quantum cohomology of the springer resolution abstract let g denote a complex semi simple simply connected group and b its associated flag variety we identify the equivariant quantum differential equation for the cotangent bundle t b with the affine knizhnik zamolodchikov connection of cherednik and matsuo this recovers kim s description of quantum cohomology of b as a limiting case a parallel result is proven for resolutions of the slodowy slices extension to arbitrary symplectic resolutions is discussed

boundary controllability of parabolic coupled equations abstract this paper is concerned with the boundary controllability of non scalar linear parabolic systems more precisely two coupled one dimensional parabolic equations are considered we show that in this framework boundary controllability is not equivalent and is more complex than distributed controllability in our main result we provide necessary and sufficient conditions for the null controllability

decay rates for the three dimensional linear system of thermoelasticity we consider the two and three dimensional system of linear thermoelasticity in a bounded smooth domain with dirichlet boundary conditions we analyze whether the energy of solutions decays exponentially uniformly to zero as t to infty first of all by a decoupling method we reduce the problem to an observability inequality for the lame system in linear elasticity and more precisely to whether the total energy of the solutions can be estimated in terms of the energy concentrated on its longitudinal component we show that when the domain is convex the decay rate is never uniform in fact the lack of uniform decay holds in a more general class of domains in which there exist rays of geometric optics of arbitrarily large length that are always reflected perpendicularly or almost tangentially on the boundary we also show that in three space dimensions the lack of uniform decay may also be due to a critical polarization of the energy on the transversal component of the displacement in two space dimensions we prove a sufficient and almost necessary condition for the uniform decay to hold in terms of the propagation of the transversal characteristic rays under the further assumption that the boundary of the domain does not have contacts of infinite order with its tangents we also give an example due to d hulin in which these geometric properties hold in three space dimensions we indicate without proof how a careful analysis of the polarization of singularities may lead to sharp sufficient conditions for the uniform decay to hold in two space dimensions we prove that smooth solutions decay polynomially in the energy space to a finite dimensional subspace of solutions except when the domain is a ball or an annulus finally we discuss some closely related controllability and spectral issues

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

boundary control of the timoshenko beam it is shown that the timoshenko beam can be uniformly stabilized by means of a boundary control a numerical study on the spectrum is also presented

a note on the controllability for the wave equation in nonsmooth plane domains we study exact boundary controllability for a two dimensional wave equation in a region which is an angular sector of a circle or an angular sector of an annular region the control of neumann type acts on the curved part of the boundary while in the straight part we impose homogeneous dirichlet boundary condition the initial state has finite energy and the control is square integrable

transverse vibrations of the elastically connected rectangular double membrane compound system abstract the free and forced vibrations of a system of two rectangular membranes attached together by a winkler elastic layer are studied analytically the motion of the system is described by two non homogeneous partial differential equations the solutions of the free vibrations are obtained by the bernoulli fourier method solving the boundary value problem the natural frequencies and the mode shape functions are found the initial value problem is also solved the free vibrations of the double membrane system are realised by synchronous and asynchronous deflections the forced vibrations of membranes subjected to arbitrarily distributed continuous loads are determined by using the classical method of the expansion in a series of the normal modes of vibrations discussing the vibrations caused by the harmonic exciting forces it is shown that the dynamic absorption phenomenon appears therefore the double membrane system can be used as a dynamic vibration absorber as a numerical example the vibrations of the system consisting of two identical membranes subjected to harmonic uniform distributed load are treated in detail

energy decay for the linear klein gordon equation and boundary control abstract in this work we study the asymptotic behavior of the solutions of the linear klein gordon equation in r n n 1 we prove that local energy of solutions to the cauchy problem decays polynomially afterwards we use the local decay of energy to study exact boundary controllability for the linear klein gordon equation in general bounded domains of r n n 1

rapid boundary stabilization of linear distributed systems we prove that under rather general assumptions an exactly controllable problem is uniformly stabilizable with arbitrarily prescribed decay rates our approach is direct and constructive and avoids many of the technical difficulties associated with the usual methods based on riccati equations we give several applications for the wave equation and for petrovsky systems

generalized terminal state constraint for model predictive control a terminal state equality constraint for model predictive control mpc laws is investigated where the terminal state input pair is not fixed a priori but it is a free variable in the optimization the approach named generalized terminal state constraint can be used for both tracking mpc i e when the objective is to track a given steady state and economic mpc i e when the objective is to minimize a cost function which does not necessarily attains its minimum at a steady state it is shown that the proposed technique provides in general a larger feasibility set with respect to the existing approaches given the same prediction horizon moreover a new receding horizon strategy is introduced exploiting the generalized terminal state constraint under mild assumptions the new strategy is guaranteed to converge in finite time with arbitrarily good accuracy to an mpc law with an optimally chosen terminal state constraint while still enjoying a larger feasibility set the features of the new technique are illustrated by an inverted pendulum example in both the tracking and the economic contexts

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

ingham type theorems for vector valued functions and observability of coupled linear systems we propose a new approach to study the observability of coupled linear distributed systems it is based on a generalization of some classical theorems of nonharmonic analysis to vector valued functions applying this method we answer some questions of j l lions controlabilite exacte et stabilisation de systems distribues vol 2 masson 1988 on the boundary observability of coupled linear distributed systems in particular cases

indirect boundary stabilization of weakly coupled hyperbolic systems this work is concerned with the boundary stabilization of an abstract system of two coupled second order evolution equations wherein only one of the equations is stabilized indirect damping see e g j math anal appl 173 1993 pp 339 358 we show that under a condition on the operators of each equation and on the boundary feedback operator the energy of smooth solutions of this system decays polynomially at infty we then apply this abstract result to several systems of partial differential equations wave wave systems kirchhoff petrowsky systems and wave petrowsky systems

stabilization with discounted optimal control abstract we provide a condition under which infinite horizon discounted optimal control problems can be used in order to obtain stabilizing controls for nonlinear systems the paper gives a mathematical analysis of the problem as well as an illustration by a numerical example

on exact boundary controllability for linearly coupled wave equations in this paper we study exact boundary controllability for a system of two linear wave equations coupled by lower order terms we obtain square integrable control of neuman type for initial state with finite energy in nonsmooth domains of the plane

cohomology classes of conormal bundles of schubert varieties and yangian weight functions we consider the conormal bundle of a schubert variety s i in the cotangent bundle t mathrm mathrm gr of the grassmannian mathrm mathrm gr of k planes in mathrm mathbb c n this conormal bundle has a fundamental class kappa i in the equivariant cohomology h mathrm mathbb t t mathrm mathrm gr here mathrm mathbb t mathrm mathbb c n times mathrm mathbb c the torus mathrm mathbb c n acts on t mathrm mathrm gr in the standard way and the last factor mathrm mathbb c acts by multiplication on fibers of the bundle we express this fundamental class as a sum y i of the yangian y mathfrak gl 2 weight functions w j j we describe a relation of y i with the double schur polynomial s i a modified version of the kappa i classes named kappa i satisfy an orthogonality relation with respect to an inner product induced by integration on the non compact manifold t mathrm mathrm gr this orthogonality is analogous to the well known orthogonality satisfied by the classes of schubert varieties with respect to integration on mathrm mathrm gr the classes kappa i i form a basis in the suitably localized equivariant cohomology h mathrm mathbb t t mathrm mathrm gr this basis depends on the choice of the coordinate flag in mathrm mathbb c n we show that the bases corresponding to different coordinate flags are related by the yangian r matrix

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

on the finite time performance of model predictive control abstract this paper addresses the finite time performance of model predictive control mpc for linear time invariant lti systems without constraints the performance of mpc is compared with that of finite horizon optimal control to find out how well model predictive control can perform relative to the optimal performance with the same or different horizons by exploring the properties of the riccati difference equation rde an upper and a lower bound of the ratio between the finite time performance of mpc and finite horizon optimal cost are obtained it is possible to extend the obtained results to more complicated systems such as nonlinear dynamic systems with constraints with appropriate generalizations simulation example supports our results

the turnpike property in finite dimensional nonlinear optimal control turnpike properties have been established long time ago in finite dimensional optimal control problems arising in econometry they refer to the fact that under quite general assumptions the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces the first and the last of which being transient short time arcs and the middle piece being a long time arc staying exponentially close to the optimal steady state solution of an associated static optimal control problem we provide in this paper a general version of a turnpike theorem valuable for nonlinear dynamics without any specific assumption and for very general terminal conditions not only the optimal trajectory is shown to remain exponentially close to a steady state but also the corresponding adjoint vector of the pontryagin maximum principle the exponential closedness is quantified with the use of appropriate normal forms of riccati equations we show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method or a shooting method in particular we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector not at the initial time but at the middle of the trajectory

exact controllability of wave equations in rn coupled in parallel abstract we consider a system of two parallel identical wave equations in r n coupled by a distributed internal dissipator it is shown that despite the distributed damping there exists a subspace of states which is unstable we prove an exact controllability result with dirichlet boundary controls with no restrictions on the coupling parameters with a control time that is independent of the coupling constants and dictated by geometric optics

a two level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems this work is concerned with the boundary observability of an abstract system of two coupled second order evolution equations the coupling operator being a compact perturbation of the uncoupled system we assume that only one of the two components of the unknown is observed this is indirect observability we prove that by observing only one component one can get back a full weakened energy of both components under a compatibility condition linking the operators of each equation and for small coupling using the hilbert uniqueness method we then establish an indirect exact controllability result we apply this abstract result to several coupled systems of partial differential equations wave wave coupled elastodynamic systems petrowsky petrowsky and wave petrowsky systems

an exponential turnpike theorem for dissipative discrete time optimal control problems we investigate the exponential turnpike property for finite horizon undiscounted discrete time optimal control problems without any terminal constraints considering a class of strictly dissipative systems we derive a boundedness condition for an auxiliary optimal value function which implies the exponential turnpike property two theorems illustrate how this boundedness condition can be concluded from structural properties like controllability and stabilizability of the control system under consideration

the role of sampling for stability and performance in unconstrained nonlinear model predictive control we investigate the impact of sampling on stability and performance estimates in nonlinear model predictive control without stabilizing terminal constraints or costs interpreting the sampling period as a discretization parameter the relation between continuous and discrete time estimates depending on this parameter is analyzed the technique presented in this paper allows us to determine the sampling rate required in order to approximate the continuous time suboptimality bound arbitrarily well and thus gives insight into the trade off between sampling time and guaranteed performance

some results on the controllability of coupled semilinear wave equations the desensitizing control case first we consider a semilinear hyperbolic equation with partially known initial data in a bounded domain for this system we construct a locally distributed control that desensitizes a certain norm of the state this result is new and the method for proving it combines a judicious application of the carleman estimate and a localization technique then we discuss possible extensions of our result finally we apply our method to the exact controllability of two hyperbolic equations one semilinear and the other one linear coupled in cascade through the boundary the control acting in one equation only

internal controllability of first order quasi linear hyperbolic systems with a reduced number of controls in this paper we investigate the exact controllability of n times n first order one dimensional quasi linear hyperbolic systems by m n internal controls that are localized in space in some part of the domain we distinguish two situations the first one is when the equations of the system have the same speed in this case we can use the method of characteristics and obtain a simple and complete characterization for linear systems thanks to a linear test this also provides some sufficient conditions for the local exact controllability around the trajectories of semilinear systems however when the speed of the equations is not the same we see that we encounter the problem of loss of derivatives if we try to control quasi linear systems with a reduced number of controls to solve this problem as in a prior article by coron and lissy on a navier stokes control system we first use the notion of algebraic solvability due to gromov however in contrast with this prior article where a standard fixed

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

frequency domain approach for the polynomial stability of a system of partially damped wave equations in this paper we study the stability of a system of wave equations which are weakly coupled and partially damped using a frequency domain approach based on the growth of the resolvent on the imaginary axis we establish the polynomial energy decay rate for smooth initial data we show that the behavior of the system is sensitive to the arithmetic property of the ratio of the wave propagation speeds of the two equations

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

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

explicit observability inequalities for the wave equation with lower order terms by means of carleman inequalities in this paper by means of carleman estimates and the usual energy estimate we obtain directly two observability inequalities for the linear wave equation with time variant nonsmooth lower order terms we do not need any unique continuation property of the linear equation a priori since this is actually one of the by products of our analysis furthermore the constant in the observability inequality is estimated by an explicit function of the norm of the involved coefficients in the equation also we apply our observability estimates to exact controllability for wave equations

unconstrained model predictive control and suboptimality estimates for nonlinear continuous time systems this paper presents a continuous time version of recent results on unconstrained nonlinear model predictive control mpc schemes based on a controllability assumption and a corresponding infinite dimensional optimization problem performance estimates and stability conditions are derived in terms of the prediction horizon and the sampling time of the mpc controller moreover improved estimates for small sampling times are discussed and a comparison to the application of the discrete time results in a sampled data context is provided