the velocity tracking problem for navier stokes flows with bounded distributed controls we present some systematic approaches to the mathematical analysis and numerical approximation of the time dependent optimal control problem of tracking the velocity for navier stokes flows in bounded two dimensional domains with bounded distributed controls we study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined we also define and analyze semidiscrete in time and fully space time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system the results of some computational experiments are provided

a penalized neumann control approach for solving an optimal dirichlet control problem for the navier stokes equations we introduce a penalized neumann boundary control approach for solving an optimal dirichlet boundary control problem associated with the two or three dimensional steady state navier stokes equations we prove the convergence of the solutions of the penalized neumann control problem the suboptimality of the limit and the optimality of the limit under further restrictions on the data we describe the numerical algorithm for solving the penalized neumann control problem and report some numerical results

maximum principle of state constrained optimal control governed by fluid dynamic systems this work deals with maximum principle of optimal control governed by some fluid dynamic systems which include three dimensional navier stokes equations some type of state constraint is involved

optimal controls of 3 dimensional navier stokes equations with state constraints this work is concerned with the maximum principles for optimal control problems governed by 3 dimensional navier stokes equations some types of state constraints time variables are considered

analysis and approximation for linear feedback control for tracking the velocity in navier stokes flows abstract some systematic approaches to the mathematical formulation and numerical resolution of the linear feedback control problem for tracking the velocity in navier stokes flows in a bounded two dimensional domain with bounded distributed control are presented semidiscrete in time and full space time discrete approximations are also studied some computational results are presented and compared with analogous results from optimal control theory

unipotent elements tilting modules and saturation the results in this paper were motivated by work of serre 19 where semisimplicity results for representations of arbitrary groups in positive characteristic were established a key ingredient in the arguments was the notion of saturation where one embeds a unipotent element of prime order of a simple algebraic group in a 1 dimensional unipotent subgroup one of our goals here is to show that saturation can be achieved rather generally and to establish a uniqueness result for the resulting unipotent subgroups as a by product we show for a simple algebraic group in good characteristic a unipotent element of prime order is contained in a particularly nice subgroup of type a1 and that the lie algebra of the ambient algebraic group usually affords a tilting module for this subgroup the tilting decompositions are applied to show that the centralizers of the unipotent element a corresponding 1 dimensional unipotent group and its lie algebra all coincide and are closely related to the centralizer of the a1 subgroup we also establish a convenient factorization of the centralizer of the unipotent element results for finite groups of lie type are also obtained

pontryagin s maximum principle for optimal control of the stationary navier stokes equations this paper deals with pontryagin s maximum principle of the optimal control governed by stationary navier stokes equation some kind of state constraint is involved

pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint over all y u y mad 0 t u subject to the semilinear equation y t ay t by t du t f t a e t 0 t 1 1 with state constraint y 0 y t s 1 2 the following hypotheses will be e9ective throughout this paper h1 v and h are real hilbert spaces with norms and moreover v h v algebraically and topologically with compact injections where v denotes the dual of

pontryagin maximum principle for optimal control of variational inequalities in this paper we investigate optimal control problems governed by variational inequalities we present a method for deriving optimality conditions in the form of pontryagin s principle the main tools used are the ekeland s variational principle combined with penalization and spike variation techniques

optimal control of navier stokes equations by oseen approximation a non standard sequential quadratic programming type iterational process based on oseen s approximation is proposed and analyzed to solve an optimal control problem for the steady state navier stokes equations further numerical approximation by a finite element method and sample computational experiments are presented too

extensions for frobenius kernels 1 1 let g be a simple algebraic group over an algebraically closed field k of characteristic p 0 for r 1 let gr be the rth frobenius kernel of g it is well known that the representations for g1 are equivalent to the restricted representations for lie g historically the cohomology for frobenius kernels has been best understood for large primes friedlander and parshall fp first computed the cohomology ring h g1 k for p 3 h 1 where h is the coxeter number of the underlying root system they proved that the cohomology ring can be identified with the coordinate algebra of the nullcone andersen and jantzen aj later verified this fact for p h furthermore they generalized this calculation by looking at h g1 h0 where h0 indb for p h their results had some restrictions on the type of root system involved kumar lauritzen and thomsen klt removed the restrictions on the root systems through the use of frobenius splittings the cohomology ring h g1 k modulo nilpotents can be identified in general with the coordinate algebra of the restricted nullcone n1 x lie g x p 0 for good primes nakano parshall and vella npv proved that this variety is irreducible and can be identified with the closure of some richardson orbit recently carlson lin nakano and parshall clnp have given an explicit description of n1 these recent results provide some indication that one can systematically study extensions of frobenius kernels for small primes by using general formulas which exhibit generic behavior for large primes

existence of optimal controls for viscous flow problems a class of optimal control problems in viscous flow is studied main result is the existence theorem for optimal control three typical flow control problems are formulated within this general class

extensions for finite chevalley groups ii let g be a semisimple simply connected algebraic group defined and split over the field fp with p elements let g fq be the finite chevalley group consisting of the fq rational points of g where q p r and let g r be the rth frobenius kernel the purpose of this paper is to relate extensions between modules in mod g f q and mod g r with extensions between modules in mod g among the results obtained are the following for r 2 and p 3 h 1 the g f q extensions between two simple g f q modules are isomorphic to the g extensions between two simple p r restricted g modules with suitably twisted highest weights for p 3 h 1 we provide a complete characterization of h 1 g f q h0 where h 0 ind g b a and a is p r restricted furthermore for p 3 h 1 necessary and sufficient bounds on the size of the highest weight of a g module v are given to insure that the restriction map h 1 g v h 1 g f q v is an isomorphism finally it is shown that the extensions between two simple p r restricted g modules coincide in all three categories provided the highest weights are close together

stability enhancement by boundary control in 2 d channel flow in this paper we stabilize the parabolic equilibrium profile in a two dimensional 2 d channel flow using actuators and sensors only at the wall the control of channel flow was previously considered by speyer and coworkers and bewley and coworkers who derived feedback laws based on linear optimal control and implemented by wall normal actuation with an objective to achieve global lyapunov stabilization we arrive at a feedback law using tangential actuation using teamed pairs of synthetic jets or rotating disks and only local measurements of wall shear stress allowing to embed the feedback in microelectromechanical systems mems hardware without need for wiring this feedback is shown to guarantee global stability in at least h sup 2 norm which by sobolev s embedding theorem implies continuity in space and time of both the flow field and the control as well as their convergence to the desired steady state the theoretical results are limited to low values of reynolds number however we present simulations that demonstrate the effectiveness of the proposed feedback for values five order of magnitude higher

h control theory of fluid dynamics robust or h control theory has been an expanding subject in recent years because of its wide applications and its close connection to operator theory on hardy spaces we develop an hcontrol theor

pontryagin s principle for state constrained control problems governed by parabolic equations with unbounded controls this paper deals with optimal control problems governed by semilinear parabolic equations with pointwise state constraints and unbounded controls under some strong stability assumption we obtain necessary optimality conditions in the form of a pontryagin s minimum principle in qualified form a pontryagin s principle in nonqualified form is also proved without any stability condition

an optimal control problem in exterior hydrodynamics in this paper we consider the problem of accelerating an obstacle in an incompressible viscous fluid from rest to a given speed in a given time with minimum energy expenditure an existence theorem for the speed trajectory which corresponds to the absolute minimum is provided the results are valid for arbitrary reynolds numbers

necessary and sufficient conditions for optimal controls in viscous flow problems a class of optimal control problems in viscous flow is studied main results are the pontryagin maximum principle and the verification theorem for the hamilton jacobi bellman equation characterising the feedback problem the maximum principle is established by two quite different methods

the time optimal control of navier stokes equations the maximum principle for the time optimal control of the navier stokes equations in 2 d is established

an analogue of the jacobson morozov theorem for lie algebras of reductive groups of good characteristics let g be the lie algebra of a connected reductive group g over an algebraically closed field of characteristic p 0 suppose that g 1 is simply connected and p is good for the root system of g given a one dimensional torus a c g let g a i denote the weight component of ad i corresponding to weight i e x i z it is proved in the paper that for any nonzero nilpotent element e e g there is a one dimentional torus ae c g such that e e g ae 2 and kerade c fi og ae i

irreducible representations of lie algebras of reductive groups and the kac weisfeiler conjecture letg be the lie algebra of a connected reductive groupg over an algebraically closed field of characteristicp 0 suppose thatg 1 is simply connected andp is good for the root system ofg ifp 2 suppose in addition thatg admits a nondegenerateg invariant trace form letv be an irreducible and faithfulg module withp character g it is proved in the paper that dimv is divisible byp1 2dim where stands for the orbit of under the coadjoint action ofg

optimal controls of boussinesq equations with state constraints abstract this work is concerned with the maximum principle for an optimal control problem governed by boussinesq equations some integral type state constraints are considered

a1 type overgroups of elements of order p in semisimple algebraic groups and the associated finite groups keywords semisimple algebraic groups closed connected subgroups finite groups fixed point subgroups reference ctg article 1995 001doi 10 1006 jabr 1995 1285 record created on 2008 12 16 modified on 2017 05 12

viscosity solutions of dynamic programming equations for the optimal control of the two dimensional navier stokes equations in this paper we study the infinite dimensional hamilton jacobi equation associated with the optimal feedback control of viscous hydrodynamics we resolve the global unique solvability problem of this equation by showing that the value function is the unique viscosity solution

the pontryagin maximum principle from dynamic programming and viscosity solutions to first order partial differential equations on demontre le principe de maximum de pontryagin pour le probleme de lagrange de commande optimale en utilisant le fait que la fonction valeur du probleme est la solution viscosite de l equation de hamilton jacobi bellman associee

parametrizing nilpotent orbits via bruhat tits theory let k denote a field with nontrivial discrete valuation we assume that k is complete with perfect residue field let g be the group of k rational points of a reductive linear algebraic group defined over k let g denote the lie algebra of g fix r r subject to some restrictions we show that the set of distinguished degenerate moy prasad cosets of depth r up to an equivalence relation parametrizes the nilpotent orbits in g

dynamic programming of the navier stokes equations a class of optimal control problems in viscous flow is studied main results are the pontryagin maximum principle and the verification theorem for the hamilton jacobi bellman equation

the necessary conditions for optimal control in hilbert spaces obtention du principe maximum de pontryagin pour une classe generale de problemes de commande de type lagragien afin qu associe avec un ensemble de conditions necessaires une commande soit satisfaite le principe est demontre en utilisant la programmation dynamique et l equation resultante d hamilton jacobi bellman de dimension finie

robin type boundary control problems for the nonlinear boussinesq type equations abstract in this paper we study a linear and a nonlinear boundary control problems arising from viscous flows the equations are of nonlinear navier stokes type for the velocity and pressure of transport diffusion type for the temperature and the salinity the essential difficulties are due to the nonlinear nature of a part of the boundary conditions and to the nature of the equations time dependent coupled and nonlinear the existence and the conditions of the uniqueness of the solution for the variational problem are studied the control is of linear or nonlinear robin type and acts on a part of the boundary during a time t the cost function measures the distance between the observed and the computed vorticity the existence of an optimal control in the admissible set of states and controls is proved a first order necessary conditions of optimality are obtained

optimal boundary control of a simplified ericksen leslie system for nematic liquid crystal flows in 2d in this paper we investigate an optimal boundary control problem for a two dimensional simplified ericksen leslie system modelling the incompressible nematic liquid crystal flows the hydrodynamic system consists of the navier stokes equations for the fluid velocity coupled with a convective ginzburg landau type equation for the averaged molecular orientation the fluid velocity is assumed to satisfy a no slip boundary condition while the molecular orientation is subject to a time dependent dirichlet boundary condition that corresponds to the strong anchoring condition for liquid crystals we first establish the existence of optimal boundary controls then we show that the control to state operator is frechet differentiable between appropriate banach spaces and derive first order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables

nilpotent orbits in good characteristic and the kempf rousseau theory let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 image and suppose that p is a good prime for the root system of g in this paper we give a fairly short conceptual proof of pommerening s theorem pommerening j algebra 49 1977 525 536 j algebra 65 1980 373 398 which states that any nilpotent element in image is richardson in a distinguished parabolic subalgebra of the lie algebra of a levi subgroup of g as a by product we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent g orbits in image for earlier proofs of this theorem see kawanaka invent math 84 1986 575 616 premet trans amer math soc 347 1995 2961 2988 spaltenstein j fac sci univ tokyo sect ia math 31 1984 283 286 we extend recent results of sommers internal math res notices 11 1998 539 562 to reductive lie algebras of good characteristic thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in image and unipotent elements in g earlier computations of these groups in positive characteristics relied mostly on work of mizuno j fac sci univ tokyo sect ia math 24 1977 525 563 tokyo j math 3 1980 391 459 our approach is based on the theory of optimal parabolic subgroups for g unstable vectors also known as the kempf rousseau theory which provides a good substitute for the image theory prominent in the characteristic zero case

a criterion for burn in that balances mean residual life and residual variance optimum burn in times have been determined for a variety of criteria such as mean residual life and conditional survival in this paper we consider a residual coefficient of variation that balances mean residual life with residual variance to study this quantity we develop a general result concerning the preservation ofbathtub distributions using this result we give a condition so that the residual coefficient of variation is bathtub shaped furthermore we show that it attains its optimum value at a time that occurs after the mean residual life function attains its optimum value but not necessarily before the change point of the failure rate function

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

boundary value problems and optimal boundary control for the navier stokes system the two dimensional case we study optimal boundary control problems for the two dimensional navier stokes equations in an unbounded domain control is effected through the dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with minimal regularity satisfying the energy estimates an objective of interest is the drag functional we first establish three important results for inhomogeneous boundary value problems for the navier stokes equations namely we identify the trace space for the velocity fields possessing finite energy we prove the existence of a solution for the navier stokes equations with boundary data belonging to the trace space and we identify the space in which the stress vector along the boundary of admissible solutions is well defined then we prove the existence of an optimal solution over the control set finally we justify the use of lagrange multiplier principles derive an optimality system of equations in the weak sense from which optimal states and controls may be determined and prove that the optimality system of equations satisfies in appropriate senses a system of partial differential equations with boundary values

existence and characterization of an optimal control for the problem of long waves in a shallow water model in this paper we present a method of optimal control developed in order to calculate the current corresponding to the observed sea level in a fluid domain omega and during a time t the control is the external stress f the cost function measures the distance between the observed and computed sea levels the equations satisfied by the depth and the depth averaged velocity are of nonlinear shallow water type the existence and uniqueness of a solution for the direct problem are studied in the case of dirichlet nonhomogeneous boundary conditions we prove by means of minimizing sequences the existence of an optimal control f u in the case of the small data and a very viscous fluid to characterize it we build a sequence of problems corresponding to a linearization of the direct problem we obtain the necessary conditions of optimality the set of equations and the inequality characterizing the optimal control f u is obtained as the limit of the penalization

pontryagin s principle for local solutions of control problems with mixed control state constraints this paper deals with optimal control problems of semilinear parabolic equations with pointwise state constraints and coupled integral state control constraints we obtain necessary optimality conditions in the form of a pontryagin s minimum principle for local solutions in the sense of l p p leq infty

ext 1 quivers for the witt algebra w 1 1 let g be the finite dimensional witt algebra w 1 1 over an algebraically closed field of characteristic p 3 it is well known that all simple w 1 1 modules are finite dimensional each simple module admits a character g given g one can form the finite dimensional reduced enveloping algebra u g the simple modules for u g are precisely those simple w 1 1 modules admitting the character in this paper the authors compute ext1 between pairs of simple modules for u g

inequalities for nbue and nwue life distributions this paper presents a procedure for developing reliability inequalities for several known classes of failure distributions the inequalities are rational functions of the moments and conditional tail moments of the underlying failure distribution the introduction of four new classes of failure distributions based on the variance behavior facilitates the development of certain inequalities for the nbue and nwue distributions