ground state solutions for choquard equations with hardy littlewood sobolev upper critical growth and potential vanishing at infinity abstract in this paper we investigate the following non autonomous choquard equation u v x u i u 2 k x f u 2 u 2 2 u k x f u in r n where n 3 i is the riesz potential of order 0 n 2 n n 2 is the upper critical exponent due to the hardy littlewood sobolev inequality the functions v k c r n r may vanish at infinity and the function f c r r is noncritical and f t 0 t f s d s based on variational methods and the hardy type inequalities using the mountain pass theorem and the nehari manifold approach we prove the existence of ground state solution

constrained receding horizon predictive control for systems with disturbances a receding horizon predictive control method for systems with input constraints and disturbances is proposed a polyhedral feasible set of states which is invariant with respect to a given state feedback control law is derived in the presence of bounded disturbances the proposed predicted control algorithm deploys a strategy in which the current state is steered into the polyhedral invariant feasible set within a finite number n of feasible control moves despite the presence of disturbances the future control moves over the horizon n are represented as the sum of the state feedback control and a perturbation the perturbation term provides the degrees of freedom with which to enlarge the stabilizable set of initial states the control algorithm is formulated in linear matrix inequalities so that it can be solved using semidefinite programming

brief paper convergence properties of constrained linear system under mpc control law using affine disturbance feedback this paper shows new convergence properties of constrained linear discrete time system with bounded disturbances under model predictive control mpc law the mpc control law is obtained using an affine disturbance feedback parametrization with an additional linear state feedback term this parametrization has the same representative ability as some recent disturbance feedback parametrization but its choice together with an appropriate cost function results in a different closed loop convergence property more exactly the state of the closed loop system converges to a minimal invariant set with probability one deterministic convergence to the same minimal invariant set is also possible if a less intuitive cost function is used numerical experiments are provided that validate the results

existence and concentration of ground state solutions for choquard equations involving critical growth and steep potential well abstract in the present paper we are interested in the following choquard type equation u v x u i u 2 u 2 2 u u p 2 u in r 3 where p 4 6 r r is a constant such that the operator l v x is non degenerate i is a riesz potential of order 0 3 2 3 is the upper critical exponent due to the hardy littlewood sobolev inequality the function v c r 3 r is nonnegative and has a potential well int v 1 0 additionally the operator has a sequence of dirichlet eigenvalues in h 0 1 expressed as 0 1 2 n n if 1 via the nehari manifold techniques and the ekeland variational principle we prove the existence and concentration of positive ground state solutions for sufficiently large if 1 and j for all j n employing the nehari pankov manifold methods and the constrained minimization arguments we obtain the existence of ground state solution for large enough and verify the asymptotic behaviour of ground state solutions as

singularly perturbed critical choquard equations abstract in this paper we study the semiclassical limit for the singularly perturbed choquard equation e 2 u v x u e 3 r 3 q y g u y x y d y q x g u in r 3 where 0 3 e is a positive parameter v q are two continuous real function on r 3 and g is the primitive of g which is of critical growth due to the hardy littlewood sobolev inequality under suitable assumptions on g we first establish the existence of ground states for the critical choquard equation with constant coefficients next we establish existence and multiplicity of semi classical solutions and characterize the concentration behavior by variational methods

groundstates for a local nonlinear perturbation of the choquard equations with lower critical exponent abstract we prove the existence of ground state solutions by variational methods to the nonlinear choquard equations with a nonlinear perturbation u u i u n 1 u n 1 u f x u in r n where n 1 i is the riesz potential of order 0 n the exponent n 1 is critical with respect to the hardy littlewood sobolev inequality and the nonlinear perturbation f satisfies suitable growth and structural assumptions

brief paper adaptive model predictive control for a class of constrained linear systems based on the comparison model this paper proposes an adaptive model predictive control mpc algorithm for a class of constrained linear systems which estimates system parameters on line and produces the control input satisfying input state constraints for possible parameter estimation errors the key idea is to combine the robust mpc method based on the comparison model with an adaptive parameter estimation method suitable for mpc to this end first a new parameter update method based on the moving horizon estimation is proposed which allows to predict an estimation error bound over the prediction horizon second an adaptive mpc algorithm is developed by combining the on line parameter estimation with an mpc method based on the comparison model suitably modified to cope with the time varying case this method guarantees feasibility and stability of the closed loop system in the presence of state input constraints a numerical example is given to demonstrate its effectiveness

efficient robust optimization for robust control with constraints this paper proposes an efficient computational technique for the optimal control of linear discrete time systems subject to bounded disturbances with mixed linear constraints on the states and inputs the problem of computing an optimal state feedback control policy given the current state is non convex a recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program while the reparameterized problem is theoretically tractable the number of variables is quadratic in the number of stages or horizon length n and has no apparent exploitable structure leading to computational time of mathcal o n 6 per iteration of an interior point method we focus on the case when the disturbance set is norm bounded or the linear map of a hypercube and the cost function involves the minimization of a quadratic cost here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem that is the policy optimization problem may be decomposed into a set of coupled finite horizon control problems this decomposition can then be formulated as a highly structured quadratic program solvable by primal dual interior point methods in which each iteration requires mathcal o n 3 time this cubic iteration time can be guaranteed using a riccati based block factorization technique which is standard in discrete time optimal control numerical results are presented using a standard sparse primal dual interior point solver that illustrate the efficiency of this approach

causal state feedback parameterizations in robust model predictive control in this paper we investigate the problem of nonlinearity and non convexity typically associated with linear state feedback parameterizations in the robust model predictive control rmpc for uncertain systems in particular we propose two tractable approaches to compute an rmpc controller consisting of both a causal state feedback gain and a control perturbation component for linear discrete time systems involving bounded disturbances and norm bounded structured model uncertainties along with hard constraints on the input and state both the state feedback gain and the control perturbation are explicitly considered as decision variables in the online optimization while avoiding nonlinearity and non convexity in the formulation the proposed rmpc controller computed through lmi optimizations is responsible for steering the uncertain system state to a terminal invariant set numerical examples from the literature demonstrate the advantages of the proposed scheme

robust constrained predictive control using comparison model this paper proposes a quadratic programming qp approach to robust model predictive control mpc for constrained linear systems having both model uncertainties and bounded disturbances to this end we construct an additional comparison model for worst case analysis based on a robust control lyapunov function rclf for the unconstrained system not necessarily an rclf in the presence of constraints this comparison model enables us to transform the given robust mpc problem into a nominal one without uncertain terms based on a terminal constraint obtained from the comparison model we derive a condition for initial states under which the ultimate boundedness of the closed loop is guaranteed without violating state and control constraints since this terminal condition is described by linear constraints the control optimization can be reduced to a qp problem

input to state stability of robust receding horizon control with an expected value cost this paper is concerned with the stability of a class of receding horizon control rhc laws for constrained linear discrete time systems subject to bounded state disturbances and convex state and input constraints the paper considers the class of finite horizon feedback control policies parameterized as affine functions of the system state calculation of which can be shown to be tractable via a convex reparameterization when minimizing the expected value of a finite horizon quadratic cost we show that the value function is convex when solving this optimal control problem at each time step and implementing the result in a receding horizon fashion we provide sufficient conditions under which the closed loop system is input to state stable iss

parameterized tube model predictive control this paper develops a parameterized tube model predictive control mpc synthesis method the most relevant novel feature of our proposal is the online use of a single tractable linear program that optimizes parameterized minkowski decomposable state and control tubes and an associated fully separable nonlinear control policy the induced control policy enjoys a higher degree of nonlinearity than existing tube mpc and robust mpc using disturbance affine control policy our proposal offers greater generality than the state of the art robust mpc methods it is conjectured and also established in three cases that our proposal is equivalent feasibility wise to dynamic programming dp it is also shown that under natural assumptions our method is computationally efficient while it possesses rather strong system theoretic properties

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

model predictive control using segregated disturbance feedback this paper proposes a new control parametrization under the model predictive control mpc framework for constrained linear discrete time systems with bounded disturbances the proposed parametrization takes the form of a special piecewise affine disturbance feedback in an effort to reduce conservatism it is a generalization of linear disturbance feedback parametrization introduced in the recent literature numerical computations and stability properties of the resulting mpc problem using the proposed parametrization are discussed when the disturbance set and the problem data satisfy mild assumptions the associated finite horizon optimization can be computed efficiently and exactly the advantage of the proposed parametrization over linear disturbance feedback is illustrated via numerical examples

ground state solutions for a class of choquard equations with potential vanishing at infinity abstract this paper is dedicated to studying the following nonlinear choquard equation u v x u r n q y f u y x y d y q x f u u d 1 2 r n where n 3 0 n v c r n 0 q c r n 0 f c r r and f t 0 t f s d s by combining the non nehari manifold approach with some new inequalities we prove that the above equation has a ground state solution in the case when v x 0 as x where the strict monotonicity condition on f is not required moreover by using perturbation method we obtain the existence of a least energy solution in the zero mass case i e v 0 where f satisfies the condition that f t 0 0 for some t 0 r instead of the usual ambrosetti rabinowitz type condition these results extend the ones in alves figueiredo and yang 2015 1 and some related literature

classification of positive solitary solutions of the nonlinear choquard equation in this paper we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary choquard equation delta u u 2u left frac 1 x u 2 right 0 quad u in h 1 mathbb r 3 which can be considered as a certain approximation of the hartree fock theory for a one component plasma as explained in lieb and lieb simon s papers starting from 1970s we first prove that all the positive solutions of this equation must be radially symmetric and monotone decreasing about some fixed point interestingly to use the new method of moving planes introduced by chen li ou we deduce the problem into an elliptic system as a key step we transform this differential system into a system of integral equations with the help of riesz and bessel potentials and then use the method of a moving plane in an integral form next using radial symmetry we deduce the uniqueness result from lieb s work our argument can be adapted well to study the radial symmetry of positive solutions of the equation in the generalized form u k 1 f left u k 2 u right

receding horizon output feedback control for constrained uncertain systems using periodic invariance in this article we consider a receding horizon output feedback control rhoc method for linear discrete time systems with polytopic model uncertainties and input constraints first we derive a set of estimator gains and then we obtain on the basis of the periodic invariance a series of state feedback gains stabilising the augmented output feedback system with these estimator gains these procedures are formulated as linear matrix inequalities an rhoc strategy is proposed based on these state feedback and state estimator gains in conjunction with their corresponding periodically invariant sets the proposed rhoc strategy enhances the performance in comparison with the case in which static periodic gains are used and increases the size of the stabilisable region by introducing a degree of freedom to steer the augmented state into periodically invariant sets

an approach to symmetrization via polarization we prove that the steiner symmetrization of a function can be approximated in l p r n by a sequence of very simple rearrangements which are called polarizations this result is exploited to develop elementary proofs of many inequalities including the isoperimetric inequality in euclidean space in this way we also obtain new symmetry results for solutions of some varia tional problems furthermore we compare the solutions of two boundary value problems one of them having a polarized geometry and we show some point wise inequalities between the solutions this leads to new proofs of well known functional inequalities which compare the solutions of two elliptic or parabolic problems one of them having a steiner symmetrized geometry the method also allows us to investigate the case of equality in the inequalities roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry

a linear programming approach to constrained robust predictive control a receding horizon predictive control algorithm for systems with model uncertainty and input constraints is developed the proposed algorithm adopts the receding horizon dual mode i e free control moves and invariant set paradigm the approach is novel in that it provides a convenient way of combining predictions of control moves which are optimal in the sense of worst case performance with large target invariant sets thus the proposed algorithm has a large stabilizable set of states corresponding to a cautious state feedback law while enjoying the good performance of a tightly tuned but robust control law unlike earlier approaches which are based on qp or semidefinite programming here computational complexity is reduced through the use of lp

infinite horizon performance bounds for uncertain constrained systems we present a new method to bound the performance of controllers for uncertain linear systems with mixed state and input constraints and bounded disturbances we take as a performance metric either an expected value or minimax discounted cost over an infinite horizon and provide a method for computing a lower bound on the achievable performance of any causal control policy in either case our lower bound is compared to an upper performance bound provided by restricting the choice of controller to one that is affine in the observed disturbances and we show that the two bounds are closely related in particular the lower bounds have a natural interpretation in terms of affine control policies that are optimal for a problem with a restricted disturbance set we show that our performance bounds can be computed via solution of a finite dimensional convex optimization problem and provide numerical examples to illustrate the efficacy of our method

striped parameterized tube model predictive control a modification of the parameterized tube model predictive control ptmpc strategy for linear systems with additive disturbances is proposed which reduces the dependence of the number of optimization variables on horizon length from quadratic to linear by using a triangular striped prediction structure unlike ptmpc which assumes a fixed linear terminal feedback law for predictions the proposed prediction scheme allows disturbance compensation to extend beyond the initial n step prediction horizon the resulting scheme can potentially outperform ptmpc in terms of the size of the domain of attraction and allows for a longer horizon n for the same computational demand

output feedback receding horizon control of constrained systems this paper considers output feedback control of linear discrete time systems with convex state and input constraints which are subject to bounded state disturbances and output measurement errors we show that the non convex problem of finding a constraint admissible affine output feedback policy over a finite horizon to be used in conjunction with a fixed linear state observer can be converted to an equivalent convex problem when used in the design of a time varying robust receding horizon control law we derive conditions under which the resulting closed loop system is guaranteed to satisfy the system constraints for all time given an initial state estimate and bound on the state estimation error when the state estimation error bound matches the minimal robust positively invariant mrpi set for the system error dynamics we show that this control law is time invariant but its calculation generally requires solution of an infinite dimensional optimization problem finally using an invariant outer

existence of solutions for a class of nonlinear schrodinger equations with potential vanishing at infinity abstract in this paper we investigate the existence of positive ground state solution for the following class of elliptic equations u v x u k x f u in r n where n 3 v k are nonnegative continuous functions and f is a continuous function with a quasicritical growth here we prove a hardy type inequality and use it together with variational method to get a ground state solution

optimization over state feedback policies for robust control with constraints this paper is concerned with the optimal control of linear discrete time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input it is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence this implies that a broad class of constrained finite horizon robust and optimal control problems where the optimization is over affine state feedback policies can be solved in a computationally efficient fashion using convex optimization methods this equivalence result is used to design a robust receding horizon control rhc state feedback policy such that the closed loop system is input to state stable iss and the constraints are satisfied for all time and all allowable disturbance sequences the cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance free state and input sequences the value of the receding horizon control law can be calculated at each sample instant using a single tractable and convex quadratic program qp if the disturbance set is polytopic or a tractable second order cone program socp if the disturbance set is given by a 2 norm bound

a positive ground state solution for a class of asymptotically periodic schrodinger equations in this paper by using the reformative conditions a class of asymptotically periodic schrodinger equations are studied via the variational method a positive ground state solution is obtained

monotone optimal control of permutable gsmps we consider markovian gsmps generalized semi markov processes in which the rates of events are subject to control a control is monotone if the rate of one event is increasing or decreasing in the number of occurrences of other events we give general conditions for the existence of monotone optimal controls the conditions are functional properties for the one step cost functions and more importantly structural properties for the gsmp the main conditions on costs are submodularity or supermodularity with respect to pairs of events the key structural condition is strong permutability requiring that the state at any time be determined by the number of events of each type that have occurred regardless of their order this permits a reformulation of the original control problem into one based only on event counting processes this reformulation leads to a unified treatment of a broad class of models and to meaningful generality beyond existing results

control of admission and routing in loss networks hybrid dynamic programming equations call admission and routing control decisions in stochastic loss circuit switched networks with semi markovian multi class call arrival and general connection time processes are formulated as optimal stochastic control problems the resulting so called hybrid dynamic programming equation systems take the form of vectors of partial differential equations with each component associated to a distinct distribution of routed calls over the network i e distinct occupation states this framework reduces to that of a markov decision process when the traffic is poisson and the associated computational limitations are approximately those of linear programs examples are provided of i network state space constructions and controlled state transition processes ii a new closed form solution for a simple network and iii the analysis and illustrative numerical results for a three link network

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

existence of groundstates for a class of nonlinear choquard equations we prove the existence of a nontrivial solution h n to the nonlinear choquard equation i in n where i is a riesz potential under almost necessary conditions on the nonlinearity in the spirit of berestycki and lions this solution is a groundstate if moreover is even and monotone on 0 then is of constant sign and radially symmetric

optimal control of two interacting service stations optimal controls described by switching curves in the two dimensional state space are shown to exist for the optimal control of a markov network with two service stations and linear cost the controls govern routing and service priorities finite horizon and long run average cost problems are considered and value iteration is a key tool nonconvex value functions are shown to exist for slightly more general networks nonconvex value functions are also shown to arise for a simple single station control problem in which the instantaneous cost is convex but not monotone nevertheless optimality of threshold policies is established for the single station problem the proof is based on a novel use of stochastic coupling and policy iteration

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

large population cost coupled lqg problems with nonuniform agents individual mass behavior and decentralized varepsilon nash equilibria we consider linear quadratic gaussian lqg games in large population systems where the agents evolve according to nonuniform dynamics and are coupled via their individual costs a state aggregation technique is developed to obtain a set of decentralized control laws for the individuals which possesses an epsiv nash equilibrium property a stability property of the mass behavior is established and the effect of inaccurate population statistics on an isolated agent is also analyzed by variational techniques