a boundary value problem for the minimum time function a natural boundary value problem for the dynamic programming partial differential equation associated with the minimum time problem is proposed the minimum time function is shown to be the unique viscosity solution of this boundary value problem

transmutation techniques and observability for time discrete approximation schemes of conservative systems in this article we consider abstract linear conservative systems and their time discrete counterparts our main result is a representation formula expressing solutions of the continuous model through the solution of the corresponding time discrete one as an application we show how observability properties for the time continuous model yield uniform with respect to the time step observability results for its time discrete approximation counterparts provided the initial data are suitably filtered the main output of this approach is the estimate on the time under which we can guarantee uniform observability for the time discrete models besides using a reverse representation formula we also prove that this estimate on the time of uniform observability for the time discrete models is sharp we then conclude with some general comments and open problems

real algebraic hypersurfaces in complex projective varieties let m be a compact affine nonsingular real algebraic variety of dimension m for the background material on real algebraic geometry the reader may refer to the book 2 by an affine real algebraic variety we mean a locally ringed space isomorphic to an algebraic subset of ir for some n equipped with the sheaf of n valued regular functions cf 2 sect 3 2 denote by halgi m z 2 the subgroup of the homology group h i m z 2 which consists of all homology classes represented by zariski closed real algebraic hypersurfaces of m cf 2 sect 11 3 let h2ig m z 2 be the image of h g l m z 2 under the poincar6 duality isomorphism hm i m 7z 2 hi m z 2 the group h2ig m 7 2 is an important invariant with numerous interesting applications it has been recently extensively studied in 4 cf also the cited paper for a survey of earlier results and applications in this paper we investigate the group hlalg m z 2 where m is the underlying real algebraic variety of a complex projective nonsingular variety more precisely every complex projective variety v that is up to isomorphism an algebraic subset of cp n for some n endowed with the sheaf of c valued regular functions can be in the obvious way considered as a real algebraic variety and as such will be denoted by v of course dim v 2 dim v moreover v is an affine real algebraic variety cf 2 proposition 3 4 8 the very interesting and natural class of real varieties of type v has never been the subject of any serious investigation with a recent exception 3 7 7 if v is nonsingular then v is also nonsingular and we can consider the group hlalg v z 2 and in particular define d v dimz 2 h21g v e 2

smooth regularization of bang bang optimal control problems consider the minimal time control problem for a single input control affine system x x x u1y1 x in rn where the scalar control u1 satisfies the constraint lu1 l when applying a shooting method for solving this kind of optimal control problem one may encounter numerical problems due to the fact that the shooting function is not smooth whenever the control is bang bang in this article we propose the following smoothing procedure for e 0 small we consider the minimal time problem for the control system x x x u1ey1 x e i 2m uieyi x where the scalar controls uie i 1 m with m 2 satisfy the constraint i 1m uie uie t 2 1 we prove under appropriate assumptions a strong convergence result of the solution of the regularized problem to the solution of the initial problem

real algebraic cycles on complex projective varieties if we identify cn with r2n then to any quasi affine complex algebraic variety v of dimension d corresponds in a natural way a real algebraic variety of dimension 2d in the sense of bcr which we will denote by vr since each complex morphism v v can be considered as a real morphism vr v r and all complex algebraic varieties are obtained by glueing affine pieces together we can extend this construction in a natural way to arbitrary complex algebraic varieties x in order to obtain xr the underlying real algebraic structure of x if f x x is a morphism of complex algebraic varieties then it is also in a natural way a regular mapping of real structures xr x r which by abuse of notation will be denoted by f as well if y is a real algebraic variety of dimension n and hk y z 2z is the singular homology group in dimension k with respect to the strong topology then for all k n there is a subgroup denoted by h alg k y z 2z consisting of the fundamental classes of k dimensional real algebraic subsets of y see bcr for a definition of the fundamental class of a real algebraic set and bk1 for more information on h alg k we will study the group h alg 2d 1 y z 2z for y xr where x is a nonsingular projective complex algebraic variety of dimension d in particular y is a nonsingular real algebraic variety of dimension n 2d that is compact for the strong topology it will be more convenient to consider h 1 alg y z 2z which is by definition the image of h alg n 1 y z 2z under the poincare duality isomorphism hn 1 y z 2z h 1 y z 2z then the homomorphism

constrained kaczmarz extended algorithm for image reconstruction abstract in this paper we present a constrained version of kaczmarz extended algorithm for improving image reconstruction from projections in computerized tomography we prove convergence of our algorithm in the general inconsistent case to a constrained least squares solution of the reconstruction problem under weaker hypothesis than those proposed in a previous paper by koltracht and lancaster for classical kaczmarz s projection method numerical experiments and comparisons are also presented on some model problems from the field of electromagnetic geotomography

2d grushin type equations minimal time and null controllable data abstract we study internal null controllability for degenerate parabolic equations of grushin type g x x 2 x 2 y y 2 0 in the rectangle x y 1 1 0 1 previous works proved that null controllability holds for weak degeneracies small and fails for strong degeneracies large moreover in the transition regime and with strip shaped control domains a positive minimal time is required in this paper we work with controls acting on two strips symmetric with respect to the degeneracy we give the explicit value of the minimal time and we characterize some initial data that can be steered to zero in time t when the system is not null controllable their regularity depends on the control domain and the time t we also prove that with a control that acts on one strip touching the degeneracy line x 0 then grushin type equations are null controllable in any time t 0 and for any degeneracy 0 our approach is based on a precise study of the observability property for the one dimensional heat equations satisfied by the fourier coefficients in variable y this precise study is done through a transmutation process on the resulting one dimensional wave equations by lateral propagation of energy method

on optimal location of diffusion and related optimal control for null controllable heat equation abstract we consider optimal locations of heat diffusion and related optimal control to achieve null controllability for multi dimensional heat equations both time optimal control and norm optimal control problems are considered the reason behind combining these two problems together is that these two problems are actually equivalent the energy to be used to drive the system to zero in minimal time interval is actually the minimal energy of driving the system to zero in this minimal time interval and visa versa we formulate the optimal locations for time optimal control and norm optimal control into two types of shape optimization problems one is seeking the optimal domain of heat diffusion with a fixed interior actuator domain this can be considered as a domain perturbation problem in shape optimization another is to seek both the optimal locations of the optimal heat diffusion domain and the related optimal actuator domain the existences of these two types of shape optimization problems over some class of open sets in general r n space have been proved separately

an algorithm for approximate calculation of the attainable sets of the nonlinear control systems with integral constraint on controls a b s t r a c t in this paper an algorithm for approximate calculation of the attainable sets of nonlinear control systems with integral constraint on controls is presented a numerical example of the computer application of this algorithm is also given

note on the cost of the approximate controllability for the heat equation with potential abstract we prove the approximate controllability for the heat equation with potential with a cost of order ec e when the target is in h 0 1 with a precision in l 2 norm also a quantification estimate of the unique continuation for initial data in l 2 of the heat equation with potential is established

on the minimum time function and the minimum energy problem a nonlinear case abstract for a semilinear control system in general banach spaces we prove results on exact and approximate controllability regularity of the minimum time function connections between the minimum time function and the minimum energy

on the observability of time discrete conservative linear systems abstract we consider various time discretization schemes of abstract conservative evolution equations of the form z a z where a is a skew adjoint operator we analyze the problem of observability through an operator b more precisely we assume that the pair a b is exactly observable for the continuous model and we derive uniform observability inequalities for suitable time discretization schemes within the class of conveniently filtered initial data the method we use is mainly based on the resolvent estimate given by burq and zworski in n burq m zworski geometric control in the presence of a black box j amer math soc 17 2 2004 443 471 electronic we present some applications of our results to time discrete schemes for wave schrodinger and kdv equations and fully discrete approximation schemes for wave equations

an observability estimate for the heat equation from a product of two measurable sets this paper presents an observability estimate for the heat equation in 0 t where has a real analytic boundary the observation region is e where and e are positive measurable subsets of and 0 t respectively the building of this estimate is based on a slightly modified version of the new estimate established in apraiz and escauriaza 2012 15 the technique developed in phung and wang 2012 11 and the shortcut of the lebeau robbiano strategy provided in miller 2010 3

on the controllability of parabolic systems with a nonlinear term involving the state and the gradient we present some results concerning the controllability of a quasi linear parabolic equation with linear principal part in a bounded domain of mathbb r n with dirichlet boundary conditions we analyze the controllability problem with distributed controls supported on a small open subset and boundary controls supported on a small part of the boundary we prove that the system is null and approximately controllable at any time if the nonlinear term f y nabla y grows slower than y log 3 2 1 y nabla y nabla y log 1 2 1 y nabla y at infinity generally in this case in the absence of control blow up occurs the proofs use global carleman estimates parabolic regularity and the fixed point method

on the continuity property of lp balls and an application abstract in this paper continuity properties of the set valued map p b p 0 p 1 are considered where b p 0 is the closed ball of the space l p t 0 r m centered at the origin with radius 0 it is proved that the set valued map p b p 0 p 1 is continuous applying obtained results the attainable set of the nonlinear control system with integral constraint on the control is studied the admissible control functions are chosen from b p 0 it is shown that the attainable set of the system is continuous with respect to p

continuous dependence of optimal control to controlled domain of actuator for heat equation abstract in this paper we consider continuous dependence of the optimal control with respect to the actuator domain which is varying as open subset in the spatial domain for a multi dimensional heat equation both time optimal control and norm optimal control problems are considered the reason behind combining these two problems together is that these two problems are actually equivalent the energy to be used to drive the system to target set in minimal time interval is actually the minimal energy of driving the system to target set in this minimal time interval and visa versa it is shown that both optimal control and optimal cost are continuous with respect to open controlled actuator domain under the lebesgue measure

approximation of the attainable sets of the nonlinear control systems with integral constraint on controls abstract in this article the attainable sets of a control system are investigated it is assumed that the behavior of the control system is described by a differential equation which is nonlinear with respect to the phase state vector and control vector the admissible control functions are chosen from the closed ball centered at the origin with radius 0 in l p t 0 r m with p 1 an approximation method has been obtained for numerical construction of the attainable sets

the time optimal control with constraints of the rectangular type for linear time varying odes in this paper we study a time optimal control problem of some linear time varying ordinary differential equations where the control constraint set is of the rectangular type we derive a necessary and sufficient condition and provide an algorithm for the optimal time as well as the optimal control for this problem the basis consists of an equivalence theorem between the time optimal control problem and an associated norm optimal control problem and the variational characterization of the latter problem

minimum time and minimum energy for linear control systems abstract in this paper we give estimates concerning regularity properties of the minimum time function and the minimum norm controls involving the controllability function given by the observability inequalities we further study the continuity of the minimum time function in two variables the initial point and the radius of the ball that limits the control we prove a maximum principle for both time and norm optimal control problems

time optimal boundary controls for the heat equation the fact that the time optimal controls for parabolic equations have the bang bang property has been recently proved for controls distributed inside the considered domain interior control the main result in this article asserts that the boundary controls for the heat equation have the same property at least in rectangular domains this result is proved by combining methods from traditionally distinct fields the lebeau robbiano strategy for null controllability and estimates of the controllability cost in small time for parabolic systems on one side and a remez type inequality for muntz spaces and a generalization of turan s inequality on the other side

quantitative unique continuation for the semilinear heat equation in a convex domain in this paper we study certain unique continuation properties for solutions of the semilinear heat equation t u u g u with the homogeneous dirichlet boundary condition over 0 t is a bounded convex open subset of r d with a smooth boundary for the subset the function g r r satisfies certain conditions we establish some observation estimates for u v w hereu and v are two solutions to the above mentioned equation the observation is made over t w here is any non empty open subset of a ndt is a positive number such that both u and v exist on the interval 0 t at least two results can be derived from these estimates i if u v t l2 t hen u v t l2 c where constants c 0a nd 0 1 can be independent of u and v in certain cases ii if two solutions of the above equation hold the same value over t then they coincide over 0 t m tm indicates the maximum number such that these two solutions exist on 0 t m

on conditions for asymptotic stability of dissipative infinite dimensional systems with intermittent damping abstract we study the asymptotic stability of a dissipative evolution in a hilbert space subject to intermittent damping we observe that even if the intermittence satisfies a persistent excitation condition if the hilbert space is infinite dimensional then the system needs not being asymptotically stable not even in the weak sense exponential stability is recovered under a generalized observability inequality allowing for time domains that are not intervals weak asymptotic stability is obtained under a similarly generalized unique continuation principle finally strong asymptotic stability is proved for intermittences that do not necessarily satisfy some persistent excitation condition evaluating their total contribution to the decay of the trajectories of the damped system our results are discussed using the example of the wave equation schrodinger s equation and for strong stability also the special case of finite dimensional systems

perturbations of time optimal control problems for a class of abstract parabolic systems in this work we study the asymptotic behavior of the solutions of a class of abstract parabolic time optimal control problems when the generators converge in an appropriate sense to a given strictly negative operator our main application to pdes systems concerns the behavior of optimal time and of the associated optimal controls for parabolic equations with highly oscillating coefficients as we encounter in homogenization theory our main results assert that provided that the target is a closed ball centered at the origin and of positive radius the solutions of the time optimal control problems for the systems with oscillating coefficients converge in the usual norms to the solution of the corresponding problem for the homogenized system in order to prove our main theorem we provide several new results which could be of a broader interest on time and norm optimal control problems

on the existence of time optimal controls with constraints of the rectangular type for heat equations this paper presents a time optimal control problem with control constraints of the rectangular type for internally controlled heat equations an existence result of time optimal controls for such a problem is established the rectangular type of control constraints originates from the study of time optimal control problems for ordinary differential equations in the finite dimensional case there is no difference between such problems with control constraints of the rectangular type and those of the ball type from the perspective of the study on the existence of optimal controls interestingly in the infinite dimensional case the problem with control constraints of the rectangular type differs essentially from that with control constraints of the ball type for infinite dimensional systems the existence for time optimal controls with constraints of the ball type has already been discussed in the literature while the study of the rectangular type has not been touched upon as far as we know

on image reconstruction algorithms for binary electromagnetic geotomography in this paper we discuss the selected image reconstruction methods of binary tomography in the context of their application to geophysical imaging we restrict our considerations to a discrete version of high frequency electromagnetic geotomography which we label as binary electromagnetic geotomography beg basically such an imaging technique may be applied to detect subsurface anomalies air filled voids whose attenuation coefficient is very low nearly zero value and considerably different from that for the background the assumption for a binary representation of the image to be reconstructed substantially relaxes image reconstruction problems related to ill posedness that comes from an intrinsic limitation of an angular range of projections we test two algorithms for binary tomography where the penalty term is based on the markov random field mrf model the mean field reference distribution and mean field annealing are applied to estimate the global maximizer of the gibbs boltzmann distribution associated with the objective function we also apply the projected gradient algorithm that uses a binary steering very efficient implementations of the algorithms are also given the numerical results are presented for noise free noisy and real data

regularity of the minimum time function and minimum energy problems the linear case we prove new results on the continuity properties and on reachable sets of the minimum time function associated with a linear dynamical system in a separable hilbert space h part of the results are new also in the finite dimensional case the regularity results are stated thanks to the result we obtain on the connection between the minimum time function and the minimum energy

equivalent conditions on periodic feedback stabilization for linear periodic evolution equations abstract this paper studies the periodic feedback stabilization for a class of linear t periodic evolution equations several equivalent conditions on the linear periodic feedback stabilization are obtained these conditions are related to the following subjects the attainable subspace of the controlled evolution equation under consideration the unstable subspace of the evolution equation with the null control provided by the kato projection the poincare map associated with the evolution equation with the null control and two unique continuation properties for the dual equations on different time horizons 0 t and 0 n 0 t where n 0 is the sum of algebraic multiplicities of distinct unstable eigenvalues of the poincare map it is also proved that a t periodic controlled evolution equation is linear t periodic feedback stabilizable if and only if it is linear t periodic feedback stabilizable with respect to a finite dimensional subspace some applications to heat equations with time periodic potentials are presented

the minimum time function for semilinear evolutions we study the bellman equation for the minimum time problem associated with a semilinear evolution system from the point of view of contingent solutions for this purpose new contingent derivatives are defined which involve functions instead of vectors as directions our analysis is based on flow invariance results for appropriate differential inclusions

equivalence of three different kinds of optimal control problems for heat equations and its applications this paper presents an equivalence theorem for three different kinds of optimal control problems which are optimal target control problems optimal time control problems and optimal norm control problems controlled systems in this study are internally controlled heat equations with the aid of this theorem we establish an optimal norm feedback law and build up some explicit algorithms for solutions of optimal norm and optimal time control problems

time varying bang bang property of time optimal controls for heat equation and its application abstract in this paper the bang bang property of time optimal controls for heat equation is established compared with the existing results on these problems the bound of control variables is not a constant but a time varying function as an application of the bang bang property some kind of relation between the time optimal control problem and its corresponding target optimal control problem is considered

the null controllability of thermoelastic plates and singularity of the associated minimal energy function abstract we consider the null controllability problem for thermoelastic plates defined on a two dimensional domain and subject to hinged clamped or free boundary conditions the uncontrolled partial differential equation system generates an analytic semigroup on the space of finite energy consequently the concept of null controllability is indeed appropriate for consideration here it is shown that all finite energy states can be driven to zero by means of just one l 2 0 t control be it either mechanical or thermal the singularity as t 0 of the associated minimal energy function is the main object studied in the paper singularity and blow up rates for minimal energy function are not only of interest in their own right but are also of critical importance in stochastic pdes in this paper we establish the optimal blow up rate o t 5 2 for this function it is shown that the rate of singularity is the same as for finite dimensional truncations of the model in view of sharp estimates available in the finite dimensional setting math control signals systems 9 1997 327 the singularity rates provided in this paper are optimal

spectral inequalities for non selfadjoint elliptic operators and application to the null controllability of parabolic systems we consider elliptic operators a on a bounded domain that are compact perturbations of a selfadjoint operator we first recall some spectral properties of such operators localization of the spectrum and resolvent estimates we then derive a spectral inequality that measures the norm of finite sums of root vectors of a through an observation with an exponential cost following the strategy of g lebeau and l robbiano 1995 we deduce the construction of a control for the non selfadjoint parabolic problem partial t u a u b g in particular the l 2 norm of the control that achieves the extinction of the lower modes of a is estimated examples and applications are provided for systems of weakly coupled parabolic equations and for the measurement of the level sets of finite sums of root functions of a

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 approach to the optimal time for a time optimal control problem of an internally controlled heat equation in this paper we study the optimal time for a time optimal control problem mathcal p where the state system is an internally controlled heat equation by projecting the original problem via the finite element method we obtain another time optimal control problem mathcal p hl governed by a linear system of ordinary differential equations here h and l are the mesh sizes of the finite element spaces of the state space and the control space respectively the purpose of this study is to approach the optimal time for the problem mathcal p through the optimal time for the problem mathcal p hl we prove that the optimal time for the problem mathcal p hl converges to the optimal time for the problem mathcal p when h and l are approaching zero more significantly we obtain error estimates between the optimal times in terms of h and l for certain cases

approximation of time optimal controls for heat equations with perturbations in the system potential in this paper we study a certain approximation property for a time optimal control problem of the heat equation with l infty potential we prove that the optimal time and the optimal control to the same time optimal control problem for the heat equation where the potential has a small perturbation are close to those for the original problem we also verify that for the heat equation with a small perturbation in the potential one can construct a new time optimal control problem which has the same target as that of the original problem but has a different control constraint bound from that of the original problem such that the new and the original problems share the same optimal time and meanwhile the optimal control of the new problem is close to that of the original one the main idea to approach such approximation is an appropriate use of an equivalence theorem of minimal norm and minimal time control problems for the heat equations under consideration this theorem was first established by wa

time optimal controls of the linear fitzhugh nagumo equation with pointwise control constraints time optimal control governed by the internally controlled linear fitzhugh nagumo equation with pointwise control constraint is considered making use of ekeland s variational principle we obtain pontryagin s maximum principle for a time optimal control problem using the maximum principle the bang bang property of the optimal controls is established under appropriate assumptions

the dynamic programming equation for the time optimal control problem in infinite dimensions the existence and uniqueness of a viscosity solution for the bellman equation associated with the time optimal control problem for a semilinear evolution equation in hilbert space is provided applications to time optimal control problems governed by parabolic equations are given

convergence properties of art and sor algorithms art algorithms with relaxation parameters are studied for general consistent or inconsistent linear algebraic systemsrx f and a general convergence theorem is formulated the advantage of severe underrelaxation is reexamined and clarified the relationship to solutions obtained by applying sor methods to the equationrr t y f is investigated

long time behavior of a coupled heat wave system arising in fluid structure interaction this paper is devoted to the analysis of the long time behavior of a coupled wave heat system in which a wave and a heat equation evolve in two bounded domains with natural transmission conditions at a common interface these conditions couple in particular the heat unknown with the velocity of the wave solution this model may be viewed as a simplified version of linearized models which arise in fluid structure interaction first we show the strong asymptotic stability of solutions to this system then based on the construction of ray like solutions by means of geometric optics expansions and a careful analysis of the transfer of energy at the interface we show the lack of uniform decay in general domains further we obtain a polynomial decay result for smooth solutions of the system under a suitable geometric assumption which guarantees that the heat domain envelopes the wave domain finally in the absence of geometric conditions we show a logarithmic decay result for the same system but with simplified transmission conditions at the interface we also analyze the difficulty there is to extend this result to the more natural transmission conditions

semismooth newton methods for time optimal control for a class of odes time optimal control problems for a class of linear multi input systems are considered the problems are regularized and the asymptotic and monotone behavior of the regularization procedure is investigated for the regularized problems the applicability of semismooth newton methods is verified first numerical tests are presented which show that the proposed approach is different from other methods in that it does not rely on a priori information about the switching structure

on the equivalence of minimal time and minimal norm controls for internally controlled heat equations we prove the equivalence of the minimal time and minimal norm control problems for heat equations on bounded smooth domains of the euclidean space with homogeneous dirichlet boundary conditions and controls distributed internally on an open subset of the domain where the equation evolves we consider the problem of null controllability whose aim is to drive solutions to rest in a finite final time as a consequence of this equivalence using the well known variational characterization of minimal norm controls we establish necessary and sufficient conditions for the minimal time and the corresponding control

blowup time optimal control for ordinary differential equations this paper studies a time optimal control problem governed by a classical controlled system of nonlinear ordinary differential equations corresponding to certain controls the solutions of the system have the behavior of blowup namely they blow up at a finite time the goal for controlling the system is to minimize the blowup time therefore the time optimal control problem studied in this paper holds its target set outside of its state space from this perspective it essentially differentiates from the most time optimal control problems in studies from past publications the purpose of this study is to develop a new method for proving the existence of time optimal controls for such kinds of time optimal control problems through analyzing a typical problem we also aim to create a new method for establishing the pontryagin maximum principle of optimal controls the main results derived from this paper can be extended to controlled systems where nonlinear terms have a more general form by the two methods mentioned above

on the convergence of asynchronous paracontractions with application to tomographic reconstruction from incomplete data abstract convergence of iterative processes in c k of the form x i r i j i x 1 r i 1 1 p j i x i where j i 1 2 n i 1 2 is analyzed it is shown that if the matrices p 1 p n are paracontracting in the same smooth strictly convex norm and if the sequence j i i 1 has certain regularity properties then the above iterates converge this result implies the convergence of a parallel asynchronous implementation of the algebraic reconstruction technique art algorithm often used in tomographic reconstruction from incomplete data

row action methods for huge and sparse systems and their applications this paper brings together and discusses theory and applications of methods identified and labelled as row action methods for linear feasibility problems find x in bf r n such that ax leqq b linearly constrained optimization problems minimize f x subject to ax leqq b and some interval convex programming problems minimize f x subject to c leqq ax leqq b the main feature of row action methods is that they are iterative procedures which without making any changes to the original matrix a use the rows of a one row at a time such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern apart from a high degree of sparaseness fields of application where row action methods are used in various ways include image reconstruction from projection operations research and game theory learning theory pattern recognition and transportation theory a row action method for the

attainable subspaces and the bang bang property of time optimal controls for heat equations in this paper we study two subjects on internally controlled heat equations with time varying potentials the attainable subspaces and the bang bang property for some time optimal control problems we present some equivalent characterizations on the attainable subspaces and provide a sufficient condition to ensure the bang bang property both the above mentioned characterizations and the sufficient condition are closely related to some function spaces consisting of some solutions to the adjoint equations it seems for us that the existing ways to derive the bang bang property for heat equations with time invariant potentials see for instance h o fattorini infinite dimensional linear control systems the time optimal and norm optimal problems elsevier amsterdam 2005 f gozzi and p loreti siam j control optim 37 1999 pp 1195 1221 v mizel and t seidman siam j control optim 35 1997 pp 1204 1216 and g wang siam j control optim 47 2008 pp 1701 1720 do not work

the minimal time function in infinite dimensions the goal of this paper is to study regularity properties of the minimal time function t cdot associated with linear control systems in infinite dimensions this paper proves that various regularity properties of t cdot hold on the whole reachable set provided they hold around the target some methods to obtain estimates on t cdot around the target are given and applications to both distributed and boundary control problems are presented

l infty null controllability for the heat equation and its consequences for the time optimal control problem in this paper we establish a certain l infty null controllability for the internally controlled heat equation in omega times 0 t with the control restricted to a product set of an open nonempty subset in omega and a subset of positive measure in the interval 0 t based on this we obtain a bang bang principle for the time optimal control of the heat equation with controls taken from the set mathcal u ad u cdot t 0 infty rightarrow l 2 omega measurable u cdot t in u a e in t where u is a closed and bounded subset of l 2 omega namely each optimal control u cdot t of the problem satisfies necessarily the bang bang property u cdot t in partial u for almost all t in 0 t where partial u denotes the boundary of the set u and t is the optimal time we also get the uniqueness of the optimal control when the target set s is convex and the control set u is a closed ball

an abstract bang bang principle and time optimal boundary control of the heat equation a principal technical result of this paper is that the one dimensional heat equation with boundary control is exactly null controllable with control restricted to an arbitrary set e subset 0 t of positive measure a general abstract argument is presented to show that in contrast to previous results this implies the bang bang property for time optimal controls i e such a control can take only extreme values of the hull of the constraint set without imposing any condition regarding the target state

on the continuity properties of the attainable sets of control systems with integral constraints on control in the paper the attainable sets of the control systems with integral constraints on control are studied it is assumed that the system is nonlinear with respect to the phase state vector and linear with respect to the control vector the admissible control functions are chosen in lp from the ball centered at the origin with given radius it is proved that the attainable sets of the system are continuous with respect to p

minimum time with bounded energy minimum energy with bounded time necessary and sufficient conditions for the regularity of the minimum time function and minimum energy function for a control system with controls in l p 0 infty bbb r m and p ge 1 are given in terms of topological properties of the reachable sets in particular standard local controllability assumptions are sufficient to yield the continuity of both value functions for linear systems and p ge1 and the holder continuity of the minimum time function for nonlinear systems and p 1

optimal actuator location of minimum norm controls for heat equation with general controlled domain abstract in this paper we study optimal actuator location of the minimum norm controls for a multi dimensional heat equation with control defined in the space l 2 0 t the actuator domain is time varying in the sense that it is only required to have a prescribed lebesgue measure for any moment we select an optimal actuator location so that the optimal control takes its minimal norm over all possible actuator domains we build a framework of finding the nash equilibrium so that we can develop a sufficient and necessary condition to characterize the optimal relaxed solutions for both actuator location and corresponding optimal control of the open loop system the existence and uniqueness of the optimal classical solutions are therefore concluded as a result we synthesize both optimal actuator location and corresponding optimal control into a time varying feedbacks

algebraic models of smooth manifolds in fact we conjecture that such a family can be chosen uncountable in view of this theorem it seems natural and interesting to investigate algebrogeometric properties of various algebraic models of a given smooth manifold let us now describe one of these problems which will be the main object of study in this paper for notions definitions and results of real algebraic geometry we refer the reader to the book 9 given a compact nonsingular real affine algebraic variety x denote by h ig x 7z 2 the subgroup of nk x z 2 of homology classes represented by zariski closed k dimensional algebraic subvarieties of x cf 9 chap 1 1 let h lg x z 2 be the image of h a k x 71 2 d dimx under the poincar6 duality isomorphism hd k x 71 2 o hk x 71 2 these groups h ig x 71 2 are undoubtedly one of the most important invariants of an algebraic variety x still our knowledge of the behavior of h g x 71 2 is rather limited the main open problem is the following

some results on the controllability of forward stochastic heat equations with control on the drift abstract in this paper we establish the null approximate controllability for forward stochastic heat equations with control on the drift the null controllability is obtained by a time iteration method and an observability estimate on partial sums of eigenfunctions for elliptic operators as a consequence of the null controllability we obtain the observability estimate for backward stochastic heat equations which leads to a unique continuation property for backward stochastic heat equations and hence the desired approximate controllability for forward stochastic heat equations it deserves to point out that one needs to introduce a little stronger assumption on the controller for the approximate controllability of forward stochastic heat equations than that for the null controllability this is a new phenomenon arising in the study of the controllability problem for stochastic heat equations

the structure of some matrices arising in tomography linear algebraic systems rx y are considered which are generated by typical limited angle tomographic problems a detailed examination is made of vectors x for which rx 0 and which are responsible for ghost images frequently referred to in the literature in particular the construction of an orthogonal basis for the subspace of all such vectors x is investigated and is complete in some special cases it is shown how advantage can be taken of this information

null controllability with vanishing energy linear null controllable systems for which an arbitrary initial state can be transferred to the origin with arbitrarily small energy are characterized theorems are stated in terms of an associated algebraic riccati equation and in terms of the spectrum of the linear part of the system the results so obtained allow us to determine ornstein uhlenbeck operators for which the liouville theorem about bounded harmonic functions holds

sharp observability estimates for heat equations the goal of this article is to derive new estimates for the cost of observability of heat equations we have developed a new method allowing one to show that when the corresponding wave equation is observable the heat equation is also observable this method allows one to describe the explicit dependence of the observability constant on the geometry of the problem the domain in which the heat process evolves and the observation subdomain we show that our estimate is sharp in some cases particularly in one space dimension and in the multi dimensional radially symmetric case our result extends those in fattorini and russell arch rational mech anal 43 272 292 1971 to the multi dimensional setting and improves those available in the literature namely those by miller j differ equ 204 1 202 226 2004 siam j control optim 45 2 762 772 2006 atti accad naz lincei cl sci fis mat natur rend lincei 9 mat appl 17 4 351 366 2006 and tenenbaum and tucsnak j differ equ 243 1 70 100 2007 our approach is based on an explicit representation formula of some solutions of the wave equation in terms of those of the heat equation in contrast to the standard application of transmutation methods which uses a reverse representation of the heat solution in terms of the wave one we shall also explain how our approach applies and yields some new estimates on the cost of observability in the particular case of the unit square observed from one side we will also comment on the applications of our techniques to controllability properties of heat type equations