new algorithms for unconstrained nonlinear optimal control problems a family of new first order algorithms for solving continuous time optimal control problems is presented the algorithms make use of the riccati matrix differential equation and are capable of solving the linear quadratic problem in one step the paper includes an analysis of the convergence of the proposed algorithms in the space of relaxed controls as well as the proof of the reduction of the cost functional at each iteration and numerical examples

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

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

dual variables in semigeostrophic theory this paper is a study of the duality structure present in the semigeostrophic equations of meterology we explore a new viewpoint represented by legendre transformations between alternative choices of independent variables including space coordinates geostrophic coordinates and isentropic coordinates detailed examples are given for both uniform and non uniform potential vorticity of the modelling of an atmospheric front at a given time in stable flow in each case the front is a half line of gradient discontinuity on a continuous convex surface obtained as the self intersection on a swallowtail surface after the latter is convexified so that non physical multivaluedness is removed

consistent approximations for optimal control problems based on runge kutta integration this paper explores the use of runge kutta integration methods in the construction of families of finite dimensional consistent approximations to nonsmooth control and state constrained optimal control problems consistency is defined in terms of epiconvergence of the approximating problems and hypoconvergence of their optimality functions a significant consequence of this concept of consistency is that stationary points and global solutions of the approximating discrete time optimal control problems can only converge to stationary points and global solutions of the original optimal control problem the construction of consistent approximations requires the introduction of appropriate finite dimensional subspaces of the space of controls and the extension of the standard runge kutta methods to piecewise continuous functions it is shown that in solving discrete time optimal control problems that result from runge kutta integration a non euclidean inner product and norm must be used on the control space to avoid potentially serious ill conditioning effects

on the discrete adjoints of adaptive time stepping algorithms we investigate the behavior of adaptive time stepping numerical algorithms under the reverse mode of automatic differentiation ad by differentiating the time step controller and the error estimator of the original algorithm reverse mode ad generates spurious adjoint derivatives of the time steps the resulting discrete adjoint models become inconsistent with the adjoint ode and yield incorrect derivatives to regain consistency one has to cancel out the contributions of the non physical derivatives in the discrete adjoint model we demonstrate that the discrete adjoint models of one step explicit adaptive algorithms such as the runge kutta schemes can be made consistent with their continuous counterparts using simple code modifications furthermore we extend the analysis to cover second order adjoint models derived through an extra forward mode differentiation of the discrete adjoint code several numerical examples support the mathematical derivations

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

vortex motion on surfaces with constant curvature vortex motion on two dimensional riemannian surfaces with constant curvature is formulated by way of the stereographic projection the relation and difference between the vortex motion on a sphere s 2 and on a hyperbolic plane h 2 can be clearly analysed the hamiltonian formalism is presented for the motion of point vortices on s 2 and h 2 the set of first integrals for each hamiltonian shows a corresponding algebraic property in terms of the poisson bracket defined respectively for s 2 and h 2 as an example of analytic solutions the motion of a vortex pair dipole is considered it is shown that a dipole draws a geodesic curve as its trajectory on s 2 and h 2

an effective pseudospectral method for constraint dynamic optimisation problems with characteristic times dynamic optimisation problem with characteristic times widely existing in many areas is one of the frontiers and hotspots of dynamic optimisation researches this paper considers a class of dynam

on l 2 sufficient conditions and the gradient projection method for optimal control problems mbox sf l 2 local convergence and active constraint identification theorems are proved for gradient projection iterates in sets of mbox sf l infty functions on 0 1 with range in a polyhedron u subset bbb r m these theorems extend earlier results for u 0 infty subset bbb r 1 and are based on an infinite dimensional variant of the karush kuhn tucker second order sufficient conditions in polyhedral subsets of bbb r n the new sufficient conditions and convergence results proved here are directly applicable to continuous time optimal control problems with smooth nonconvex nonquadratic objective functions and hamiltonians that are quadratic in the control input vector u in particular these theorems apply to nonconvex nonquadratic regulator problems with control linear state equations and vector valued inputs u t satisfying unqualified affine inequality constraints at almost all t in 0 1

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

dynamic optimization for robust path planning of horizontal oil wells this paper considers the three dimensional path planning problem for horizontal oil wells the decision variables in this problem are the curvature tool face angle and switching points for each turn segment in the path and the optimization objective is to minimize the path length and target error the optimal curvatures tool face angles and switching points can be readily determined using existing gradient based dynamic optimization techniques however in a real drilling process the actual curvatures and tool face angles will inevitably deviate from the planned optimal values thus causing an unexpected increase in the target error this is a critical challenge that must be overcome for successful practical implementation accordingly this paper introduces a sensitivity function that measures the rate of change in the target error with respect to the curvature and tool face angle of each turn segment based on the sensitivity function we propose a new optimization problem in which the switching points are adjusted to minimize target error sensitivity subject to continuous state inequality constraints arising from engineering specifications and an additional constraint specifying the maximum allowable increase in the path length from the optimal value our main result shows that the sensitivity function can be evaluated by solving a set of auxiliary dynamic systems by combining this result with the well known time scaling transformation we obtain an equivalent transformed problem that can be solved using standard nonlinear programming algorithms finally the paper concludes with a numerical example involving a practical path planning problem for a ci 16 cp146 well

on the time discretization of control systems this paper develops an approach for obtaining discrete approximations to nonlinear affine control systems that are of higher than first order of accuracy with respect to the discretization step h the approach consists of two parts first the set cal u of measurable admissible controls is replaced by an appropriate finite dimensional subset cal u n then the differential equations corresponding to controls from cal u n which are in a reasonable sense regular are discretized by single step discretization methods the main result estimates the accuracy in the first part measured in terms of a prescribed collection of performance indexes the result can be interpreted both in the context of approximation of optimal control problems and in the context of approximation of the reachable set in the first case accuracy o h2 is proven for appropriate runge kutta type discretization methods without explicitly or implicitly requiring any regularity of the optimal solutions in the case of a convex reachable set we obtain o h2 approximation with respect to the hausdorff distance and o h1 5 accuracy in the nonconvex case an application to the time aggregation of discrete time control systems is also presented

the motion of a vortex on a closed surface of constant negative curvature the purpose of this work is to present an algorithm to determine the motion of a single hydrodynamic vortex on a closed surface of constant curvature and of genus greater than one the algorithm is based on a relation between the laplace beltrami green function and the heat kernel the algorithm is used to compute the motion of a vortex on the bolza surface this is the first determination of the orbits of a vortex on a closed surface of genus greater than one the numerical results show that all the 46 vortex equilibria can be explicitly computed using the symmetries of the bolza surface some of these equilibria allow for the construction of the first two examples of infinite vortex crystals on the hyperbolic disc the following theorem is proved a weierstrass point of a hyperellitic surface of constant curvature is always a vortex equilibrium

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

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

adjoint sensitivity analysis of regional air quality models the task of providing an optimal analysis of the state of the atmosphere requires the development of efficient computational tools that facilitate an efficient integration of observational data into models in a variational approach the data assimilation problem is posed as a minimization problem which requires the sensitivity derivatives of a cost functional with respect to problem parameters the direct decoupled method has been extensively applied for sensitivity studies of air pollution adjoint sensitivity is a complementary approach which efficiently calculates the derivatives of a functional with respect to a large number of parameters in this paper we discuss the mathematical foundations of the adjoint sensitivity method applied to air pollution models and present a complete set of computational tools for performing three dimensional adjoint sensitivity studies numerical examples show that three dimensional adjoint sensitivity analysis provides information on influence areas which cannot be obtained solely by an inverse analysis of the meteorological fields several illustrative data assimilation results in a twin experiments framework as well as the assimilation of a real data set are also presented

optimal control computation for nonlinear systems with state dependent stopping criteria in this paper we consider a challenging optimal control problem in which the terminal time is determined by a stopping criterion this stopping criterion is defined by a smooth surface in the state space when the state trajectory hits this surface the governing dynamic system stops by restricting the controls to piecewise constant functions we derive a finite dimensional approximation of the optimal control problem we then develop an efficient computational method based on nonlinear programming for solving the approximate problem we conclude the paper with four numerical examples

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

iterative learning control design of nonlinear multiple time delay systems abstract most available results on iterative learning control address trajectory tracking problem for systems without multiple time delay this paper is concerned with iterative learning control design for nonlinear systems with multiple delays in which external disturbances and output measurement noises are involved we obtain some new and interesting criteria to guarantee the convergence of the tracking error in the sense of the norm it will be shown that the convergence of the system outputs to the desired trajectory is ensured in the absence of disturbances and output measurement noises in the presence of disturbance and measurement noises we estimate the upper bound of the tracking error in order to confirm the validity of the present results numerical simulation is also presented

n body dynamics on closed surfaces the axioms of mechanics a major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of n body and n vortex dynamics on riemann surfaces in this paper we show how the two problems are in fact closely related when considering the role played by the intrinsic geometry of the surface this enables a straightforward deduction of the dynamics of point masses using recently derived results for point vortices on general closed differentiable surfaces m endowed with a metric g we find generally that kepler s laws do not hold what is more even newton s first law the law of inertia fails on closed surfaces with variable curvature e g the ellipsoid

nonlinear stability of a latitudinal ring of point vortices on a nonrotating sphere we study the nonlinear stability of relative equilibria of configurations of identical point vortices on the surface of a sphere in particular we study how the stability changes as a function of the colatitude theta and of the number of vortices n by using the integrals of motion we view the system in asuitable corotating frame where the polygonal vortex configuration is at rest then after a sufficient criterion due to dirichlet the stability ranges are the theta intervals for which the hessian of the hamiltonian evaluated at the equilibrium configuration is positive or negative definite we find that the stability intervals coincide with those for linear stability determined by polvani and dritschel j fluid mech 255 1993 pp 35 64 for n 3 we recover the result previously established by pekarsky and marsden j math phys 39 1998 pp 5894 5907

solution differentiability for nonlinear parametric control problems perturbed nonlinear control problems with data depending on a vector parameter are considered using second order sufficient optimality conditions it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter the proof exploits the close connections between solutions of a riccati differential equation and shooting methods for solving the associated boundary value problem solution differentiability provides a firm theoretical basis for numerical feedback schemes that have been developed for computing neighbouring extremals the results are illustrated by an example that admits two extremal solutions second order sufficient conditions single out one optimal solution for which a sensitivity analysis is carried out

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

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

approximations to the multiplier method we analyze approximations to the multiplier method for solving an equality constrained optimization problem the multiplier method replaces the constrained problem by the unconstrained optimization of a penalized lagrangian over a hilbert space we consider the effect of replacing this hilbert space by a subspace the analysis is closely connected to an analysis of the galerkin approximation to the original optimization problem our estimates are essentially corollaries of a theorem which bounds the error in the galerkin approximation to a nonlinear equation some applications to problems in optimal control are discussed

sensitivity analysis for dynamic systems with time lags many problems in bioscience for which observations are reported in the literature can be modelled by suitable functional differential equations incorporating time lags other terminology delays or memory effects parameterized by scientifically meaningful constant parameters p or and variable parameters for example control functions u t it is often desirable to have information about the effect on the solution of the dynamic system of perturbing the initial data control functions time lags and other parameters appearing in the model the main purpose of this paper is to derive a general theory for sensitivity analysis of mathematical models that contain time lags in this paper we use adjoint equations and direct methods to estimate the sensitivity functions when the parameters appearing in the model are not only constants but also variables of time to illustrate the results the methodology is applied numerically to an example of a delay differential model

an a priori estimate for discrete approximations in nonlinear optimal control we examine the convergence of an approximate discretization applied to the first order optimality conditions for a nonlinear optimal control problem with convex control constraints under an assumption of the coercivity type we prove the existence of an optimal control for the original problem such that its l infty distance from the approximating sequence is proportional to the error in a corollary we give conditions for riemann integrability of the optimal control

a second order algorithm for continuous time nonlinear optimal control problems a second order algorithm is presented for the solution of continuous time nonlinear optimal control problems the algorithm is an adaptation of the trust region modifications of newton s method and solves at each iteration a linear quadratic control problem with an additional constraint under some assumptions the proposed algorithm is shown to possess a global convergence property a numerical example is presented to illustrate the method

the approximation for the boundary optimal control problem of burgers fisher equation with constraints this paper deals with the numerical approximation with meshless method for the boundary optimal control problem with some control and state constraints governed by the burgers fisher equation which is a nonlinear evolution equation and is the prototype model for the reaction convection and diffusion phenomena arising in many spatial temporal processes by making use of the element free galerkin efg method the original optimal control problem is discretized spatially to a semi discrete optimal control problem governed by a system of nonlinear ordinary differential equations then by using the control parameterization method the original problem can be reduced to an optimal parameter selection problem governed by a lumped parameter system which can be solved as a nonlinear optimization problems by using the sequential quadratic programming sqp algorithm the numerical simulations are given to illustrate the effectiveness of the proposed numerical approximation method

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

conservation of generalized momentum maps in mechanical optimal control problems with symmetry summary in this paper a structure preserving direct method for the optimal control of mechanical systems is developed the new method accommodates a large class of one step integrators for the underlying state equations the state equations under consideration govern the motion of affine hamiltonian control systems if the optimal control problem has symmetry associated generalized momentum maps are conserved along an optimal path this is in accordance with an extension of noether s theorem to the realm of optimal control problems in the present work we focus on optimal control problems with rotational symmetries the newly proposed direct approach is capable of exactly conserving generalized momentum maps associated with rotational symmetries of the optimal control problem this is true for a variety of one step integrators used for the discretization of the state equations examples are the one step theta method a partitioned variant of the theta method and energy momentum em consistent integrators numerical investigations confirm the theoretical findings copyright 2016 john wiley sons ltd

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

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

variants of the kuhn tucker sufficient conditions in cones of nonnegative functions second order sufficient conditions of the kuhn tucker type are proved for certain constrained minimization problems on sets of nonnegative mathcal l p functions with p in 2 infty the objective functions for these problems have specially structured bilinear second gateaux differentials that are bounded with respect to the mathcal l 2 norm and vary continuously with respect to the mathcal l 2 norm on mathcal l p structure and smoothness conditions of this sort are satisfied by nontrivial classes of constrained input bolza optimal control problems and in this context the associated kuhn tucker sufficient conditions yield a partial extension of the classical weak sufficiency theory in the calculus of variations

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

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

complete decomposition algorithm for nonconvex separable optimization problems and applications abstract in this paper we present a complete decomposition algorithm for nonconvex separable optimization problems applied in the optimal control problems this complete decomposition algorithm combines recursive quadratic programming with the dual method when our algorithm is applied to discretized optimal control problems a simple and parallel computation and a simple and regular data flow pattern between consecutive computational steps results this paper also suggests an approach for developing a hardware implementation of our algorithm and gives an estimation of the execution time needed to solve a practical example

optimal control of switched systems and its parallel optimization algorithm in this paper a class of optimal switching control problems is considered in which the mode sequence of active subsystems and the number of mode switchings are not pre specified and both the switching sequence and the control inputs are to be chosen such that the cost functional is minimized for solving this problem we discuss the necessary conditions for optimality and the construction method for suboptimal solutions and develop a few sufficient conditions of judgment on the optimal or suboptimal solutions of the switched system according to the sufficient conditions a parallel computational algorithm is constructed to find optimal or suboptimal solutions for illustration two examples are solved using the proposed algorithm

fatode a library for forward adjoint and tangent linear integration of odes fatode is a fortran library for the integration of ordinary differential equations with direct and adjoint sensitivity analysis capabilities the paper describes the capabilities implementation code organization and usage of this package fatode implements four families of methods explicit runge kutta for nonstiff problems fully implicit runge kutta singly diagonally implicit runge kutta and rosenbrock for stiff problems each family contains several methods with different orders of accuracy users can add new methods by simply providing their coefficients for each family the forward adjoint and tangent linear models are implemented general purpose solvers for dense and sparse linear algebra are used users can easily incorporate problem tailored linear algebra routines the performance of the package is demonstrated on several test problems to the best of our knowledge fatode is the first publicly available general purpose package that offers forward and adjoint sensitivity analysis capabi

families of lift and contact transformations a study of the contact transformation is carried out with special emphasis on the lift transformation the latter is the more basic concept because it applies to a single curve rather than to pairs of them in the first instance from that viewpoint the lift transformation can also transform an unfamiliar system of partial differential equations to a familiar one without reference to contact we illustrate this by showing how the semi geostrophic equations of meterology transform to hamiltonian form general and special lift transformations are distinguished new and very wide families of such transformations are derived these give a desirable perspective to particular cases such as the legendre transformation and the geostrophic transformation a brief study of generalized duality is described the distinction between regular and singular lifted curves is discussed in general terms and their participation in a contact transformation is illustrated by an accessible physical example

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

symplectic runge kutta schemes for adjoint equations automatic differentiation optimal control and more the study of the sensitivity of the solution of a system of differential equations with respect to changes in the initial conditions leads to the introduction of an adjoint system whose discretization is related to reverse accumulation in automatic differentiation similar adjoint systems arise in optimal control and other areas including classical mechanics adjoint systems are introduced in such a way that they exactly preserve a relevant quadratic invariant more precisely an inner product symplectic runge kutta and partitioned runge kutta methods are defined through the exact conservation of a differential geometric structure but may be characterized by the fact that they preserve exactly quadratic invariants of the system being integrated therefore the symplecticness or lack of symplecticness of a runge kutta or partitioned runge kutta integrator should be relevant to understanding its performance when applied to the computation of sensitivities to optimal control problems and in othe

symbolic numeric efficient solution of optimal control problems for multibody systems this paper presents an efficient symbolic numerical approach for generating and solving the boundary value problem differential algebraic equation bvp dae originating from the variational form of the optimal control problem ocp this paper presents the method for the symbolic derivation by means of symbolic manipulation software maple of the equations of the ocp applied to a generic multibody system the constrained problem is transformed into a nonconstrained problem by means of the lagrange multipliers and penalty functions from the first variation of the nonconstrained problem a bvp dae is obtained and the finite difference discretization yields a nonlinear systems for the numerical solution of the nonlinear system a damped newton scheme is used the sparse and structured jacobians is quickly inverted by exploiting the sparsity pattern in the solution strategy the proposed method is implemented in an object oriented fashion and coded in c language efficiency is ensured in core routines by using lapack and blas for linear algebra

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

space time adaptive solution of inverse problems with the discrete adjoint method this paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent adaptive grid adaptive step models discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation however this approach brings several important challenges the space time adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations a reduction in accuracy of the discrete adjoint sensitivities may appear due to the inter grid transfer operators moreover the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations this work shows that several of these potential issues can be avoided through a multi level optimization strategy using discontinuous galerkin dg hp adaptive discretizations paired with runge kutta rk time integration we extend the concept of dual adjoint consistency to space time rk dg discretizations which are then shown to be well suited for the adaptive solution of time dependent inverse problems furthermore we prove that dg mesh transfer operators on general meshes are also dual consistent this allows the simultaneous derivation of the discrete adjoint for both the numerical solver and the mesh transfer logic with an automatic code generation mechanism such as algorithmic differentiation ad potentially speeding up development of large scale simulation codes the theoretical analysis is supported by numerical results reported for a two dimensional non stationary inverse problem

discrete second order adjoints in atmospheric chemical transport modeling atmospheric chemical transport models ctms are essential tools for the study of air pollution for environmental policy decisions for the interpretation of observational data and for producing air quality forecasts many air quality studies require sensitivity analyses i e the computation of derivatives of the model output with respect to model parameters the derivatives of a cost functional defined on the model output with respect to a large number of model parameters can be calculated efficiently through adjoint sensitivity analysis while the traditional first order adjoint models give the gradient of the cost functional with respect to parameters second order adjoint models give second derivative information in the form of products between the hessian of the cost functional and a vector representing a perturbation in sensitivity analysis a search direction in optimization an eigenvector etc in this paper we discuss the mathematical foundations of the discrete second order adjoint sensitivity method and present a complete set of computational tools for performing second order sensitivity studies in three dimensional atmospheric ctms the tools include discrete second order adjoints of runge kutta and of rosenbrock time stepping methods for stiff equations together with efficient implementation strategies numerical examples illustrate the use of these computational tools in important applications like sensitivity analysis optimization uncertainty quantification and the calculation of directions of maximal error growth in three dimensional atmospheric ctms

holomorphic structures in hydrodynamical models of nearly geostrophic flow we study complex structures arising in hamiltonian models of nearly geostrophic flows in hydrodynamics in many of these models an elliptic monge ampere equation defines the relationship between a balanced velocity field defined by a constraint in the hamiltonian formalism and the materially conserved potential vorticity elliptic monge ampere operators define an almost complex structure and in this paper we show that a natural extension of the so called geostrophic momentum transformation of semi geostrophic theory which has a special importance in theoretical meteorology defines kahler and special kahler structures on phase space furthermore analogues of the geostrophic momentum coordinates are shown to be special lagrangian coordinates under conditions which depend upon the physical approximations under consideration certain duality properties of the operators are studied within the framework of the kahler geometry

brief paper lossless convexification of a class of optimal control problems with non convex control constraints we consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost convex state constraints and non convex control constraints we propose a convex relaxation of the non convex control constraints and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem which is referred to as the lossless convexification of the optimal control problem the lossless convexification enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non convex optimal control problem the solution approach is demonstrated on a number of planetary soft landing optimal control problems

brief paper a suboptimal feedback control for nonlinear time varying systems with continuous inequality constraints a feedback control law which is constructed as a first order approximation to the optimal control is proposed for nonlinear time varying systems subject to continuous inequality constraints on the control and state this control law is effective under small state perturbations caused by changes on initial conditions and or modeling uncertainty

vortex dynamics in complex domains on a spherical surface we consider the motion of both point vortices and uniform vortex patches in arbitrary possibly multiply connected regions bounded by impenetrable walls on the surface of a sphere by exploiting knowledge of the functional form of the relevant green s function in a pre image circular domain that is conformally equivalent to a stereographic projection of the fluid domain on the spherical surface we first generalize kirchhoff routh theory for point vortex motion in the plane to point vortex motion on a spherical shell next we study vortex patch motion and show that there is a contour dynamics formulation for the evolution of uniform vortex patches in any finitely connected domain on a spherical shell bounded by impenetrable walls we describe a novel numerical scheme whereby this motion can be computed some illustrative calculations are shown

robust suboptimal control of nonlinear systems in this paper we consider a nonlinear dynamic system with uncertain parameters our goal is to choose a control function for this system that balances two competing objectives i the system should operate efficiently and ii the system s performance should be robust with respect to changes in the uncertain parameters with this in mind we introduce an optimal control problem with a cost function penalizing both the system cost a function of the final state reached by the system and the system sensitivity the derivative of the system cost with respect to the uncertain parameters we then show that the system sensitivity can be computed by solving an auxiliary initial value problem this result allows one to convert the optimal control problem into a standard mayer problem which can be solved directly using conventional techniques we illustrate this approach by solving two example problems using the software miser3

point vortex dynamics a classical mathematics playground the idealization of a two dimensional ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics several examples are given including the integrability of the three vortex problem the interplay of relative equilibria of identical vortices and the roots of certain polynomials addition formulas for the cotangent and the weierstras function projective geometry and other topics the hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community

forward and adjoint sensitivity analysis with continuous explicit runge kutta schemes abstract we study the numerical solution of tangent linear first and second order adjoint models with high order explicit continuous runge kutta pairs the approaches currently implemented in popular packages such as sundials or daspkadjoint are based on linear multistep methods for adaptive time integration of nonlinear models interpolation of the forward model solution is required during the adjoint model simulation we propose to use the dense output mechanism built in the continuous runge kutta schemes as a highly accurate and cost efficient interpolation method in the inverse problem run we implement our approach in a fortran library called denserks which is found to compare well to other similar software on a number of test problems

an optimization approach to state delay identification we consider a nonlinear delay differential system with unknown state delays our goal is to identify these state delays using experimental data to this end we formulate a dynamic optimization problem in which the state delays are decision variables and the cost function measures the discrepancy between predicted and observed system output we then show that the gradient of this problem s cost function can be computed by solving an auxiliary delay differential system by exploiting this result the state delay identification problem can be solved efficiently using a gradient based optimization method

comparison of optimization algorithms the measured convergence rates of several algorithms belonging to the class of 1 gradient methods in function space are compared the steepest descent conjugate gradient and quasi newton algorithms and the effect of different types of linear search routine and initial guess parameters on their convergence rates are considered it is concluded that conjugate gradient methods do offer significant advantages even for non linear system models and bounded controls

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

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

thomson s heptagon a case of bifurcation at infinity abstract vortex modeling has a long history descartes 1644 used it as a model for the solar systems j j thomsom 1883 used it as a model for the atom we consider point vortex systems which can be regarded as discrete solutions of the euler equation their dynamics is described by a hamiltonian system of equations we are interested in polygonal configurations and how their stability depends upon various dynamical variables in the plane a polygon with seven vortices has been shown to be a special boundary case polygons with n 7 vortices are linearly and nonlinearly stable while polygons with n 7 vortices are unstable why should n 7 be special celestial mechanics helped us to simplify a problem that has been studied for over a century and to show that the case of thomson s heptagon is actually a case of bifurcation at infinity this becomes particularly clear when considering the corresponding problem of a ring on a sphere with two polar vortices of variable intensities n and s at the north and south pole respectively

non commutative discretize then optimize algorithms for elliptic pde constrained optimal control problems abstract in this paper we analyze the convergence of several optimize then discretize and discretize then optimize algorithms based on either a second order or a fourth order finite difference discretization for solving elliptic pde constrained optimal control problems to ensure the convergence of a discretize then optimize algorithm one well accepted criterion is to design the discretization scheme such that the resulting discretize then optimize algorithm commutes with the corresponding optimize then discretize algorithm in other words both algorithms should give rise to exactly the same discrete optimality system however such a restrictive criterion is not trivial to fulfill by investigating a distributed control problem governed by an elliptic equation we first show that enforcing such a stringent condition of commutative property is only sufficient but not necessary for achieving the desired convergence we then introduce some suitable h 1 semi norm penalty regularization terms to recover the lost convergence due to the inconsistency caused by the loss of commutativity numerical experiments are carried out to verify our theoretical analysis and also validate the effectiveness of our proposed regularization techniques

examples of quaternionic and kahler structures in hamiltonian models of nearly geostrophic flow we study 2 forms on phase spaces of hamiltonian models of nearly geostrophic flows a quaternionic structure is identified and the complex part of a symplectic representation of this structure corresponds to an elliptic monge ampere equation the real part is an invariant kahler structure

h norm computation for ltv systems using nonlinear optimal control algorithms we discuss the computation for 2 induced norms of finite horizon linear time varying systems it is well known that 2 induced norm calculating problems can be converted into optimal control problems so with the help of optimal control algorithms we propose a numerical technique for 2 induced norm computation our approach is as follows first we present a new computational method for nonlinear optimal control problems that is numerically reliable stable and robust analysing the convergence property under some assumptions next by modifying the presented method we propose an algorithm for the 2 induced norm calculation of finite horizon linear time varying systems which is essentially different from all other existing methods finally the effectiveness of our algorithm is demonstrated by simulation

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

brief paper initial condition of costate in linear optimal control using convex analysis the pontryagin maximum principle is one of the most important results in optimal control and provides necessary conditions for optimality in the form of a mixed initial terminal boundary condition on a pair of differential equations for the system state and its conjugate costate unfortunately this mixed boundary value problem is usually difficult to solve since the pontryagin maximum principle does not give any information on the initial value of the costate in this paper we explore an optimal control problem with linear and convex structure and derive the associated dual optimization problem using convex duality which is often much easier to solve than the original optimal control problem we present that the solution to the dual optimization problem supplements the necessary conditions of the pontryagin maximum principle and elaborate the procedure of constructing the optimal control and its corresponding state trajectory in terms of the solution to the dual problem

distributed optimal control for multi agent trajectory optimization this paper presents a novel optimal control problem referred to as distributed optimal control that is applicable to multiscale dynamical systems comprised of numerous interacting agents the system performance is represented by an integral cost function of the macroscopic state that is optimized subject to a hyperbolic partial differential equation known as the advection equation the microscopic control laws are derived from the optimal macroscopic description using a potential function approach the optimality conditions of the distributed optimal control problem are first derived analytically and then demonstrated numerically through a multi agent trajectory optimization problem

optimal control of the temperature in a catalytic converter abstract this paper is concerned with the optimization of the dynamics of the gas flow in an exhaust pipe as a prototypical question the start up heating of the catalytic converter is considered reaching the optimal temperature in the converter rapidly competes against the costs i e releasing too much unburnt fuel in the exhaust gases the underlying model is a one dimensional small mach number asymptotic gas dynamic model for a mixture of burnt and unburnt gases formulated on a network consisting of the various pieces of the exhaust tube for the optimization adjoint calculus is applied on the continuous level the resulting system is then discretized finally numerical examples show the validity of this approach

statistics of point vortex turbulence in non neutral flows and in flows with translational and rotational symmetries a theory esler and ashbee in j fluid mech 779 275 308 2015 describing the statistics of n freely evolving point vortices in a bounded two dimensional domain is extended first the case of a non neutral vortex gas is addressed and it is shown that the density of states function can be identified with the probability density function of an infinite sum of independent non central chi squared random variables the details of which depend only on the shape of the domain equations for the equilibrium energy spectrum and other statistical quantities follow the validity of which are verified against direct numerical simulations of the equations of motion second domains with additional conserved quantities associated with a symmetry e g circle periodic channel are investigated and it is shown that the treatment of the non neutral case can be modified to account for the additional constraint

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

brief paper optimal control problems with multiple characteristic time points in the objective and constraints in this paper we develop a computational method for a class of optimal control problems where the objective and constraint functionals depend on two or more discrete time points these time points can be either fixed or variable using the control parametrization technique and a time scaling transformation this type of optimal control problem is approximated by a sequence of approximate optimal parameter selection problems each of these approximate problems can be viewed as a finite dimensional optimization problem new gradient formulae for the cost and constraint functions are derived with these gradient formulae standard gradient based optimization methods can be applied to solve each approximate optimal parameter selection problem for illustration two numerical examples are solved

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

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

optimal boundary control for water hammer suppression in fluid transmission pipelines when fluid flow in a pipeline is suddenly halted a pressure surge or wave is created within the pipeline this phenomenon called water hammer can cause major damage to pipelines including pipeline ruptures in this paper we model the problem of mitigating water hammer during valve closure by an optimal boundary control problem involving a nonlinear hyperbolic pde system that describes the fluid flow along the pipeline the control variable in this system represents the valve boundary actuation implemented at the pipeline terminus to solve the boundary control problem we first use the method of lines to obtain a finite dimensional ode model based on the original pde system then for the boundary control design we apply the control parameterization method to obtain an approximate optimal parameter selection problem that can be solved using nonlinear optimization techniques such as sequential quadratic programming sqp we conclude the paper with simulation results demonstrating the capability of optimal boundary control to significantly reduce flow fluctuation

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

numerical solution of some initial optimal control problems using the reproducing kernel hilbert space technique in this paper we present a general technique for solving a class of linear nonlinear optimal control problems in fact an analytical solution of the state variable is represented in the form of a

the motion of point vortices on closed surfaces we develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere when the sum of the vortex circulations is non ze