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

a duality approach for solving control constrained linear quadratic optimal control problems we use a fenchel duality scheme for solving control constrained linear quadratic optimal control problems we derive the dual of the optimal control problem explicitly where the control constraints are embedded in the dual objective functional which turns out to be continuously differentiable we specifically prove that strong duality and saddle point properties hold we carry out numerical experiments with the discretized primal and dual formulations of the problem for which we implement powerful existing finite dimensional optimization techniques and associated software we illustrate that by solving the dual of the optimal control problem instead of the primal one significant computational savings can be achieved other numerical advantages are also discussed

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

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

higher order moreau s sweeping process mathematical formulation and numerical simulation in this paper we present an extension of moreau s sweeping process for higher order systems the dynamical framework is carefully introduced qualitative dissipativity stability existence regularity and uniqueness results are given the time discretization of these nonsmooth systems with a time stepping algorithm is also presented this differential inclusion can be seen as a mathematical formulation of complementarity dynamical systems with arbitrary dimension and arbitrary relative degree between the complementary slackness variables applications of such high order sweeping processes can be found in dynamic optimization under state constraints and electrical circuits with ideal diodes

an inverse theorem for the gowers u s 1 n norm we prove the inverse conjecture for the gowers u s 1 n norm for all s 1 this is new for s 4 more precisely we establish that if f n 1 1 is a function with f u s 1 n d then there is a bounded complexity s step nilsequence f g n g that correlates with f where the bounds on the complexity and correlation depend only on s and d from previous results this conjecture implies the hardy littlewood prime tuples conjecture for any linear system of finite complexity

nonconventional ergodic averages and nilmanifolds we study the l2 convergence of two types of ergodic averages the first is the average of a product of functions evaluated at return times along arithmetic progressions such as the expressions appearing in furstenberg fs proof of szemer ledi fs theorem the second average is taken along cubes whose sizes tend to for each average we show that it is sufficient to prove the convergence for special systems the characteristic factors we build these factors in a general way independent of the type of the average to each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold from the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density

minimizing the transition time in lasers by optimal control methods single mode semiconductor laser with homogeneous transverse profile the introduction of patterned current fronts that steer the transition between directly modulated optical bits allows for a strong modification in the shape of the trajectory evolving from the initial to the final state in semiconductor lasers the problem of minimal transition time between two stationary points of the system will be formulated in the framework of optimal control theory in this approach the patterned current fronts serve as time dependent control functions for which upper and lower bounds are imposed we first calculate the optimal control on a lotka volterra model which qualitatively well describes the dynamic response of several types of lasers the optimal control is of bang bang type and switches from the upper to the lower value of the control bounds then we consider a more specific semiconductor laser model and show again that the optimal control is bang bang and calculate specific trajectories for the transition between the below threshold state and one where the laser is emitting the sensitivity of the trajectory to external perturbations is also studied

the adjoint arc in nonsmooth optimization we extend the theory of necessary conditions for nonsmooth problems of bolza in three ways first we incorporate state constraints of the intrinsic type x t e x t for all t second we make no assumption of calmness or normality and third we show that a single adjoint function of bounded variation simultaneously satisfies the hamiltonian inclusion the euler lagrange inclusion and the weierstrass pontryagin maximum condition along with the usual transversality relations

nonlinear optimal control via occupation measures and lmi relaxations we consider the class of nonlinear optimal control problems ocps with polynomial data i e the differential equation state and control constraints and cost are all described by polynomials and more generally for ocps with smooth data in addition state constraints as well as state and or action constraints are allowed we provide a simple hierarchy of lmi linear matrix inequality relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value under some convexity assumptions the sequence converges to the optimal value of the ocp preliminary results show that good approximations are obtained with few moments

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

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

lossless convexification of non convex optimal control problems for state constrained linear systems this paper analyzes a class of finite horizon optimal control problems with mixed non convex and convex control constraints and linear state constraints a convex relaxation of the problem is proposed and it is proved that a solution of the relaxed problem is also a solution of the original problem this process is called lossless convexification and its generalization for problems with state constraints is the primary contribution of the paper doing so enables the use of interior point methods of convex optimization to obtain global optimal solutions of the original non convex problem the approach is also demonstrated on example problems

global convergence of a nonsmooth newton method for control state constrained optimal control problems we investigate a nonsmooth newton method for the numerical solution of optimal control problems subject to mixed control state constraints the necessary conditions are stated in terms of a local minimum principle by use of the fischer burmeister function the local minimum principle is transformed into an equivalent nonlinear and nonsmooth equation in appropriate banach spaces this nonlinear and nonsmooth equation is solved by a nonsmooth newton s method we prove the global convergence and the locally superlinear convergence under certain regularity conditions the globalized method is based on the minimization of the squared residual norm numerical examples for the rayleigh problem conclude the article

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

brief paper optimal control problems with a continuous inequality constraint on the state and the control we consider an optimal control problem with a nonlinear continuous inequality constraint both the state and the control are allowed to appear explicitly in this constraint by discretizing the control space and applying a novel transformation a corresponding class of semi infinite programming problems is derived a solution of each problem in this class furnishes a suboptimal control for the original problem furthermore we show that such a solution can be computed efficiently using a penalty function method on the basis of these two ideas an algorithm that computes a sequence of suboptimal controls for the original problem is proposed our main result shows that the cost of these suboptimal controls converges to the minimum cost for illustration an example problem is solved

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

direct optimization methods for solving a complex state constrained optimal control problem in microeconomics abstract we analyze and solve a complex optimal control problem in microeconomics which has been investigated earlier in the literature the complexity of the control problem originates from four control variables appearing linearly in the dynamics and several state inequality constraints thus the control problem offers a considerable challenge to the numerical analyst we implement a hybrid optimization approach which combines two direct optimization methods the first step consists in solving the discretized control problem by nonlinear programming methods the second step is a refinement step where in addition to the discretized control and state variables the junction times between bang bang singular and boundary subarcs are optimized the computed solutions are shown to satisfy precisely the necessary optimality conditions of the maximum principle where the state constraints are directly adjoined to the hamiltonian despite the complexity of the control structure we are able to verify sufficient optimality conditions which are based on the concavity of the maximized hamiltonian

nil bohr0 sets and polynomial recurrence abstract in huang et al 17 it was proved that for any nild bohr0 set a there are a minimal system x t and a non empty open subset u of x with a n z u t n u t d n u 0 and for any minimal system x t and any open non empty u x the set n z u t n u t d n u 0 is an almost nild bohr0 set the polynomial form of this problem is considered in this paper it is shown that the latter is still true in the polynomial case while the former is not in general we also consider the special case when the system is a nilsystem

stability analysis of optimal control problems with a second order state constraint this paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second order the strengthened legendre clebsch condition is assumed to hold and no assumption on the structure of the contact set is made under a weak second order sufficient condition taking into account the active constraints we show that the solutions are lipschitz continuous w r t the perturbation parameter in the l 2 norm and holder continuous in the l infty norm we use a generalized implicit function theorem in metric spaces by dontchev and hager siam j control optim 36 1998 pp 698 718 the difficulty is that multipliers associated with second order state constraints have a low regularity they are only bounded measures we obtain lipschitz stability of a primitive of the state constraint multiplier

ipr recurrence and nilsystems abstract by a result due to furstenberg a homeomorphism t of a compact space is distal if and only if it possesses the property of ip recurrence meaning that for any x 0 x for any open neighborhood u of x 0 and for any sequence n i in z the set r u x 0 n z t n x 0 u has a non trivial intersection with the set of finite sums n i 1 n i 2 n i s i 1 i 2 i s s n we show that translations on compact nilmanifolds which are known to be distal are characterized by a stronger property of ip r recurrence which asserts that for any x 0 x and any neighborhood u of x 0 there exists r n such that for any r element sequence n 1 n r in z the set r u x 0 has a non trivial intersection with the set n i 1 n i 2 n i s i 1 i 2 i s s r we also show that the property of ip r recurrence is equivalent to an ostensibly much stronger property of polynomial ip r recurrence this should be juxtaposed with the fact that for general distal transformations the polynomial ip recurrence is strictly stronger than the ip recurrence

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

a model for cancer chemotherapy with state space constraints abstract the analysis of optimal control problems is significantly more difficult when state space constraints are imposed and thus in many modelling approaches explicit constraints are handled by including penalty terms in the objective in this paper we use the example of a simple model for cancer chemotherapy to compare the structure of extremals when upper limits on the total number of cancer cells are imposed a implicitly through a penalty term in the objective and b explicitly in the form of state space constraints

receding horizon boundary control of nonlinear conservation laws with shock avoidance this paper deals with the boundary control for scalar non linear hyperbolic systems of conservation laws one of the issues arising from these systems is the occurrence of singularities called shocks in the domain we will show that to avoid the shock a set of constraints on both the control and the state at the boundary has to be fulfilled then a proof of the exponential stability of the system is established provided that there is no shock and that the state at the boundary is exponentially stable these conditions are shown to be achieved by the receding horizon optimal control approach a simulation is finally carried out with the freeway traffic model to demonstrate the effectiveness of the here proposed control

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

rate limited time optimal control of a planar pendulum abstract time optimal control problems often admit as solution bang bang controls but they present several drawbacks in real applications where additional degrees of regularity are desirable in this paper the rate limited time optimal swing up of an inverted pendulum is considered and necessary conditions that have to be satisfied by optimal trajectories in the presence of rate constraints are derived a comparison between this approach and the results from the standard time optimal problem is reported

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

a proof of a general maximum principle for optimal controls via a multiplier rule on metric space a unified proof is given of the maximum principle for optimal control with various kinds of constraints by using a multiplier rule on metric spaces

a combinatorial graph based solution method for a class of continuous time optimal control problems the paper addresses a class of continuous time optimal control problems whose solutions are typically characterized by both bang bang and singular control regimes analytical study and numerical computation of such solutions are very difficult and far from complete when only techniques from control theory are used this paper solves optimal control problems by reducing them to the combinatorial search for the shortest path in a specially constructed graph since the nodes of the graph are weighted in a sequence dependent manner we extend the classical shortest path algorithm to our case the proposed solution method is currently limited to single state problems with multiple control functions a production planning problem and a train operation problem are optimally solved to illustrate the method

application of optimal control method in buckling analysis of constrained elastica problems abstract main concern of the paper is to illustrate the generic application of the optimal control method in the analysis of a constrained buckling problem and its simple and elegant formulation when compared to other techniques thus an optimal control methodology is adopted to investigate the buckling behavior of thin elastic structures under the presence of unilateral constraints we particularly explore the post buckling response of a constant or variable length elastica constrained by rigid walls the equivalence between the calculus of variations and the optimal control is first demonstrated with a main focus on the post buckling behavior of a constant length elastica which is constrained by horizontal walls the gradual construction of the constrained buckling problem as an optimal control problem from its equivalent lagrangian form clearly shows its advantage when compared to the calculus of variations the necessary optimality conditions which constitute the pontryagin s minimum or maximum principle are also derived by considering a direct adjoining approach the solution of the optimal control problem is then performed by applying a direct method validation of the methodology is first achieved by reproducing examples available in the literature then the effects of factors such as the geometry of the walls and the variability in the bending stiffness of the elastica on its buckling response are analyzed these constrained buckling problems are investigated for the first time while through them it is also readily shown that the presence of geometric or material nonlinearity does not introduce any essential complexity in the buckling analysis when the optimal control methodology is adopted

multiplier methods for nonlinear optimal control error estimates are derived for an augmented lagrangian approximation to an optimal control problem convex control constraints are treated explicitly while a lagrange multiplier is introduced for the nonlinear system dynamics the nonlinear optimal control problem does not fit the classical theory for estimating the error in multiplier approximations since the natural coercivity assumption is formulated in a hilbert space where the cost is not differentiable this discrepancy between the function space setting needed for coercivity and that needed for differentiability is compensated for by regularity results associated with the necessary conditions the paper concludes with an analysis of the optimal penalty parameter corresponding to a given finite element discretization

multiple recurrence and nilsequences aiming at a simultaneous extension of khintchine s and furstenberg s recurrence theorems we address the question if for a measure preserving system x mathcal x mu t and a set a in mathcal x of positive measure the set of integers n such that mu a cap t n a cap t 2n a cap ldots cap t kn a mu a k 1 epsilon is syndetic the size of this set surprisingly enough depends on the length k 1 of the arithmetic progression under consideration in an ergodic system for k 2 and k 3 this set is syndetic while for k 4 it is not the main tool is a decomposition result for the multicorrelation sequence int f x f t n x f t 2n x ldots f t kn x d mu x where k and n are positive integers and f is a bounded measurable function we also derive combinatorial consequences of these results for example showing that for a set of integers e with upper banach density d e 0 and for all e 0 the set big n in mathbb z colon d big e cap e n cap e 2n cap e 3n big d e 4 epsilon big is syndetic

high speed switch on of a semiconductor gas discharge image converter using optimal control methods abstract experiments evidence that optical commutation of current in semiconductor gas discharge devices from a low to a high value may be accompanied by an oscillatory mode of such a transient and by a long effective time for the transition for a simple two component model of transport processes in these nonlinear systems which encounter the main observed features in dynamics the problem of minimizing the time of switching on the high current state is considered this problem can be formulated as an optimal control problem with control function provided by a proper temporal variation of the feeding voltage it is shown that the optimal control can substantially shorten the effective transient time of the process and totally suppress the occurrence of an overshooting in a transient optimal control strategies for constraints on control and state variables and for different parameters are presented

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

on reproducing kernels and quantization of states quantization of a mechanical system with the phase space a kahler manifold is studied it is shown that the calculation of the feynman path integral for such a system is equivalent to finding the reproducing kernel function the proposed approach is applied to a scalar massive conformal particle interacting with an external field which is described by deformation of a hermitian line bundle structure

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

an optimal control model of oil discovery and extraction starting with hotelling 1931 7 the stock of non renewable resources have been treated as fixed along the line of pindyck 1978 8 and greiner and semmler in press 5 we treat the stock of oil resources as time varying depending on new discoveries the resource is finite and only a part of the resource is known while the rest has not yet been discovered the discovery leads to a rise of known oil resource which can then be optimally exploited the optimal control model has two state variables the known stock of the resource and the cumulated past extraction the control variable is the optimal extraction rate the optimal control model assumes a monopolistic resource owner who maximizes intertemporal profits from exploiting the resource where the price of the resource depends on the extraction rate the known stock of the resource and the cumulated past extraction the model is solved for a finite time horizon using nudocccs a numerical solution method to solve finite horizon optimal control problems various parameter constellations are explored for certain parameter constellations the price path becomes u shaped as some empirical research see greiner and semmler in press 5 have found to hold for actual price data this holds if the stock of the initially known resource is small

on the structure of lagrange multipliers for state constrained optimal control problems abstract state constrained optimal control problems represent severe analytical and numerical challenges it is shown that under certain conditions on the active set the measure representing the lagrange multiplier associated to the state constraint can be decomposed in a distributed part in l2 with support on the active set and a boundary measure concentrated on the interface between active and inactive sets

sequences from zero entropy noncommutative toral automorphisms and sarnak conjecture abstract in this paper we study c algebra version of sarnak conjecture for noncommutative toral automorphisms let a be a noncommutative torus and be the noncommutative toral automorphism arising from a matrix s g l d z we show that if the voiculescu brown entropy of is zero then the sequence n u n z is a sum of a nilsequence and a zero density sequence where u a and is any state on a then by a result of green and tao 9 this sequence is linearly disjoint from the mobius function

sufficient optimality conditions for optimal control subject to state constraints strong second order sufficient optimality conditions ssc are derived for optimal control problems of systems described by nonlinear odes subject to mixed control state and pure state constraints the obtained ssc are expressed in terms of a modified legendre clebsch condition and the associated riccati equation the role of ssc in stability analysis of solutions to parametric optimal control problems is briefly discussed

optimal control problems with mixed control state constraints noindent we consider control problems governed by semilinear parabolic equations in the presence of pointwise mixed control state constraints we obtain optimality conditions with finitely additive measures as multipliers associated to the mixed constraints we study the regularity of the multipliers for different problems in particular when a monotonicity or a separation condition is satisfied we prove that multipliers are bounded measurable functions

poincare recurrence and number theory thirty years later hillel furstenberg s 1981 article in the bulletin gives an elegant introduction to the interplay between dynamics and number theory summarizing the major developments that occurred in the few years after his landmark paper 21 the field has evolved over the past thirty years with major advances on the structural analysis of dynamical systems and new results in combinatorics and number theory furstenberg s article continues to be a beautiful introduction to the subject drawing together ideas from seemingly distant fields furstenberg s article 21 gave a general correspondence between regularity properties of subsets of the integers and recurrence properties in dynamical systems now dubbed the furstenberg correspondence principle he then showed that such recurrence properties always hold proving what is now referred to as the multiple recurrence theorem combined these results gave a new proof of szemeredi s theorem 45 if s z has positive upper density then s contains arbitrarily long arithmetic progressions this proof lead to an explosion of activity in ergodic theory and topological dynamics beginning with new proofs of classic results of ramsey theory and ultimately leading to significant new combinatorial and number theoretic results the full implications of these connections have yet to be understood the approach harks back to the earliest results on recurrence in the measurable setting and in the topological setting a measure preserving system is a quadruple x b t where x denotes a set b is a algebra on x is a probability measure on x b and t x x is a measurable transformation such that t 1 a a for all a b poincare recurrence states that if x b t is a measure preserving system and a b with a 0 there exists n n such that a t na 0 a topological dynamical system is a pair x t where x is a compact metric space and t x x is a continuous map one can show that any such topological space admits a borel probability measure that preserves t in particular poincare recurrence implies recurrence in the topological setting birkhoff 13 gave a direct proof of

dual approximations in optimal control we analyze a dual approximation for the solution to an optimal control problem the differential equation is handled with a lagrange multiplier while other constraints are treated explicitly an algorithm for solving the dual problem is presented

hyperfunction solutions of the zero rest mass field equations in this paper it is shown how the penrose transform maps tangential hyperfunction dolbeault groups with coefficients in a power of the hyperplane section bundle on the hyperquadric of null twistors in projective twistor space isomorphically to all hyperfunction solutions of the massless field equations of nonnegative helicity on compactified minkowski space this is an extension of the penrose transform which generated real analytic solutions of the same field equations on the same space cf eastwood m penrose r wells r o 10 in additions one obtains the result that each hyperfunction solution of the massless field equations of nonnegative helicity is the sum of massless fields of positive and negative frequency a generalization of the usual fourier decomposition for solutions with appropriate growth conditions

time optimal zermelo s navigation problem with moving and fixed obstacles in this paper we consider a time optimal zermelo s navigation problem znp with moving and fixed obstacles this problem can be formulated as an optimal control problem with continuous inequality constraints and terminal state constraints by using the control parametrization technique together with the time scaling transform the problem is transformed into a sequence of optimal parameters selection problems with continuous inequality constraints and terminal state constraints for each problem an exact penalty function method is used to append all the constraints to the objective function yielding a new unconstrained optimal parameters selection problem it is solved as a nonlinear optimization problem different scenarios are considered in the simulation and the results obtained show that the proposed method is effective

a hardy field extension of szemeredi s theorem abstract in 1975 szemeredi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions bergelson and leibman showed in 1996 that the common difference of the arithmetic progression can be a square a cube or more generally of the form p n where p n is any integer polynomial with zero constant term we produce a variety of new results of this type related to sequences that are not polynomial we show that the common difference of the progression in szemeredi s theorem can be of the form n where is any positive real number and x denotes the integer part of x more generally the common difference can be of the form a n where a x is any function that is a member of a hardy field and satisfies a x x k and a x x k 1 0 for some non negative integer k the proof combines a new structural result for hardy sequences techniques from ergodic theory and some recent equidistribution results of sequences on nilmanifolds

method of evolving junctions a new approach to optimal control with constraints abstract we propose a new strategy called method of evolving junctions mej to compute the solutions for a class of optimal control problems with constraints on both state and control variables our main idea is that by leveraging the geometric structures of the optimal solutions we recast the infinite dimensional optimal control problem into an optimization problem depending on a finite number of points called junctions then using a modified gradient flow method whose dimension can change dynamically we find local solutions for the optimal control problem we also employ intermittent diffusion a global optimization method based on stochastic differential equations to obtain the global optimal solution we demonstrate via a numerical example that mej can effectively solve path planning problems in dynamical environments

a conformal holomorphic field theory a formulation of a field theory on the complex minkowski space in terms of complex differential geometry is proposed it is also shown that our model of field theory differs from the standard model on the real minkowski space only in the limit of high energy

necessary conditions in nonsmooth optimization the paper contains four theorems concerning first order necessary conditions for a minimum in nonsmooth optimization problems in banach spaces a lagrange multiplier rule for a mathematical programming problem in which an infinite dimensional equality constraint is included in the constraints a general maximum principle for nonsmooth optimal control problems with state constraints and a kind of multiplier rule for mathematical programming problems which applies when only finitely many equality constraints are present but when the lipschitz continuity assumptions are removed a summary of relevant background results from analysis is provided

error estimates for the euler discretization of an optimal control problem with first order state constraints we propose some error estimates for the discrete solution of an optimal control problem with first order state constraints where the trajectories are approximated with a classical euler scheme we obtain order 1 approximation results in the l infty norm as opposed to the order 2 3 results obtained in the literature we assume either a strong second order optimality condition or a weaker formulation in the case where the state constraint is scalar and satisfies some hypotheses for junction points and where the time step is constant our technique is based on some homotopy path of discrete optimal control problems that we study using perturbation analysis of nonlinear programming problems

metric regularity and stability of optimal control problems for linear systems this paper studies stability properties of the solutions of optimal control problems for linear systems the analysis is based on an adapted concept of metric regularity the strong bi metric regularity which is introduced and investigated in the paper it allows one to give a more precise description of the effect of perturbations on the optimal solutions in terms of a holder type estimate and to investigate the robustness of this estimate the holder exponent depends on a natural number k which is known as the controllability index of the reference solution an inverse function theorem for strongly bi metrically regular mappings is obtained which is used in the case k 1 for proving stability of the solution of the considered optimal control problem under small nonlinear perturbations moreover a new stability result with respect to perturbations in the matrices of the system is proved in the general case k geq 1

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

optimal control approach for system of ordinary differential equations a method is proposed for solving a system of ordinary differential equation using parametric optimal control solution the boundary conditions of the problem are used as control variables and the objective functional is considered as a certain experimental function necessary conditions of the investigated problem are presented the examples illustrate the robustness accuracy and efficiency of the proposed numerical technique

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

nilsequences and a structure theorem for topological dynamical systems abstract we characterize inverse limits of nilsystems in topological dynamics via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure preserving systems we provide two applications of the structure the first is to nilsequences which have played an important role in recent developments in ergodic theory and additive combinatorics we give a characterization that detects if a given sequence is a nilsequence by only testing properties locally meaning on finite intervals the second application is the construction of the maximal nilfactor of any order in a distal minimal topological dynamical system we show that this factor can be defined via a certain generalization of the regionally proximal relation that is used to produce the maximal equicontinuous factor and corresponds to the case of order 1

sequential virtual motion camouflage method for nonlinear constrained optimal trajectory control nonlinear constrained optimal trajectory planning is a challenging and fundamental area of research this paper proposes bio inspired fast time approaches for this type of problems based on the inspiration drawn from the natural phenomenon known as the motion camouflage two algorithms are proposed the virtual motion camouflage vmc subspace method and the sequential vmc method as a hybrid approach the sequential vmc method works through a two step structure in each iteration first the vmc subspace method will solve for an optimal solution over a selected subspace second an algorithm consisting of a linear programming and a line search will vary the subspace so that the next vmc subspace result will be guaranteed not to be worse than that of the current step the dimension and time complexities of the algorithms will be analyzed and the optimality of the solution via the sequential vmc approach will be studied through the vmc approaches the state and control variables in the kinematics or dynamics models of vehicles in the selected subspace can be represented by a single degree of freedom vector called the path control parameter vector the reduction in dimension and no involvement of equality constraints will in practice make the convergence faster and easier and a much smaller computational cost is expected two simulation examples the breakwell problem and a minimum time robot obstacle avoidance problem with different numbers of obstacles are used to demonstrate the capabilities of the algorithms

a characteristic method for fully convex bolza problems over arcs of bounded variation the aim of this paper is to study the value function of a fully convex bolza problem with state constraints and under no coercivity assumptions this requires that the state trajectories be of boun

stability analysis for nonlinear optimal control problems subject to state constraints there is considered a family of nonlinear optimal control problems depending on a functional parameter the problems are subject to state constraints conditions are derived under which the solutions and lagrange multipliers are locally lipschitz continuous functions of the parameter those conditions consist of constraint qualifications and weakened coercivity conditions

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

some results on optimal control with unilateral state constraints in this paper we study the problem of quadratic optimal control with state variables unilateral constraints for linear time invariant systems the necessary conditions are formulated as a linear invariant system with complementary slackness conditions some structural properties of this system are examined then it is shown that the problem can benefit from the higher order moreau s sweeping process that is a specific distributional differential inclusion and from ten dam s geometric theory for the partitioning of the admissible domain boundary in fact the first step may be also seen as follows does the higher order moreau s sweeping process developed in cite va bb dg jj2003 correspond to the necessary conditions of some optimal control problem with an adapted integral action the knowledge of the qualitative behaviour of optimal trajectories is improved with the approach which also paves the way towards efficient time stepping numerical algorithms to solve the optimal control boundary value problem

second order necessary conditions in pontryagin form for optimal control problems in this article we state and prove first and second order necessary conditions in pontryagin form for optimal control problems with pure state and mixed control state constraints we say that a lagrange multiplier of an optimal control problem is a pontryagin multiplier if it is such that pontryagin s minimum principle holds and we call optimality conditions in pontryagin form those which only involve pontryagin multipliers our conditions rely on a technique of partial relaxation and apply to pontryagin local minima

a hamiltonian approach to compute an energy efficient trajectory for a servomotor system this paper considers a nonlinear constrained optimal control problem ncocp originated from energy optimal trajectory planning of servomotor systems solving the exact optimal solution is challenging because of the nonlinear and switching cost function and various constraints this paper proposes a method to manage the switching cost function to establish a set of necessary conditions of an ncocp specifically a concept sub trajectory is introduced to match multiple hamiltonian due to switches in the cost function necessary conditions on the optimal trajectory are established as a union of conditions for all sub trajectories and weierstrass erdmann corner conditions between sub trajectories the set of feasible structures of optimal trajectories is further identified and represented by various state transition diagrams for the servomotor application a decomposition based shooting method is proposed to compute an optimal trajectory by solving multi point boundary value problems simulations and experiments validate the effectiveness of the methodology and the energy saving benefit

regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence abstract inverse limits of nilsystems in topologically dynamical systems were characterized by host kra and maass recently namely for each d n a certain generalization of the regionally proximal relation was introduced and for a distal minimal system it was shown that such a relation is an equivalence one which determines the maximal d step nilfactor one of the main results in this article is to show that the above results hold for a general minimal system a combinatorial consequence is also deduced which is the topological correspondence of the result obtained by host and kra for positive upper banach density subsets using ergodic methods

open loop optimal controller design using variational iteration method this article presents a design approach of a finite time open loop optimal controller using pontryagin s minimum principle the resulting equations constitute a two point boundary value problem which is generally impossible to solve analytically and furthermore the numerical solution is difficult to obtain due to the coupled nature of the solutions in this paper the variational iteration method is adopted to easily solve hamilton equations by use of iteration formulas derived from the correction functionals corresponding to hamilton equations the proposed approach allows to derive the numerical solution of the optimal control problem but an analytical or approximate expression of the optimal control law can often be obtained as a function of the time variable depending on the nature of the control problem which is simple to implement the different possible forms of control law that can be attained following the proposed design approach are illustrated by four application examples

coherent states and geometric quantization based on the concept of generalized coherent states a theory of mechanical systems is formulated in a way which naturally exhibits the mutual relation of classical and quantum aspects of physical phenomena

higher order regionally proximal equivalence relations for general minimal group actions abstract we introduce higher order regionally proximal relations suitable for an arbitrary acting group for minimal abelian group actions these relations coincide with the ones introduced by host kra and maass our main result is that these relations are equivalence relations whenever the action is minimal this was known for abelian actions by a result of shao and ye we also show that these relations lift through extensions between minimal systems answering a question by tao given a minimal system we prove that the regionally proximal equivalence relation of order d corresponds to the maximal dynamical antolin camarena szegedy nilspace factor of order at most d in particular the regionally proximal equivalence relation of order one corresponds to the maximal abelian group factor finally by using a result of gutman manners and varju under some restrictions on the acting group it follows that the regionally proximal equivalence relation of order d corresponds to the maximal pronilfactor of order at most d a factor which is an inverse limit of nilsystems of order at most d

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

consistent approximation of a nonlinear optimal control problem with uncertain parameters this paper focuses on a non standard constrained nonlinear optimal control problem in which the objective functional involves an integration over a space of stochastic parameters as well as an integration over the time domain the research is inspired by the problem of optimizing the trajectories of multiple searchers attempting to detect non evading moving targets in this paper we propose a framework based on the approximation of the integral in the parameter space for the considered uncertain optimal control problem the framework is proved to produce a zeroth order consistent approximation in the sense that accumulation points of a sequence of optimal solutions to the approximate problem are optimal solutions of the original problem in addition we demonstrate the convergence of the corresponding adjoint variables the accumulation points of a sequence of optimal state adjoint pairs for the approximate problem satisfy a necessary condition of pontryagin minimum principle type which facilitates assessment of the optimality of numerical solutions

well posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control this paper deals with the shooting algorithm for optimal control problems with a scalar control and a regular scalar state constraint additional conditions are displayed under which the so called alternative formulation is equivalent to pontryagin s minimum principle the shooting algorithm appears to be well posed invertible jacobian iff i the no gap second order sufficient optimality condition holds and ii when the constraint is of order q geq 3 there is no boundary arc stability and sensitivity results without strict complementarity at touch points are derived using robinson s strong regularity theory under a minimal second order sufficient condition the directional derivatives of the control and state are obtained as solutions of a linear quadratic problem

directional input adaptation in parametric optimal control problems this paper deals with input adaptation in dynamic processes in order to guarantee feasible and optimal operation despite the presence of uncertainty for those optimal control problems having terminal and mixed control state path constraints two orthogonal sets of adaptation directions can be distinguished in the input space the sensitivity seeking directions along which a small variation from an optimal nominal solution will not affect the respective active constraints and the complementary constraint seeking directions along which a variation will affect the respective constraints it follows that selective input adaptation strategies can be defined namely adaptation in the sensitivity and constraint seeking directions this paper proves the important result that for small parametric perturbations the cost variation resulting from adaptation in the sensitivity seeking directions over no input adaptation is typically smaller than the cost variation due to adaptation in the constraint seeking

the structure theory of nilspaces iii inverse limit representations and topological dynamics abstract this paper forms the third part of a series by the authors 12 11 concerning the structure theory of nilspaces a nilspace is a compact space x together with closed collections of cubes c n x x 2 n n 1 2 satisfying some natural axioms our goal is to extend the structure theory of nilspaces obtained by antolin camarena and szegedy and to provide new proofs our main result is that under the technical assumption that c n x is a connected space for all n then x is isomorphic in a strong sense to an inverse limit of nilmanifolds this is a direct and slight generalization of the main result of antolin camarena and szegedy we also apply our methods to obtain structure theorems in the setting of topological dynamics specifically if h is a group subject to very mild topological assumptions and h x is a minimal dynamical system then we give a simple characterization of the maximal pronilfactor of x this generalizes the case h z which is a theorem of host kra and maass although even in that case we give a significantly different proof

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

nonnegative splines in particular of degree five we investigate splines from a variational point of view which have the following properties a they interpolate given data b they stay nonnegative when the data are positive c for a given integer k 0 they minimize the functional f x int a b x k t 2 dt for all nonnegative interpolating x in w 2 k a b we extend known results for k 2 to larger k in particular to k 3 and we find general necessary conditions for solutions of this restricted minimization problem these conditions imply that solutions are splines in an augmented grid in addition we find that the solutions are in c 2k 2 a b and consist of piecewise polynomials in pi 2k 1 with respect to the augmented grid we find that for general odd k ge3 there will be no boundary arcs which means nontrivial subintervals in which the spline is identically zero we show also that the occurrence of a boundary arc in an interval between two neighboring knots prohibits the existence of any further knot in that interval for k 3 we show that between given neighboring interpolation knots the augmented grid has at most two additional grid points in the case of two interpolation knots the local problem we develop polynomial equations for the additional grid points which can be used directly for numerical computation for the general global problem we propose an algorithm which is based on a newton iteration for the additional grid points and which uses the local spline data as an initial guess there are extensions to other types of constraints such as two sided restrictions also ones which vary from interval to interval as an illustration several numerical examples including graphs of splines manufactured by matlab and fortran programs are given

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