theory see lax 14 we have for suitable u0 u0 and its derivatives with fast decay at infinity an infinite number of conservation laws the general question is to understand the dynamics induced by such equations local existence in time of solutions of 2 in the energy space is now well understood see kato 10 ginibre and tsutsumi 8 for the h theory s 32 kenig ponce and vega 11 for the l theory in the case of equation 1 and sharp h theory for 2 and bourgain 3 and 4 for the periodic case

stable blowup dynamics for the 1 corotational energy critical harmonic heat flow we exhibit a stable finite time blowup regime for the 1 corotational energy critical harmonic heat flow from 2 into a smooth compact revolution surface of 3 that reduces to the semilinear parabolic problem for a suitable class of functions f the corresponding initial data can be chosen smooth well localized and arbitrarily close to the ground state harmonic map in the energy critical topology we give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den berg hulshof and king our approach lies in the continuation of the study of the 1 equivariant energy critical wave map and schrodinger map with 2 target by merle raphael and rodnianski 2012 wiley periodicals inc

stability of blow up profile and lower bounds for blow up rate for the critical generalized kdv equation the generalized korteweg de vries equations are a class of hamiltonian systems in infinite dimension derived from the kdv equation where the quadratic term is replaced by a higher order power term these equations have two conservation laws in the energy space h 1 l 2 norm and energy we consider in this paper the critical generalized kdv equation which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in h 1 uniform in time for all h 1 solutions and thus global existence from 15 there do exist for this equation solutions u t such that u t h1 as t t where t we call them blow up solutions the question is to describe in a qualitative way how blow up occurs for solutions with l 2 mass close to the minimal mass allowing blow up and with decay in l 2 at the right we prove after rescaling and translation which leave invariant the l 2 norm that the solution converges to a universal profile locally in space at the blow up time t from the nature of this profile we improve the standard lower bound on the blow up rate for finite time blow up solutions

determination of the blow up rate for a critical semilinear wave equation in this paper we determine the blow up rate for the semilinear wave equation with critical power nonlinearity related to the conformal invariance

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

optimal linear quadratic regulator of switched systems this paper considers the optimal control problem of linear switched systems with linear quadratic lq cost or multiple lq cost by adopting an embedding transformation the switching design problem is relaxed and transformed into a traditional optimal control problem the bang bang type solutions of the embedded optimal control problems are obtained for both the positive definite lq cost case and the multiple lq cost case which are the optimal solutions to the original problems the switching sequence of modes and the switching instants can be calculated by solving a closed form optimal switching condition the optimal state feedback control law is determined simultaneously finally numerical results are provided to illustrate the effectiveness of the proposed method

on universality of blow up profile for l 2 critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation iut u u 4 nu with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for a specific set of initial data finite time blow up with a universal sharp upper bound on the blow up rate has been proved in 22 23

existence and universality of the blow up profile for the semilinear wave equation in one space dimension in this paper we consider the semilinear wave equation with a power nonlinearity in one space dimension we exhibit a universal one parameter family of functions which stand for the blow up profile in self similar variables at a non characteristic point for general initial data the proof is done in self similar variables we first characterize all the solutions of the associated stationary problem as a one parameter family then we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm in the non characteristic case and to a finite decoupled sum of such a solution in the characteristic case finally in the case where this sum is reduced to one element which is the case for noncharacteristic points we use modulation theory coupled with a nonlinear argument to show the exponential convergence in the self similar time variable of the various parameters and conclude the proof this step provides us with a result of independent interest the trapping of the solution in self similar variables near the set of stationary solutions valid also for non characteristic points the proof of these results is based on a new analysis in the self similar variable 2007 elsevier inc all rights reserved

type ii blow up for the four dimensional energy critical semi linear heat equation abstract we consider the energy critical four dimensional semi linear heat equation t u u u 3 0 we show the existence of type ii finite time blow up solutions and give a sharp description of the corresponding singularity formation these solutions concentrate a universal bubble of energy in the critical topology u t r 1 q r t u in h 1 where the blow up profile is given by the talenti aubin soliton q r 1 1 r 2 8 and with speed t t t log t t 2 as t t our approach uses a robust energy method approach developed for the study of geometrical dispersive problems raphael and rodnianski 2012 18 merle et al 2011 15 and lies in the continuation of the study of the energy critical harmonic heat flow raphael and schweyer 2011 19 and the energy critical four dimensional wave equation hillaret and raphael 2010 5

on the hybrid optimal control problem theory and algorithms a class of hybrid optimal control problems hocp for systems with controlled and autonomous location transitions is formulated and a set of necessary conditions for hybrid system trajectory optimality is presented which together constitute generalizations of the standard maximum principle these are given for the cases of open bounded control value sets and compact control value sets the derivations in the paper employ i classical variational and needle variation techniques and ii a local controllability condition which is used to establish the adjoint and hamiltonian jump conditions in the autonomous switching case employing the hybrid minimum principle hmp necessary conditions a class of general hmp based algorithms for hybrid systems optimization are presented and analyzed for the autonomous switchings case and the controlled switchings case using results from the theory of penalty function methods and ekeland s variational principle the convergence of these algorithms is established under reasonable assumptions the efficacy of the proposed algorithms is illustrated via computational examples

the orbital stability of the ground states and the singularity formation for the gravitational vlasov poisson system we study the gravitational vlasov poisson system f t v cdot nabla x f e cdot nabla v f 0 0 where e x nabla x phi x delta x phi rho x rho x int mathbb r n f x v rm d x rm d v in dimension n 3 4 in dimension n 3 where the problem is subcritical we prove using concentration compactness techniques that every minimizing sequence to a large class of minimization problems attained on steady states solutions are up to a translation shift relatively compact in the energy space this implies in particular the orbital stability in the energy space of the spherically symmetric polytropes and improves the nonlinear stability results obtained for this class in 11 16 19 in dimension n 4 where the problem is l1 critical we obtain the polytropic steady states as best constant minimizers of a suitable sobolev type inequality relating the kinetic and the potential energy we then derive using an explicit pseudo conformal symmetry the existence of critical mass finite time blow up solutions and prove more generally a mass concentration phenomenon for finite time blow up solutions this is the first result of description of a singularity formation in a vlasov setting the global structure of the problem is reminiscent of the one for the focusing nonlinear schrodinger equation iu t u u p 1u in the energy space h 1 mathbb r n

numerical approximation of blow up of radially symmetric solutions of the nonlinear schrodinger equation we consider the initial value problem for the radially symmetric nonlinear schrodin ger equation with cubic nonlinearity nls in d 2 and 3 space dimensions to approximate smooth solutions of this problem we construct and analyze a numerical method based on a standard galerkin finite element spatial discretization with piecewise linear continuous functions and on an implicit crank nicolson type time stepping procedure we then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the nls equation that blow up at the origin as the temporal variable t tends from below to a finite value t star for the blow up of the amplitude of the solution we recover numerically the well known rate t star t 1 2 for d 3 for d 2 our numerical evidence supports the validity of the log log law ln ln frac 1 t star t t star t 1 2 for t extremely close to t star the scheme also approximates well the details of the blow up of the phase of the solution at the origin as t to t star

brief paper on optimal control of non autonomous switched systems with a fixed mode sequence we consider differentiability with respect to the switch times of the value function of an optimal control problem for a non autonomous switched system the control variables are the switch times between the modes and the input in each mode we provide a method to compute the derivative of the cost function given a nominal input then we view the optimal control problem as a parametrized optimization problem in which the switch times are the parameters and the optimization is over the set of feasible inputs of each mode from this point of view we provide conditions under which the continuity and differentiability of the optimal value function that is the cost function optimized over the inputs can be guaranteed