linear nonquadratic optimal control we consider the optimization of nonquadratic measures of the transient response we present a computational implementation of dynamic programming recursions to solve finite horizon problems in the limit the finite horizon performance converges to the infinite horizon performance we provide conditions based on finite horizon computations which only assure that a receding horizon implementation of the finite horizon optimal control is stabilizing and within a specified tolerance of the infinite horizon performance

nonlinear oblique boundary value problems for nonlinear elliptic equations on considere le probleme aux valeurs limites a derivee oblique non lineaire pour des equations uniformement elliptiques quasi lineaires et totalement non lineaires d ordre 2 les operateurs elliptiques satisfont des conditions de structure naturelle on donne des theoremes d existence pour les solutions

regularity of traveling periodic stratified water waves with vorticity abstract we prove real analyticity of all the streamlines including the free surface of a steady stratified flow of water over a flat bed in the absence of stagnation points with a holder continuous bernoulli function and a holder continuously differentiable density function furthermore we show that if the bernoulli function and the density function possess some gevrey regularity of index s then the stream function admits the same gevrey regularity throughout the fluid domain in particular if the gevrey index s equals to 1 then we obtain analyticity of the stream function the regularity results hold for three distinct physical regimes capillary capillary gravity and gravity water waves

steady periodic waves and formal stability for fixed depth rotational equatorial flows abstract in this paper we apply the bifurcation theory to study the steady two dimensional periodic equatorial water waves for the f plane approximation we consider the waves with vorticity which propagate with a specified fixed depth between the thermocline and the upper boundary of the centre layer moreover we present a functional j and its first variation corresponds to the exact equations in the transformed variables using the second variation of j we prove a formal stability result for the bifurcation inducing the laminar solution

on the pressure transfer function for solitary water waves with vorticity in this paper we analyse the role which the pressure function on the sea bed plays in determining solitary waves with vorticity we prove that the pressure function on the flat bed determines a unique real analytic solitary wave solution to the governing equations given a real analytic vorticity distribution in particular the pressure function on the flat bed prescribes a unique surface profile for the resulting solitary water wave

contraction based nonlinear model predictive control formulation without stability related terminal constraints contraction based nonlinear model predictive control nmpc formulations are attractive because they generally require short prediction horizons and there is no need for the terminal set computation and reinforcement that are common requirements to guarantee stability however the inclusion of the contraction constraint in the definition of the underlying optimization problem often leads to non standard features such as a need for the multi step open loop application of control sequences or the use of multi step memorization of the contraction level which may cause unfeasibility in the presence of unexpected disturbances in this study we propose a new contraction based nmpc formulation where no contraction constraint is explicitly involved the convergence of the resulting closed loop behavior is proved under mild assumptions an illustrative example is provided in order to demonstrate the relevance of the proposed formulation

particle paths in small amplitude solitary waves with negative vorticity abstract we investigate the particle trajectories in solitary waves with vorticity where the vorticity is assumed to be negative and decrease with depth we show that the individual particle moves in a similar way as that in the irrotational case if the underlying laminar flow is favorable that is the flow is moving in the same direction as of the wave propagation throughout the fluid and show that if the underlying current is not favorable some particles in a sufficiently small solitary wave move in the opposite direction to that of wave propagation along a path with a single loop or hump

the general hyperplane section of a curve in this paper we discuss some necessary and sufficient conditions for a curve to be arithmetically cohen macaulay in terms of its general hyperplane section we obtain a characterization of the degree matrices that can occur for points in the plane that are the general plane section of a non arithmetically cohen macaulay curve of p 3 we prove that almost all the degree matrices with positive subdiagonal that occur for the general plane section of a non arithmetically cohen macaulay curve of p 3 arise also as degree matrices of some smooth integral non arithmetically cohen macaulay curve and we characterize the exceptions we give a necessary condition on the graded betti numbers of the general plane section of an arithmetically buchsbaum non arithmetically cohen macaulay curve in p for curves in p 3 we show that any set of betti numbers that satisfies that condition can be realized as the betti numbers of the general plane section of an arithmetically buchsbaum non arithmetically cohen macaulay curve we also show that the matrices that arise as a degree matrix of the general plane section of an arithmetically buchsbaum integral smooth non arithmetically cohen macaulay space curve are exactly those that arise as a degree matrix of the general plane section of an arithmetically buchsbaum non arithmetically cohen macaulay space curve and have positive subdiagonal we also prove some bounds on the dimension of the deficiency module of an arithmetically buchsbaum space curve in terms of the degree matrix of the general plane section of the curve and we prove that they are sharp

stability and feasibility of state constrained mpc without stabilizing terminal constraints abstract in this paper we investigate stability and recursive feasibility of a nonlinear receding horizon control scheme without terminal constraints and costs but imposing state and control constraints under a local controllability assumption we show that every level set of the infinite horizon optimal value function is contained in the basin of attraction of the asymptotically stable equilibrium for sufficiently large optimization horizon n for stabilizable linear systems we show the same for any compact subset of the interior of the viability kernel moreover estimates for the necessary horizon length n are given via an analysis of the optimal value function at the boundary of the viability kernel

multiple eigenvalue bifurcation for fredholm operators on fait le point sur les resultats de bifurcation locale et globale et on considere la stabilite des branches de bifurcation

turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems we analyze the connection between turnpike behaviors and strict dissipativity properties for discrete time finite dimensional linear quadratic optimal control problems we first use strict dissipativity as a sufficient condition for the turnpike property next we characterize strict dissipativity and the newly introduced property of strict pre dissipativity in terms of the system matrices related to the linear quadratic problem these characterizations then lead to new necessary conditions for the turnpike properties under consideration and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities one of the key novelties which distinguishes the results in the present paper from earlier ones on linear quadratic optimal control problems is the consideration of state and input constraints

on the existence of extreme waves and the stokes conjecture with vorticity this is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity we show that for a certain class of vorticity functions a sequence of regular waves converges to an extreme wave with stagnation points at its crests we also show that for any vorticity function the profile of an extreme wave must have either a corner of 120 or a horizontal tangent at any stagnation point about which it is supposed symmetric moreover the profile necessarily has a corner of 120 if the vorticity is nonnegative near the free surface

particle trajectories beneath small amplitude shallow water waves in constant vorticity flows abstract we investigate the particle trajectories in a constant vorticity shallow water flow over a flat bed as periodic waves propagate on the water s free surface within the framework of small amplitude waves we find the solutions of the nonlinear differential equations system which describes the particle motion in the considered case and we describe the possible particle trajectories depending on the relation between the initial data and the constant vorticity some particle trajectories are undulating curves to the right or to the left others are loops with forward drift or with backward drift others can follow some peculiar shapes

betti strata of height two ideals abstract let r k x y denote the polynomial ring in two variables over an infinite field k we study the betti strata of the family g h parametrizing graded artinian quotients of r k x y having given hilbert function h the betti stratum g h parametrizes all quotients a of having the graded betti numbers determined by h and the minimal relation degrees with i dim k tor 1 r i k i we recover that the betti strata are irreducible and we calculate their codimension in the family g h theorem the codimension of g h in g h satisfies letting i generators of degree i and i relations of degree i cod g h i i i here min i h i i 1 the initial degree of the ideals when k is algebraically closed we also show that the closure of a betti stratum is the union of more special strata and is cohen macaulay theorem 2 18 our method is to identify g h as a product of determinantal varieties ours is a more direct argument than that of a iarrobino v kanev power sums gorenstein algebras and determinantal loci lecture notes in math vol 1721 springer heidelberg 1999 345 xxvii pp theorem 5 63 which obtains the codimension of betti substrata of a postulation stratum for punctual subschemes of p 2 there we relied on a result of m boij determining the codimension of betti strata of height three gorenstein algebras m boij betti number strata of the space of codimension three gorenstein artin algebras preprint 2001 key tools throughout include properties of an invariant v the number of generators of the ancestor ideal v r and previous results concerning the projective variety g h in a iarrobino punctual hilbert schemes mem amer math soc 10 188 1977 we adapt a method of m boij that reduces the calculation of the codimension of the betti strata to showing that the most special stratum has the right codimension as applications we determine which hilbert functions are possible for artinian quotients of r given the socle type theorem 3 2 and we also determine the hilbert function of the intersection of t general enough level ideals in r each having a specified hilbert function theorem 3 6

bernoulli free boundary problems in strip like domains and a property of permanent waves on water of finite depth we study weak solutions for a class of free boundary problems which includes as a special case the classical problem of travelling gravity waves on water of finite depth we show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions we also prove that in very general situations the free boundary is necessarily the graph of a function

exact solitary water waves with vorticity the solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field the main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity each solution has a supercritical parameter value and decays exponentially at infinity the proof is based on a generalized implicit function theorem of the nash moser type the first approximation to the surface profile is given by the kdv equation with a supercritical value of the surface tension coefficient a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity

analysis and design of unconstrained nonlinear mpc schemes for finite and infinite dimensional systems we present a technique for computing stability and performance bounds for unconstrained nonlinear model predictive control mpc schemes the technique relies on controllability properties of the system under consideration and the computation can be formulated as an optimization problem whose complexity is independent of the state space dimension based on the insight obtained from the numerical solution of this problem we derive design guidelines for nonlinear mpc schemes which guarantee stability of the closed loop for small optimization horizons these guidelines are illustrated by a finite and an infinite dimensional example

local well posedness and stability of peakons for a generalized dullin gottwald holm equation abstract we establish the local well posedness for a generalized dullin gottwald holm equation by using kato s theory furthermore the orbital stability of the peaked solitary waves is also proved

on the stability of receding horizon control with a general terminal cost we study the stability and region of attraction properties of a family of receding horizon schemes for nonlinear systems using dini s theorem on the uniform convergence of functions we show that there is always a finite horizon for which the corresponding receding horizon scheme is stabilizing without the use of a terminal cost or terminal constraints after showing that optimal infinite horizon trajectories possess a uniform convergence property we show that exponential stability may also be obtained with a sufficient horizon when an upper bound on the infinite horizon cost is used as terminal cost combining these important cases together with a sandwiching argument we are able to conclude that exponential stability is obtained for input constrained receding horizon schemes with a general nonnegative terminal cost for sufficiently long horizons region of attraction estimates are also included in each of the results

the second variation of the stream function energy of water waves with vorticity abstract we compute the second variation of the stream function energy of two dimensional steady free surface gravity water waves with vorticity in the stream function formulation we prove that for nonpositive vorticity the second variation of the stream function energy at extreme waves with stokes corner asymptotics cannot be nonnegative in any small neighbourhood of a given isolated stagnation point the particular form of our second variation suggests however the possibility that certain singularities in the case of nonzero vorticity might be constructible as minimizers of the stream function energy

a new formulation of the water wave problem for stokes waves of constant vorticity we consider steady symmetric gravity water waves on finite depth with constant vorticity and a monotone surface profile between crests and troughs the problem is transformed into one concerning the vertical velocity a representation formula for the stream function in terms of the surface and the vorticity is presented and we show that the surface can be determined from the vertical velocity

modulational instability in the whitham equation with surface tension and vorticity abstract we study modulational stability and instability in the whitham equation combining the dispersion relation of water waves and a nonlinearity of the shallow water equations and modified to permit the effects of surface tension and constant vorticity when the surface tension coefficient is large we show that a periodic traveling wave of sufficiently small amplitude is unstable to long wavelength perturbations if the wave number is greater than a critical value and stable otherwise similarly to the benjamin feir instability of gravity waves in the case of weak surface tension we find intervals of stable and unstable wave numbers whose boundaries are associated with the extremum of the group velocity the resonance between the first and second harmonics the resonance between long and short waves and a resonance between dispersion and the nonlinearity for each constant vorticity we show that a periodic traveling wave of sufficiently small amplitude is unstable if the wave number is greater than a critical value and stable otherwise moreover it can be made stable for a sufficiently large vorticity the results agree with those based upon numerical computations or formal multiple scale expansions to the physical problem

examples when nonlinear model predictive control is nonrobust we consider nominal robustness of model predictive control for discrete time nonlinear systems we show by examples that when the optimization problem involves state constraints or terminal constraints coupled with short optimization horizons the asymptotic stability of the closed loop may have absolutely no robustness that is to say it is possible for arbitrarily small disturbances to keep the closed loop strictly inside the interior of the feasibility region of the optimization problem and at the same time far from the desired set point this phenomenon does not occur when using model predictive control for linear systems with convex constraint sets we emphasize that a necessary condition for the absence of nominal robustness in nonlinear model predictive control is that the value function and feedback law are discontinuous at some point s in the interior of the feasibility region

brief paper controlled invariant feasibility a general approach to enforcing strong feasibility in mpc applied to move blocking strong feasibility of mpc problems is usually enforced by constraining the state at the final prediction step to a controlled invariant set however such terminal constraints fail to enforce strong feasibility in a rich class of mpc problems for example when employing move blocking in this paper a generalized least restrictive approach for enforcing strong feasibility of mpc problems is proposed and applied to move blocking mpc the approach hinges on the novel concept of controlled invariant feasibility instead of a terminal constraint the state of an earlier prediction step is constrained to a controlled invariant feasible set controlled invariant feasibility is a generalization of controlled invariance the convergence of well known approaches for determining maximum controlled invariant sets and j step admissible sets is formally proved thus an algorithm for rigorously approximating maximum controlled invariant feasible sets is developed for situations where the exact maximum cannot be determined

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

analysis of unconstrained nonlinear mpc schemes with time varying control horizon for discrete time nonlinear systems satisfying an exponential or finite time controllability assumption we present an analytical formula for a suboptimality estimate for model predictive control schemes without stabilizing terminal constraints based on our formula we perform a detailed analysis of the impact of the optimization horizon and the possibly time varying control horizon on stability and performance of the closed loop

stability proof for nonlinear mpc design using monotonically increasing weighting profiles without terminal constraints abstract in this note a new formulation of model predictive control mpc framework with no stability related terminal constraint is proposed and its stability is proved under mild standard assumptions the novelty in the formulation lies in the use of time varying monotonically increasing stage cost penalty the main result is that the 0 reachability prediction horizon can always be made stabilizing without any terminal constraints provided that the increasing rate of the penalty is made sufficiently high moreover it is shown through an illustrative example that the time varying penalty may improve the resulting closed loop performance computed with the original stage cost when compared to the traditional mpc formulation with final constraint on the state

small amplitude stokes and solitary gravity water waves with an arbitrary distribution of vorticity this paper presents an existence theory for small amplitude stokes and solitary wave solutions to the classical water wave problem in the absence of surface tension and with an arbitrary distribution of vorticity the hydrodynamic problem is formulated as an infinite dimensional hamiltonian system in which the horizontal spatial coordinate is the time like variable a centre manifold technique is used to reduce the system to a locally equivalent hamiltonian system with one degree of freedom for values of a dimensionless parameter a near its critical value alpha the phase portrait of the reduced system contains a homoclinic orbit for alpha alpha the corresponding solutions of the water wave problem are respectively a solitary wave of elevation and a family of stokes waves c 2008 elsevier b v all rights reserved less

quadratic costs do not always work in mpc abstract we consider model predictive control mpc without terminal costs and constraints firstly we rigorously show that mpc based on quadratic stage costs may fail i e there does not exist a prediction horizon length such that a controlled equilibrium is asymptotically stable for the mpc closed loop although the system is e g finite time controllable hence stability properties of the infinite horizon optimal control problem are in general not preserved in mpc as long as purely quadratic costs are employed this shows the necessity of using the stage cost as a design parameter to achieve asymptotic stability furthermore we relax the standard controllability assumption employed in mpc without terminal costs and constraints to alleviate its verification

interaction of open and closed loop control in mpc abstract we present a novel mpc algorithm without terminal constraints and or costs for which the prediction horizon is reduced in comparison to standard mpc while asymptotic stability and inherent robustness properties are maintained to this end we derive simplified stability conditions and investigate the interplay of open and closed loop control in mpc the insight gained from this analysis allows to close the control loop more often while keeping the computational complexity of basic mpc our findings are verified by numerical simulations

model predictive control for want of a local control lyapunov function all is not lost we present stability results for unconstrained discrete time nonlinear systems controlled using finite horizon model predictive control mpc algorithms that do not require the terminal cost to be a local control lyapunov function the two key assumptions we make are that the value function is bounded by a k sub spl infin function of a state measure related to the distance of the state to the target set and that this measure is detectable from the stage cost we show that these assumptions are sufficient to guarantee closed loop asymptotic stability that is semiglobal and practical in the horizon length and robust to small perturbations if the assumptions hold with linear or locally linear k sub spl infin functions then the stability will be global or semiglobal for long enough horizon lengths in the global case we give an explicit formula for a sufficiently long horizon length we relate the upper bound assumption to exponential and asymptotic controllability using terminal and stage costs that are controllable to zero with respect to a state measure we can guarantee the required upper bound but we also require that the state measure be detectable from the stage cost to ensure stability while such costs and state measures may not be easy to construct in general we explore a class of systems called homogeneous systems for which it is straightforward to choose them in fact we show for homogeneous systems that the associated k sub spl infin functions are linear thereby guaranteeing global asymptotic stability we discuss two examples found elsewhere in the mpc literature including the discrete time nonholonomic integrator to demonstrate our methods for these systems we give a new result they can be globally asymptotically stabilized by a finite horizon mpc algorithm that has guaranteed robustness we also show that stable linear systems with control constraints can be globally exponentially stabilized using finite horizon mpc without requiring the terminal cost to be a global control lyapunov function

upper bound on the slope of steady water waves with small adverse vorticity abstract we consider the angle of inclination with respect to the horizontal of the profile of a steady 2d inviscid symmetric periodic or solitary water wave subject to gravity there is an upper bound of 31 15 in the irrotational case 1 and an upper bound of 45 in the case of favorable vorticity 13 on the other hand if the vorticity is adverse the profile can become vertical we prove here that if the adverse vorticity is sufficiently small then the angle still has an upper bound which is slightly larger than 45

exact steady periodic water waves with vorticity we consider the classical water wave problem described by the euler equations with a free surface under the influence of gravity over a flat bottom we construct two dimensional inviscid periodic traveling waves with vorticity they are symmetric waves whose profiles are monotone between each crest and trough we use bifurcation and degree theory to construct a global connected set of such solutions c 2004 wiley periodicals inc

differential inclusions and target problems where and how solutions associated to a differential inclusion can or cannot enter a given target is studied for this purpose partitions of the target boundary are associated with the dynamic of the system the behaviour of these solutions is qualitatively described in terms of viability and invariance kernels of sets these kernels determine points such that there exist respectively all solutions starting at these points remain in a given set of constraints the sets that are reached in finite time by viable solutions to the system are also studied finally some applications to control systems with one target are provided and the concept of semipermeable barriers will be generalized

brief feasibility and stability of constrained finite receding horizon control issues of feasibility and stability are considered for a finite horizon formulation of receding horizon control for linear systems under mixed linear state and control constraints we prove that given any compact set of initial conditions that is feasible for the infinite horizon problem there exists a finite horizon length above which a receding horizon policy will provide both feasibility and stability even when no end or stability constraint is imposed finally computations for determining a sufficient horizon length are carried out on a simple open loop stable example under control saturation constraints