the classical problem of the calculus of variations in the autonomous case relaxation and lipschitzianity of solutions we consider the problem of minimizing b a l x t x t dt x a a x b b under the assumption that the lagrangian l is continuous and satisfies a growth assumption that does not imply superlinear growth we provide a result on the relaxation of the functional and show that a solution to the minimum problem is lipschitzian

state constraints in optimal control the degeneracy phenomenon in this paper we study the degeneracy phenomenon arising in optimal control problems with state constraints it is shown that this phenomenon occurs because of the incompleteness of the standard variants of pontryagin s maximum principle for problems with state constraints the new maximum principle containing some additional information about the behavior of the hamiltonian at the endtimes is developed as application we obtain some sufficient and necessary conditions for nondegeneracy and pointwise nontriviality of the maximum principle the results obtained envelope the optimal control problems with systems described by differential inclusions and ordinary differential equations

degenerate optimal control problems with state constraints standard necessary conditions for optimal control problems with pathwise state constraints supply no useful information about minimizers in a number of cases of interest e g when the left endpoint of state trajectories is fixed at x0 and x0 lies in the boundary of the state constraint set in these cases a nonzero but nevertheless trivial set of multipliers exists we give conditions for the existence of nontrivial multipliers a feature of these conditions is that they allow nonconvex velocity sets and measurably time dependent data the proof techniques are based on refined estimates of the distance of a given state trajectory from the set of state trajectories satisfying the state constraint originating in the dynamic programming literature

reciprocal transformations in 2 1 dimensions reciprocal backlund transformations are introduced for nonlinear integro differential equations in 2 1 dimensions invariance under such transformations is investigated

a survey of the maximum principles for optimal control problems with state constraints this paper gives a survey of the various forms of pontryagin s maximum principle for optimal control problems with state variable inequality constraints the relations between the different sets of optimality conditions arising in these forms are shown furthermore the application of these maximum principle conditions is demonstrated by solving some illustrative examples

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

is bilevel programming a special case of a mathematical program with complementarity constraints bilevel programming problems are often reformulated using the karush kuhn tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints mpcc clearly both problems are closely related but the answer to the question posed is no even in the case when the lower level programming problem is a parametric convex optimization problem this is not obvious and concerns local optimal solutions we show that global optimal solutions of the mpcc correspond to global optimal solutions of the bilevel problem provided the lower level problem satisfies the slater s constraint qualification we also show by examples that this correspondence can fail if the slater s constraint qualification fails to hold at lower level when we consider the local solutions the relationship between the bilevel problem and its corresponding mpcc is more complicated we also demonstrate the issues relating to a local minimum through examples

the euler approximation in state constrained optimal control we analyze the euler approximation to a state constrained control problem we show that if the active constraints satisfy an independence condition and the lagrangian satisfies a coercivity condition then locally there exists a solution to the euler discretization and the error is bounded by a constant times the mesh size the proof couples recent stability results for state constrained control problems with results established here on discrete time regularity the analysis utilizes mappings of the discrete variables into continuous spaces where classical finite element estimates can be invoked

regularity properties of optimal controls for problems with time varying state and control constraints abstract in this paper we report new results on the regularity of optimal controls for dynamic optimization problems with functional inequality state constraints a convex time dependent control constraint and a coercive cost function recently it has been shown that the linear independence condition on active state constraints present in the earlier literature can be replaced by a less restrictive positive linear independence condition that requires linear independence merely with respect to non negative weighting parameters provided the control constraint set is independent of the time variable we show that if the control constraint set regarded as a time dependent multifunction is merely lipschitz continuous with respect to the time variable then optimal controls can fail to be lipschitz continuous in these circumstances however a weaker holder continuity like regularity property can be established on the other hand lipschitz continuity of optimal controls is guaranteed for time varying control sets under a positive linear independence hypothesis when the control constraint sets are described at each time by a finite collection of functional inequalities

an integral type constraint qualification to guarantee nondegeneracy of the maximum principle for optimal control problems with state constraints abstract for optimal control problems involving ordinary differential equations and functional inequality state constraints the maximum principle may degenerate producing no useful information about minimizers this is known as the degeneracy phenomenon several non degenerate forms of the maximum principle valid under different constraint qualifications have been proposed in the literature in this paper we propose a new constraint qualification under which a nondegenerate maximum principle is validated in contrast with existing results our constraint qualification is of an integral type an advantage of the proposed constraint qualification is that it is verified on a larger class of problems with nonsmooth data and convex velocity sets

normal forms of necessary conditions for dynamic optimization problems with pathwise inequality constraints abstract there has been a longstanding interest in deriving conditions under which dynamic optimization problems are normal that is the necessary conditions of optimality nco can be written with a nonzero multiplier associated with the objective function this paper builds upon previous results on nondegenerate nco for trajectory constrained optimal control problems to provide even stronger normal forms of the conditions the nco developed may address problems with nonsmooth less regular data the particular case of calculus of variations problems is here explored to show a favorable comparison with existent results

variational approach to second order optimality conditions for control problems with pure state constraints for optimal control problems with set valued control constraints and pure state constraints we propose new second order necessary optimality conditions in addition to the usual second order derivative of the hamiltonian these conditions contain extra terms involving second order tangents to the set of feasible trajectory control pairs at the extremal process under consideration the second order necessary optimality conditions of the present work are obtained by using a variational approach in particular we present a new second order variational equation this approach allows us to make direct proofs as opposed to the classical way of obtaining second order necessary conditions by using an abstract infinite dimensional optimization problem no convexity assumptions on the constraints are imposed and optimal controls are required to be merely measurable

multi resolution explicit model predictive control delta model formulation and approximation this paper deals with the explicit solution and approximation of the constrained linear finite time optimal control problem for systems with fast sampling rates to this aim the recently developed explicit model predictive control empc is reformulated and characterized using the operator to enjoy its promising advantages compared to the time shift operator using the proposed model empc formulation a systematic method is proposed for first designing a low complexity approximate empc solution and then improving its closed loop action without first determining an exact optimal solution that might be of prohibitive complexity it is shown that the stability and feasibility of the proposed sub optimal solution is guaranteed

discrete bilevel programming application to a natural gas cash out problem in this paper we present a mathematical framework for the problem of minimizing the cash out penalties of a natural gas shipper the problem is modeled as a mixed integer bilevel programming problem having one boolean variable in the lower level problem such problems are difficult to solve to obtain a more tractable problem we move the boolean variable from the lower to the upper level problem the implications of this change of the problem are investigated thoroughly the resulting lower level problem is a generalized transportation problem the formulation of conditions guaranteeing the existence of an optimal solution for this problem is also in the scope of this paper the corresponding results are then used to find a bound on the optimal function value of our initial problem

explicit solutions to the two phase stefan problem for storm type materials a reciprocal transformation is employed to reduce a two phase stefan problem in nonlinear heat conduction into a form which admits a class of exact solutions analogous to the classical neumann solution the problem is considered for materials of storm type rogers 1985 j phys a math gen 18 105 9 two related cases are considered one of them has a flux condition of the type q 0 t 1 2 q 0 0 and the existence and uniqueness of the solution is proved when q 0 satisfies a certain inequality which generalizes the work of tarzia 1981 q appl math 39 491 7 obtained for constant thermal coefficients the other one has a temperature condition on the fixed face and the existence and uniqueness is proved for all data

the riccati equation for optimal control problems with mixed state control constraints necessity and sufficiency the goal of this paper is to conduct a complete study of second order conditions for the optimal control problem with mixed state control constraints the conjugate point theory is presented and a necessary condition in terms of the corresponding riccati equation is obtained sufficiency criteria are developed in terms of strengthened necessary conditions including the riccati equation the results generalize the known ones for pure control constraints as well as for the mixed state control constraints

normality of the maximum principle for nonconvex constrained bolza problems we consider a bolza optimal control problem with state constraints it is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle in general the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls in the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate cf a v arutyanov s m aseev investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints siam j control optim 35 1997 930 952 f rampazzo r b vinter a theorem on existence of neighbouring trajectories satisfying a state constraint with applications to optimal control ima 16 4 1999 335 351 f rampazzo r b vinter degenerate optimal control problems with state constraints siam j control optim 39 4 2000 989 1007 our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle in particular we allow the initial condition to be fixed and the state constraints to be nonsmooth to prove normality we use j yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined in turn by linearization of the original set of constraints along an extremal trajectory

second order sufficient conditions for optimal control problems with free final time the riccati approach second order sufficient conditions ssc for control problems with control state constraints and free final time are presented instead of deriving such ssc from first principles we transform the control problem with free final time into an augmented control problem with fixed final time for which well known ssc exist ssc are then expressed as a condition on the positive definiteness of the second variation a convenient numerical tool for verifying this condition is based on the riccati approach where one has to find a bounded solution of an associated riccati equation satisfying specific boundary conditions the augmented riccati equations for the augmented control problem are derived and their modifications on the boundary of the control state constraint are discussed two numerical examples 1 the classical earth mars orbit transfer in minimal time and 2 the rayleigh problem in electrical engineering demonstrate that the riccati equation approach provides a viable numerical test of ssc

constrained linear quadratic regulation the paper is a contribution to the theory of the infinite horizon linear quadratic regulator lqr problem subject to inequality constraints on the inputs and states extending an approach first proposed by sznaier and damborg 1987 a solution algorithm is presented which requires solving a finite number of finite dimensional positive definite quadratic programs the constrained lqr outlined does not feature the undesirable mismatch between open loop and closed loop nominal system trajectories which is present in the other popular forms of model predictive control mpc that can be implemented with a finite quadratic programming algorithm the constrained lqr is shown to be both optimal and stabilizing the solution algorithm is guaranteed to terminate in finite time with a computational cost that has a reasonable upper bound compared to the minimal cost for computing the optimal solution inherent to the approach is the removal of a tuning parameter the control horizon which is present in other mpc approaches and for which no reliable tuning guidelines are available two examples are presented that compare constrained lqr and two other popular forms of mpc the examples demonstrate that constrained lqr achieves significantly better performance than the other forms of mpc on some plants and the computational cost is not prohibitive for online implementation

a chain code for representing 3d curves abstract a chain code for representing three dimensional 3d curves is defined any 3d continuous curve can be digitalized and represented as a 3d discrete curve this 3d discrete curve is composed of constant straight line segments thus the chain elements represent the orthogonal direction changes of the constant straight line segments of the discrete curve the proposed chain code only considers relative direction changes which allows us to have a curve descriptor invariant under translation and rotation also this curve descriptor may be starting point normalized for open and closed curves and invariant under mirroring transformation in the content of this work the main characteristics of this chain code are presented this chain code is inspired by the work of guzman mcc technical report number aca 254 87 july 13 1987 for representing 3d stick bodies finally we present some results of this chain code to represent and process 3d discrete curves as linear features over the terrain by means of digital elevation model dem data also we use this chain code for representing solids composed of voxels thus each solid represents a dem which is described by only one chain

regularity properties of solutions to linear quadratic optimal control problems with state constraints abstract we establish regularity properties of solutions of linear quadratic optimal control problems involving state inequality constraints under simply stated and directly verifiable hypotheses on the data it is shown that if the state constraint has index k 0 then the optimal control u is k times differentiable the k th derivative may be discontinuous but it is a function of bounded variation and consequently it has left and right limits at each point in its domain if on the other hand the state constraint has index k 0 then the optimal control is continuous the latter property is perhaps surprising because it implies that for the class of problems considered optimal state trajectories cannot abruptly change direction when they strike the boundary of the state constraint region these findings are significant because they justify assumptions which underlie analysis of junction conditions i e properties of state trajectories at contact points with the boundary of the state constraint set provided elsewhere in the literature

conjugate points for variational problems with equality and inequality state constraints in this paper variational problems with equality and inequality state constraints are considered the theory of conjugate points for these problems is developed and necessary conditions for weak local optimality are derived in terms of this concept and the legendre condition for the case of inequality constraints the envelope like effect is taken into consideration in the accessory problem

on computing solutions to the continuous time constrained linear quadratic regulator we propose in this note a method for computing the solution to the infinite horizon continuous time constrained linear quadratic regulator the method is based on two main ingredients a multigrid method for placing a finite number of time intervals and a piece wise linear parameterization of the input within the intervals the input values at the decision time points and slopes within the time intervals are computed via quadratic programs qps the grids are gradually refined to efficiently improve the accuracy of the solution and the required matrices and vectors for all qps are computed offline and stored to improve the online efficiency two examples are presented to show the main characteristics of the proposed method

a new chain code abstract a new chain code for shapes composed of regular cells is defined this boundary chain code is based on the numbers of cell vertices which are in touch with the bounding contour of the shape this boundary chain code is termed vertex chain code vcc the vcc is invariant under translation and rotation also it may be starting point normalized and invariant under mirroring transformation using this concept of chain code it is possible to relate the chain length to the contact perimeter which corresponds to the sum of the boundaries of neighboring cells of the shape bribiesca e comp math appl 33 11 1997 1 9 also to relate the chain nodes to the contact vertices which correspond to the vertices of neighboring cells so in this way these relations among the chain and the characteristics of interior of the shape allow us to obtain interesting properties this work is motivated by the idea of obtaining various shape features computed directly from the vcc without going to cartesian coordinate representation finally in order to illustrate the capabilities of the vcc we present some results using real shapes

approximation of the modified error function abstract in this article we obtain explicit approximations of the modified error function introduced in cho and sunderland 1974 as part of a stefan problem with a temperature dependent thermal conductivity this function depends on a parameter which is related to the thermal conductivity in the original phase change process we propose a method to obtain approximations which is based on the assumption that the modified error function admits a power series representation in accurate approximations are obtained through functions involving error and exponential functions only for the special case in which assumes small positive values we show that the modified error function presents some characteristic features of the classical error function such as monotony concavity and boundedness moreover we prove that the modified error function converges to the classical one when goes to zero

a general framework to design stabilizing nonlinear model predictive controllers abstract we propose a new model predictive control mpc framework to generate feedback controls for time varying nonlinear systems with input constraints we provide a set of conditions on the design parameters that permits to verify a priori the stabilizing properties of the control strategies considered the supplied sufficient conditions for stability can also be used to analyse the stability of most previous mpc schemes the class of nonlinear systems addressed is significantly enlarged by removing the traditional assumptions on the continuity of the optimal controls and on the stabilizability of the linearized system some important classes of nonlinear systems including some nonholonomic systems can now be stabilized by mpc in addition we can exploit increased flexibility in the choice of design parameters to reduce the constraints of the optimal control problem and thereby reduce the computational effort in the optimization algorithms used to implement mpc

explicit solutions for a two phase unidimensional lame clapeyron stefan problem with source terms in both phases a two phase stefan problem with heat source terms of a general similarity type in both liquid and solid phases for a semi infinite phase change material is studied we assume the initial temperature is a negative constant and we consider two different boundary conditions at the fixed face x 0 a constant temperature or a heat flux of the form q0 t q0 0 the internal heat source functions are given by gj x t lt j x2ajt j 1 solid phase j 2 liquid phase where j j are functions with appropriate regularity properties is the mass density l is the fusion latent heat by unit of mass aj2 is the diffusion coefficient x is the spatial variable and t is the temporal variable we obtain for both problems explicit solutions with a restriction for data only for the second boundary conditions on x 0 moreover the equivalence of the two free boundary problems is also proved we generalize the solution obtained in j l menaldi d a tarzia generalized lame clapeyron solution for a one phase source stefan problem comput appl math 12 2 1993 123 142 for the one phase stefan problem finally a particular case where j j 1 2 are of exponential type given by j x exp x dj 2 with x and dj r is also studied in details for both boundary temperature conditions at x 0 this type of heat source terms is important through the use of microwave energy following e p scott an analytical solution and sensitivity study of sublimation dehydration within a porous medium with volumetric heating j heat transfer 116 1994 686 693 we obtain a unique solution of the similarity type for any data when a temperature boundary condition at the fixed face x 0 is considered a similar result is obtained for a heat flux condition imposed on x 0 if an inequality for parameter q0 is satisfied

investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints in this paper we study the degeneracy phenomenon in optimal control problems with state constraints it is shown that this phenomenon occurs because of the incompleteness of the standard variants of pontryagin s maximum principle for problems with state constraints a new maximum principle containing additional information about the behavior of the hamiltonian at the endtimes is developed we also obtain some sufficient and necessary conditions for nondegeneracy and pointwise nontriviality of the maximum principle the results obtained pertain to optimal control problems with systems described by differential inclusions and ordinary differential equations

efficiency of chain codes to represent binary objects we present a study of compression efficiency for binary objects or bi level images for different chain code schemes chain code techniques are used for compression of bi level images because they preserve information and allow a considerable data reduction furthermore chain codes are the standard input format for numerous shape analysis algorithms in this work we apply chain codes to represent object with holes and we compare their compression efficiency for seven chain codes we have also compared all these chain codes with the jbig standard for bi level images

a geometric structure for two dimensional shapes and three dimensional surfaces abstract a geometric structure called slope change notation scn which describes two dimensional 2d shapes and three dimensional 3d surfaces in a discrete representation is presented the scn of a curve is obtained by placing constant length straight line segments around the curve the endpoints of the straight line segments always touching the curve and calculating the slope changes between contiguous segments scaled to a continuous range from 1 to 1 the scn is independent of translation and rotation due to the fact that slope changes around the curve are used and optionally of size the scn for 2d shapes is 1d this is an important characteristic because shapes with particular characteristics are easily generated by numerical sequences also it is possible to perform arithmetic operations among shapes and surfaces the scn differs from other chain codes for instance freeman chains proc natn electron conf 18 312 324 1961 since the proposed notation does not use a grid and so depends only on itself its range of slope changes varies continuously from 1 to 1 its vertices always touch the curve which produces a better description of the shape and its discrete elements always have the same length using this geometric structure only slope changes are variable the segments size of any shape is always constant at the end of the paper a related theory b that allows variable segment size as a function of slope changes is introduced these ideas are based on previous work pattern recognition13 123 137 1981 and the solutions to many problems which arose are presented in this paper

lq control for constrained continuous time systems a design method of lq optimal control law is considered for constrained continuous time systems by introducing singular value decomposition for finite time horizon linear systems the sequence of lq sub optimal control laws which converges to the exact solution is obtained based on quadratic programming it is also shown that the nonlinear sub optimal feedback gain is found by the union of affine state functions

an algorithm for compression of bilevel images this paper presents the block arithmetic coding for image compression bacic algorithm a new method for lossless bilevel image compression which can replace jbig the current standard for bilevel image compression bacic uses the block arithmetic coder bac a simple efficient easy to implement variable to fixed arithmetic coder to encode images bacic models its probability estimates adaptively based on a 12 bit context of previous pixel values the 12 bit context serves as an index into a probability table whose entries are used to compute p sub 1 the probability of a bit equaling one the probability measure bac needs to compute a codeword in contrast the joint bilevel image experts group jbig uses a patented arithmetic coder the ibm qm coder to compress image data and a predetermined probability table to estimate its probability measures jbig though has not get been commercially implemented instead jbig s predecessor the group 3 fax g3 continues to be used bacic achieves compression ratios comparable to jbig s and is introduced as an alternative to the jbig and g3 algorithms bacic s overall compression ratio is 19 0 for the eight ccitt test images compared to jbig s 19 6 and g3 s 7 7 is 16 0 for 20 additional business type documents compared to jbig s 16 0 and g3 s 6 74 and is 3 07 for halftone images compared to jbig s 2 75 and g3 s 0 50

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

the dirichlet to neumann map for a nonlinear diffusion convection equation a method for constructing the dirichlet to neumann map for a nonlinear diffusion convection equation is presented the problem is reduced to the solution of a nonlinear integral equation in one independent variable existence and uniqueness of the solution may be proven for small times via a contraction mapping technique

optimal sampled data controls with running inequality state constraints pontryagin maximum principle and bouncing trajectory phenomenon sampled data control systems have steadily been gaining interest for their applications in automatic engineering where they are implemented as digital controllers and recently results have been obtained in optimal control theory for nonlinear sampled data control systems and certain generalizations in this paper we derive a pontryagin maximum principle for general nonlinear finite dimensional optimal sampled data control problems with running inequality state constraints in particular we obtain a nonpositive averaged hamiltonian gradient condition with the adjoint vector being a function of bounded variations our proof is based on the ekeland variational principle in general optimal control problems with running inequality state constraints are difficult to solve using numerical methods due to the discontinuities the jumps and the singular part of the adjoint vector however in our case we find that under certain general hypotheses the adjoint vector only experiences jumps at most at the sampling times and moreover the trajectory only contacts the running inequality state constraints at most at the sampling times we call this behavior a bouncing trajectory phenomenon and it constitutes the second major focus of this paper finally taking advantage of the bouncing trajectory phenomenon we numerically solve three examples with different kinds of constraints and in several dimensions

the broken link between normality and regularity in the calculus of variations abstract there is a link between two well known constraint qualifications in nonlinear programming namely normality implying a positive cost multiplier and regularity the tangent cone and the set of tangential constraints coincide the former implies the latter this fact has been crucial and a cornerstone for establishing second order conditions in terms of tangential constraints for problems in the calculus of variations involving equality and inequality constraints one can derive second order conditions in terms of the tangent cone and state a conjecture based on tangential constraints which is the counterpart of that in nonlinear programming however as we show in this paper the link mentioned above between normality and regularity in the infinite dimensional case may fail to hold

a maximum principle for nonsmooth optimal control problems with state constraints abstract optimality conditions are derived in the form of a maximum principle governing solutions to an optimal control problem which involves state constraints the conditions which apply in the absence of differentiability assumptions on the data are stated in terms of clarke s generalized jacobians although not the most general available the conditions are derived by a novel method this involves removal of the state constraints by introduction of a penalty term and application of ekeland s variational principle

reciprocal transformations of two component hyperbolic systems and their invariants abstract criteria for the equivalence under reciprocal transformations of two component hyperbolic systems are established in terms of reciprocal invariants these results are applied to a 1 1 dimensional isentropic gasdynamics system to determine reciprocal equivalence to the canonical system with a polytropic gas law the four parameter model pressure density relations for which this reciprocal equivalence holds are parameterized in terms of the lagrangian signal speed

existence of an exact solution for a one phase stefan problem with nonlinear thermal coefficients from tirskii s method abstract the mathematical analysis of a one phase lame clapeyron stefan problem with nonlinear thermal coefficients following g a tirskii two exact solutions of stefan s nonlinear problem sov phys dokl 4 1959 288 292 is obtained two related cases are considered one of them has a temperature condition on the fixed face x 0 and the other one has a flux condition of the type q 0 t q 0 0 we obtain in both cases sufficient conditions for data in order to have the existence of an explicit solution of a similarity type which is given by using a double fixed point

error estimates for a discrete approximation to constrained control problems a discrete approximation provided by euler s integration scheme is analyzed for control problems with linear dynamics convex costs and inequality state and control constraints the convergence rate of the discrete optimal control to the optimal control for the continuous problem is studied first order convergence in the l 2 metric is established for a problem with vacuous state constraints this estimate is refined to the uniform metric the convergence rate of the dual variables is discussed

a new approach to lipschitz continuity in state constrained optimal control for a linear quadratic state constrained optimal control problem it is proved in 11 that under an independence condition for the active constraints the optimal control is lipschitz continuous we now give a new proof of this result based on an analysis of the euler discretization given in 9 there we exploit the lipschitz continuity of the control to estimate the error in the euler discretization here we show that the theory developed for the euler discretization can be used to derive the lipschitz continuity of the optimal control

existence and lipschitz regularity of solutions to bolza problems in optimal control in this paper we investigate the existence and lipschitz continuity of optimal trajectories for the autonomous bolza problem in control theory the main feature of our results is that they relax the usual fast growth condition for the lagrangian furthermore we show that optimal solutions do satisfy the maximum principle

convergence analysis for approximations of optimal control problems subject to higher index differential algebraic equations and mixed control state constraints this paper establishes a convergence result for implicit euler discretizations of optimal control problems with differential algebraic equations of higher index and mixed control state constraints

no gap second order optimality conditions for optimal control problems with a single state constraint and control the paper deals with optimal control problems with only one control variable and one state constraint of arbitrary order we consider the case of finitely many boundary arcs and touch times we obtain a no gap theory of second order conditions allowing to characterize second order quadratic growth

bilevel optimal control problems with pure state constraints and finite dimensional lower level this paper focuses on the development of optimality conditions for a bilevel optimal control problem with pure state constraints in the upper level and a finite dimensional parametric optimization problem in the lower level after transforming the problem into an equivalent single level problem we concentrate on the derivation of a necessary optimality condition of pontryagin type we point out some major difficulties arising from the bilevel structure of the original problem and its pure state constraints in the upper level leading to a degenerated maximum principle in the absence of constraint qualifications hence we use a partial penalization approach and a well known regularity condition for optimal control problems with pure state constraints to ensure the nondegeneracy of the derived maximum principle finally we illustrate the applicability of the derived theory by means of a small example

some comments on dae theory for irk methods and trajectory optimization it has been observed elsewhere in the literature that the activation of constraints in a trajectory optimization problem can lead to higher index daes several existing codes can handle a number of these constrained problems in this note we will discuss why the situation is more complex than just saying a higher index dae occurs the discussion is in the context of a specific industrial code socs but the observations made here have relevance for a number of methods and have implications for what types of test problems a code should be tested on

a connection between the maximum principle and dynamic programming for constrained control problems we consider the mayer optimal control problem with dynamics given by a nonconvex differential inclusion whose trajectories are constrained to a closed set and obtain necessary optimality conditions in the form of the maximum principle together with a relation between the costate and the value function this additional relation is applied in turn to show that the maximum principle is nondegenerate we also provide a sufficient condition for the normality of the maximum break principle to derive these results we use convex linearizations of differential inclusions and convex linearizations of constraints along optimal trajectories then duality theory of convex analysis is applied to derive necessary conditions for optimality in this way we extend the known relations between the maximum principle and dynamic programming from the unconstrained problems to the constrained case

mathematical programs with complementarity constraints we study mathematical programs with complementarity constraints several stationarity concepts based on a piecewise smooth formulation are presented and compared the concepts are related to stat

lipschitz continuity of optimal controls for state constrained problems this paper provides new conditions under which optimal controls are lipschitz continuous for dynamic optimization problems with functional inequality constraints a control constraint expressed in terms of a general closed convex set and a coercive cost function it is shown that the linear independence condition on active state constraints present in the earlier literature can be replaced by a less restrictive positive linear independence condition that requires linear independence merely with respect to nonnegative weighting parameters smoothness conditions on the data imposed in earlier work are also relaxed the new conditions for lipschitz continuity of optimal controls are obtained by a detailed analysis of the implications of first order optimality conditions in the form of a nonsmooth maximum principle

conditions for generic normality in optimal control abstract optimal control problems in which the necessary conditions of pontryagin s maximum principle do not involve the cost or performance index are called abnormal in this paper we study normality through perturbations of the endpoint set it is known that normality holds for all problems obtained by translating the original endpoint set in directions belonging to a dense set when the equations of motion are linear in the state variable we enlarge this set of directions to a full lebesgue measure set showing that its complement is contained in the relative boundary of a convex set for nonlinear systems we show that by enlarging the endpoint set instead of translating it normality is also guaranteed almost everywhere

optimal strategies for bilevel dynamic problems in this paper we study the bilevel dynamic problem which is a hierarchy of two dynamic optimization problems where the constraint region of the upper level problem is determined implicitly by the solutions to the lower level optimal control problem to obtain optimality conditions we reformulate the bilevel dynamic problem as a single level optimal control problem that involves the value function of the lower level problem sensitivity analysis of the lower level problem with respect to the perturbation in the upper level decision variable is given and first order necessary optimality conditions are derived by using nonsmooth analysis a constraint qualification of calmness type and a sufficient condition for the calmness are also given

a cauchy problem in nonlinear heat conduction a cauchy problem on the semiline for a nonlinear diffusion equation is considered with a boundary condition corresponding to a prescribed thermal conductivity at the origin the problem is mapped into a moving boundary problem for the linear heat equation with a robin type boundary condition such a problem is then reduced to a linear integral volterra equation of ii type which admits a unique solution

brief infinite horizon predictive control of constrained continuous time linear systems this paper considers receding horizon control strategies for constrained linear systems continuous time predictions of input output responses on an infinite horizon allow exploitation of the full range of plant behaviour in addition an infinite prediction horizon ensures the nominal stability of the receding horizon control law the problem of optimizing predictions subject to constraints is made tractable through the use of a finite dimensional prediction class and by minimizing prediction costs at discrete instants necessary and sufficient conditions on the prediction class are derived using input output stability theory and the controller structure further refined on the basis of a criterion for ensuring feasibility in a receding horizon sense constrained optimization is performed using quadratic programming

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

out of plane dynamic analysis of beams with arbitrarily varying curvature and cross section by dynamic stiffness matrix method abstract the first known dynamic stiffness matrix for noncircular curved beams with variable cross section is developed with which an exact solution of the out of plane free vibration of this type of beam is derived by using the laplace transform technique and the developed dynamic stiffness matrix and equivalent nodal force vector the highly accurate dynamic responses including the stress resultants of the curved beams subjected to various types of loading can be easily obtained the dynamic stiffness matrix and equivalent nodal force vector are derived based on the general series solution of the differential equations for the out of plane motion of the curved beams with arbitrary shapes and cross sections the validity of the present solution for free vibration is demonstrated through comparison with published data the accuracy of the present solution for transient response is also confirmed through comparison with the modal superposition solution for a simply supported circular beam subjected to a moving load with the proposed solution both the free vibration and forced vibration of non uniform parabolic curved beams with various ratios of rise to span are carried out nondimensional frequency parameters for the first five modes are presented in graphic form over a range of rise to span ratios 0 05 h l 0 75 with different variations of the cross section dynamic responses of the fixed fixed parabolic curved beam subjected to a rectangular pulse are also presented for different rise to span ratios

application of a reciprocal transformation to a two phase stefan problem a reciprocal transformation is employed to reduce a two phase stefan problem in nonlinear heat conduction to a form which admits a class of exact solutions analogous to the classical newmann solution

nonlinear self adjointness and conservation laws the general concept of nonlinear self adjointness of differential equations is introduced it includes the linear self adjointness as a particular case moreover it embraces the strict self adjointness definition 1 and quasi self adjointness introduced earlier by the author it is shown that the equations possessing nonlinear self adjointness can be written equivalently in a strictly self adjoint form by using appropriate multipliers all linear equations possess the property of nonlinear self adjointness and hence can be rewritten in a nonlinear strictly self adjoint form for example the heat equation ut u 0 becomes strictly self adjoint after multiplying by u 1 conservation laws associated with symmetries are given in an explicit form for all nonlinearly self adjoint partial differential equations and systems

necessary conditions for optimal control problems with state constraints necessary conditions of optimality are derived for optimal control problems with pathwise state constraints in which the dynamic constraint is modelled as a differential inclusion the novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid an euler lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded nonconvex sets and satisfies a mild one sided lipschitz continuity hypothesis we recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by ioffe for problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one sided lipschitz hypothesis is replaced by a milder local hypothesis a recent dualization theorem permits us to infer a strengthened form of the hamiltonian inclusion from the euler lagrange condition the necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by loewen and rockafellar to achieve related necessary conditions for state constrained problems and improve on available results in certain respects even when specialized to the state constraint free case proofs make use of recent decoupling ideas of the authors which reduce the optimization problem to one to which pontryagin s maximum principle is applicable and a refined penalization technique to deal with the dynamic constraint

sqp methods for solving optimal control problems with control and state constraints adjoint variables sensitivity analysis and real time control parametric nonlinear optimal control problems subject to control and state constraints are studied two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which sqp methods provide efficient solution methods it is shown that sqp methods can be used also for a check of second order sufficient conditions and for a postoptimal calculation of adjoint variables in addition sqp methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters numerical sensitivity analysis is the basis for real time control approximations of perturbed solutions which are obtained by evaluating a first order taylor expansion with respect to the parameter the proposed numerical methods are illustrated by the optimal control of a low thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics the examples illustrate the robustness accuracy and efficiency of the proposed numerical algorithms

continuous time linear quadratic regulator with control constraints via convex duality a continuous time infinite horizon linear quadratic regulator with input constraints is studied optimality conditions both in the open loop and feedback form and continuity and differentiability properties of the optimal value function and of the optimal feedback are shown arguments rely on basic ideas of convex conjugacy and in particular use a dual optimal control problem

regularity properties of optimal controls a class of optimal control problems is considered involving integral cost functions and linear dynamics broad verifiable hypotheses he are known under which the problems have solutions these must be supplemented for standard optimality conditions in the form of a maximum principle to apply new regularity properties of optimal controls when merely hypotheses he are in force are established a byproduct is a maximum principle governing optimal controls under such circumstances where the ostate function is possibly unbounded criteria for minimizing controls to be essentially bounded are also deduced

two dimensional object recognition using a two dimensional polar transform abstract a two dimensional 2d transform is proposed for the classification of planar objects with a centroid referenced polar representation it samples the multiple intersections of n radii with the object using the mass center translation invariance is achieved and with an appropriate amplitude normalization the transform is also made invariant to scaling the modules of its 2d fast fourier transform coefficients which are rotation invariant are used as the feature vector simultaneously translation scaling and rotation parameters are estimated the high performance of this algorithm is shown with computer simulations and its advantages and disadvantages are analysed in comparison with existing methods

variational approach to second order sufficient optimality conditions in optimal control we consider a bolza optimal control problem with convex pure state constraints and general control constraints and propose new second order sufficient optimality conditions for a weak local minimum that can be applied in a general framework in particular the reference control may be merely measurable and the dynamics may be discontinuous in the time variable our approach is based on a new decomposition result for control constraints involving second order tangents

application of reciprocal backlund transformations to a class of nonlinear boundary value problems it is shown that reciprocal backlund transformations may be used to reduce an important class of nonlinear boundary value problems for higher order nonlinear diffusion equations to linear canonical boundary value problems

lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands the study of regularity properties for minimizers of problems of the form min u u dt u e iv q r u a a u b b 1 1 has been undertaken for a long time in calculus of variations and a number of results are by now available on this subject when f t s z is a smooth coercive function problem 1 1 was considered by tonelli see 18 19 who proved that every minimizer u t of the problem 1 1 is locally lipschitz on a relatively open set of full measure in a b under some additional assumptions on f it is also possible to prove global regularity theorems see for instance cesari 3 chap 2 sect 61 in recent years much attention has been devoted to problems of the form l l on the one hand in weakening the hypotheses of tonelli s regularity results and on the other in finding counterexamples of nonregular minimizers for coercive functionals with respect to this last subject we recall that it has been shown see ball and mizel l and clarke and vinter 6 that there exist polynomial functions f t s z such that for suitable c 0 and p

on constrained infinite time linear quadratic optimal control abstract this work presents a solution to the infinite time linear quadratic optimal control itlqoc problem with state and control constraints it is shown that a single finite dimensional convex program of known size can yield this solution properties of the resulting value function with respect to initial conditions are also established and are shoen to be useful in determining the aforementioned problem size an example illustrating the method is finally presented

when is the maximum principle for state constrained problems nondegenerate abstract it is well known that a form of the pontryagin maximum principle applies to optimal control problems involving a unilateral state constraint we discuss how the need arises to solve problems of this nature when the left endpoint is fixed and when the initial state lies in the boundary of the state constraint set region in such cases previous versions of the maximum principle convey no information since a trivial choice of multipliers may be made namely one in which the state constraint multiplier is a unit measure concentrated at the left endtime and all the other multipliers are zero we prove strengthened forms of the maximum principle applicable to such situations constraint qualifications are formulated under which multipliers exist besides the trivial ones

brief paper least restrictive move blocking model predictive control move blocking lowers the computational complexity of model predictive control mpc problems by reducing the number of optimization variables however this may render states close to constraints infeasible thus move blocking generally results in control laws that are restrictive the controller domains may be unacceptably and unnecessarily small furthermore different move blocking strategies may result in controller domains of different sizes all other factors being equal in this paper an approach is proposed to design move blocking mpc control laws that are least restrictive i e the controller domain is equal to the maximum controlled invariant set the domains of different move blocking controllers are then by design equal to each other this allows comparison of differing move blocking strategies based on cost performance only without needing to consider domain size also thus this paper is a step towards being able to derive optimal move blocking mpc control laws

lipschitzian stability for state constrained nonlinear optimal control for a nonlinear optimal control problem with state constraints we give conditions under which the optimal control depends lipschitz continuously in the l2 norm on a parameter these conditions involve smoothness of the problem data uniform independence of active constraint gradients and a coercivity condition for the integral functional under these same conditions we obtain a new nonoptimal stability result for the optimal control in the l infty norm and under an additional assumption concerning the regularity of the state constraints a new tight l infty estimate is obtained our approach is based on an abstract implicit function theorem in nonlinear spaces

runge kutta methods in optimal control and the transformed adjoint system the convergence rate is determined for runge kutta discretizations of nonlinear control problems the analysis utilizes a connection between the kuhn tucker multipliers for the discrete problem and the adjoint variables associated with the continuous minimum principle this connection can also be exploited in numerical solution techniques that require the gradient of the discrete cost function

linear estimates for trajectories of state constrained differential inclusions and normality conditions in optimal control abstract we consider solutions to a differential inclusion x f x constrained to a compact convex set here f is a compact possibly non convex valued lipschitz continuous multifunction whose convex closure co f satisfies a strict inward pointing condition at every boundary point x given a reference trajectory x taking values in an e neighborhood of we prove the existence of a second approximating trajectory x 0 t which satisfies the linear estimate x x ac 0 t k e if one of the following two cases occurs i the initial datum x 0 x 0 is given but lies in a compact set containing only points where the boundary is smooth ii the initial point x 0 of the approximating trajectory x can be chosen arbitrarily subsequently we employ these linear ac estimates to establish conditions for normality of the generalized euler lagrange condition for optimal control problems with state constraints in which we have an integral term in the cost we finally provide an illustrative example which underlines the fact that if conditions i and ii above are not satisfied then we can find a degenerate minimizer

whither discrete time model predictive control this note proposes an efficient computational procedure for the continuous time input constrained infinite horizon linear quadratic regulator problem clqr to ensure satisfaction of the constraints the input is approximated as a piecewise linear function on a finite time discretization the solution of this approximate problem is a standard quadratic program a novel lower bound on the infinite dimensional clqr problem is developed and the discretization is adaptively refined until a user supplied error tolerance on the clqr cost is achieved the offline storage of the required quadrature matrices at several levels of discretization tailors the method for online use as required in model predictive control mpc the performance of the proposed algorithm is then compared with the standard discrete time mpc algorithms the proposed method is shown to be significantly more efficient than standard discrete time mpc that uses a sample time short enough to generate a cost close to the clqr solution

lipschitz continuity and local semiconcavity for exit time problems with state constraints in the classical time optimal control problem it is well known that the so called petrov condition is necessary and sufficient for the minimum time function to be locally lipschitz continuous in this paper the same regularity result is obtained in the presence of nonsmooth state constraints moreover for a special class of control systems we obtain a local semiconcavity result for the constrained minimum time function

a nondegenerate maximum principle for the impulse control problem with state constraints in this article a free time impulsive control problem with state constraints and equality and inequality constraints on the trajectory endpoints is considered a weakened maximum principle is obtained for problems with data measurable in the time variable being the time transversality conditions deduced with the help of some extra convexity assumption on the state constraints in the case of smooth problems a nondegenerate maximum principle is derived by using a penalty function method

on some continuity properties of the measure lagrange multiplier from the maximum principle for state constrained problems the continuity of the lagrange multiplier mu t from the maximum principle for control problems with state constraints is investigated it is proved under certain regularity assumptions that the function mu t is holder continuous with exponent 1 2 under regularity conditions plus the strong legendre condition this function is lipschitz continuous

second order runge kutta approximations in control constrained optimal control in this paper we analyze second order runge kutta approximations to a nonlinear optimal control problem with control constraints if the optimal control has a derivative of bounded variation and a coercivity condition holds we show that for a special class of runge kutta schemes the error in the discrete approximating control is o h2 where h is the mesh spacing

convergence of nonconvergent irk discretizations of optimal control problems with state inequality constraints it has been observed that optimization codes are sometimes able to solve inequality state constrained optimal control problems with discretizations which do not converge when used as integrators on the constrained dynamics understanding this phenomenon could lead to a more robust design for direct transcription codes as well as better test problems this paper examines how this phenomenon can occur the difference between solving index 3 differential algebraic equations daes using the trapezoid method in the context of direct transcription for optimal control problems and a straightforward implicit runge kutta irk formulation of the same trapezoidal discretization is analyzed it is shown through numerical experience and theory that the two can differ as much as o 1 h3 in the control the optimization can use a small sacrifice in the accuracy of the states in the early stages of the trapezoidal method to produce better accuracy in the control whereas more precise solutions converge to an incorrect solution convergence independent of the index of the constraints is then proven for one class of problems the theoretical results are used to explain computational observations

a two phase free boundary problem for a nonlinear diffusion convection equation a two phase free boundary problem associated with a diffusion convection equation is considered the problem is reduced to a system of nonlinear integral equations which admits a unique solution for small times the system admits an explicit two component solution corresponding to a two component shock wave of the burgers equation the stability of such a solution is also discussed

nondegenerate necessary conditions for nonconvex optimal control problems with state constraints standard versions of the maximum principle for optimal control problems with pathwise state inequality constraints are satisfied by a trivial set of multipliers in the case when the left endpoint is fixed and lies in the boundary of the state constraint set and so give no useful information about optimal controls recent papers have addressed the problem of overcoming this degenerate feature of the necessary conditions in these papers it is typically shown that if a constraint qualification is imposed requiring existence of inward pointing velocities then sets of multipliers exist in addition to the trivial ones a simple new approach for deriving nondegenerate necessary conditions is presented which permits relaxation of hypotheses previously imposed concerning data regularity and convexity of the velocity set