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We consider the problem of optimal control over differential equations with interaction. It is shown that the optimal control satisfies the maximum principle and there exists a generalized optimal control. In the analyzed problem, we encounter certain new technical features as compared with the ordinary problem of optimal control.

A procedure for obtaining the initial value of the costate in a regular, finite-horizon, nonlinear-quadratic problem is devised in dimension one. The optimal control can then be constructed from the solution to the Hamiltonian equations, integrated on-line. The initial costate is found by successively solving two first-order, quasi-linear, partial differential equations (PDEs), whose independent variables are the time-horizon duration T and the final-penalty coefficient S. These PDEs need to be integrated off-line, the solution rendering not only the initial condition for the costate sought in the particular (T, S)-situation but also additional information on the boundary values of the whole two-parameter family of control problems, that can be used for design purposes. Results are tested against exact solutions of the PDEs for linear systems and also compared with numerical solutions of the bilinear-quadratic problem obtained through a power-series' expansion approach. Bilinear systems are specially treated in their character of universal approximations of nonlinear systems with bounded controls during finite time-periods. Fil: Costanza, Vicente. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; Argentina

This paper investigates the linear quadratic regulator problem for singular systems which may contain some impulse uncontrollable modes. By transforming to the Weierstrass form, impulse controllable and uncontrollable subsystems can be separated. Sufficient conditions are proposed, under which solutions to both state and costate equations for the linear quadratic regulator problem are impulse-free. The necessary conditions for the minimization of the quadratic cost function are converted to a two-point boundary value problem, which can be solved by either computing its back-time state transition matrix or solving a Riccati equation. Two examples are provided to illustrate the main results.

Costate value plays a significant role in the application of PMP-based control strategy for PHEV. It is critical for terminal SOC of battery at destination and corresponding equivalent fuel consumption. However, it is not convenient to choose the approximate costate in real driving condition. In the paper, the optimal control problem of PHEV based on PMP has been converted to nonlinear programming problem. By means of KKT condition costate can be approximated as KKT multipliers of NLP divided by the LGL weights. A kind of general costate estimation approach is proposed for predefined driving condition in this way. Dynamic model has been established in Matlab/Simulink in order to prove the effectiveness of the method. Simulation results demonstrate that the method presented in the paper can deduce the closer value of global optimal value than constant initial costate value. This approach can be used for initial costate and jump condition estimation of PMP-based control strategy for PHEV.

This paper describes a general point of view on the subject of store-and-forward queueing networks. A systematic approach to a store-and-forward network as a whole is applied. This allows network problems which involve dynamic optimization of congestion control and routing to be treated in a common framework. The dynamic model applied is non-linear and can be handled straightforwardly by optimal control system theory. It is shown that under normal operational conditions the optimal flow pattern is given by the steady-state solution of the costate equations. Properties of the optimal flow pattern are examined in detail. Then, schemes of the operation of the network under different operating conditions are discussed at length. It is shown that the optimal control strategies depend on the total volume of traffic forwarded to the network. Algorithms are developed for different traffic demand patterns, and a numerical example is also given.

The paper outlines a procedure to derive the canonical system of equations of the classical theory of thin shells using Reissner’s variational principle and partial variational principles. The Hamiltonian form of the Reissner functional is obtained using Lagrange multipliers to include the kinematical conditions that follow from the Kirchhoff-Love hypotheses. It is shown that the canonical system of equations can be represented in three different forms: one conventional form (five equilibrium equations) and two forms that are equivalent to it. This can be proved by reducing them to the same system of three equations. For problems with separable active and passive variables, partial variational principles are formulated

A problem of reconstructing material deformation curves for thin-walled structure elements is considered. We suggest the solution of this problem by the gradient method invoking the costate equations. The derivation of equation coefficients and numerical analysis results are presented.

An analytical relationship is derived which connects the matrix of partial derivatives of the current state vector with respect to the initial conditions with the matrix of solutions to variational equations in parametric variables. These equations correspond to the equations of motion obtained using the generalized coordinate-time transformation admitting an increase of the number of unknown parameters. Particular cases of transformations leading to the variational equations in regularizing variables of Sperling-Burdet and Kustaancheimo-Stiefel are considered. All formulas necessary in these cases for determination of the matrix of isochronous derivatives are obtained.

A nonlinear stochastic optimal control strategy for minimizing the first-passage failure of quasi integrable Hamiltonian systems (multi-degree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is proposed. The equations of motion for a controlled quasi integrable Hamiltonian system are reduced to a set of averaged Ito stochastic differential equations by using the stochastic averaging method. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximization of reliability and mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The final dynamical programming equations for these control problems are determined and their relationships to the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are separately established. The conditional reliability function and the mean first-passage time of the controlled system are obtained by solving the final dynamical programming equations or their equivalent Kolmogorov and Pontryagin equations. An example is presented to illustrate the application and effectiveness of the proposed control strategy.

We treat fractional order differential equations that contain left and right Riemann-Liouville fractional derivatives. Such equations arise as the Euler-Lagrange equation in variational principles with fractional derivatives. We find solutions of such equations or construct corresponding integral equations.

In particular, in [1] one establishes formulae for the integral representation of solutions to (1), characterizing their structural properties. In [7], [8] (pp. 169–191) for (1) one considers a generalization of the Goursat problem, where the desired functions enter into the boundary conditions more equitably. In addition, the qualitative character of the solvability of the proposed problem appeared to be more diverse, than that of the Goursat problem (which is uniquely solvable). The main objective of this paper is the study of a certain analog of the problem stated in [7]–[8] for the system uxx = a1(x, y)vx + b1(x, y)u + c1(x, y)v + f1(x, y), vyy = a2(x, y)uy + b2(x, y)u+ c2(x, y)v + f2(x, y). (2)

In this paper, a numerical method based on mixed finite element method for optimal boundary control problem governed by bi-harmonic equation is presented. The system of optimality equations consisting of state and costate function is derived. Based on the system, a gradient-type optimization method using a mixed finite approximation for solving the optimal boundary control is developed. In our algorithm, the trace of vortex function of costate equation on boundary plays a key role. Finally, a local error analysis at every iteration for this gradient-type optimization method is given.

In this paper, we discuss discrete versions of the heat equations and the wave equations, which are called the ω-diffusion equations and the ω-elastic equations on graphs. After deriving some basic properties, we solve the ω-diffusion equations under (i) the condition that there is no boundary, (ii) the initial condition and (iii) the Dirichlet boundary condition. We also give some additional interesting properties on the ω-diffusion equations, such as the minimum and maximum principles, Huygens property and uniqueness via energy methods. Analogues of the ω-elastic equations on graphs are also discussed.

An existence result of multiple periodic solutions to the asymptotically linear discrete Hamiltonian systems is obtained by using the Morse index theory.

This paper presents a numerical scheme to approximate the solution of the optimal control problem for nonlinear systems with restrictions on the manipulated variable. In there exists a unique regular time-interval, it has been theoretically shown that there is a related unrestricted nonlinear problem, with unknown final state and costate, whose saturated optimal control leads to the solution of the original problem with restrictions. The method proposed here systematically reduces the cost of a seed control trajectory by using the solution of the differential Riccati equation (DRE) corresponding to the linearization of the system around the seed data. The final condition for the DRE is constructed from the variations of the final state ρΔ and costate μΔ. The updating of the control strategy is performed through the gradient method applied to (ρΔ, μΔ)and to the time instants τi when the control becomes saturated. This allows to avoid working with the Hamiltonian equations, usually unstable, and also to construct a feedback control law, robust under disturbances. The results are illustrated by optimizing a quadratic cost imposed to a two-dimensional non-linear system with bounded control values.

This paper demonstrates, how an approximation of the costate trajectory can be obtained for switched systems with state jumps, when an optimal control problem with fixed switching sequence is solved using a direct method, i.e. direct shooting or collocation. The obtained trajectories include the jump conditions for the costate given by necessary conditions for optimal control of switched systems.

Necessary conditions are derived for optimal control problems subject to index-2 differential-algebraic equations, pure state constraints, and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which arise frequently in practical applications. The structure of the optimal control problem under consideration is exploited and special emphasis is laid on the representation of the Lagrange multipliers resulting from the necessary conditions for infinite optimization problems.

Necessary conditions in terms of a local minimum principle are derived for optimal control problems subject to index-2 differential-algebraic equations, pure state constraints, and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which arise frequently in practical applications. The local minimum principle is based on the necessary optimality conditions for general infinite optimization problems. The special structure of the optimal control problem under consideration is exploited and allows us to obtain more regular representations for the multipliers involved. An additional Mangasarian-Fromowitz-like constraint qualification for the optimal control problem ensures the regularity of a local minimum. An illustrative example completes the article.

We show that, in the setting of general semigroups there is in general no relation between smoothness of the target and smoothness of the costate for time and norm optimal controls. However, there exist relations in particular cases, such as self adjoint and unitary semigroups. †Dedicated to Prof. N.U. Ahmed on the occasion of his 70th birthday.

The problem of designing strategies for optimal feedback control of non-linear processes, specially for regulation and set-point changing, is attacked in this paper. A novel procedure based on the Hamiltonian equations associated to a bilinear approximation of the dynamics and a quadratic cost is presented. The usual boundary-value situation for the coupled state–costate system is transformed into an initial-value problem through the solution of a generalized algebraic Riccati equation. This allows to integrate the Hamiltonian equations on-line, and to construct the feedback law by using the costate solution trajectory. Results are shown applied to a classical non-linear chemical reactor model, and compared against suboptimal bilinear-quadratic strategies based on power series expansions. Since state variables calculated from Hamiltonian equations may differ from the values of physical states, the proposed control strategy is suboptimal with respect to the original plant. Copyright © 2005 John Wiley & Sons, Ltd.

For an optimal control problem with an infinite time horizon, assuming various terminal state conditions (or none), terminal conditions for the costate are obtained when the state and costate tend to limits with a suitable convergence rate. Under similar hypotheses, the sensitivity of the optimum to small perturbations is analysed, and in particular the stability of the optimum when the infinite horizon is truncated to a large finite horizon. An infinite horizon version of Pontryagin's principle is also obtained. The results apply to various economic models.

A maximum principle is developed for a class of problems involving the optimal control of a system of linear hyperbolic equations in one space dimension that are not necessarily separable. An index of performance is formulated, which consists of vector functions of the state variable, their first- and second-order space derivatives and first-order time derivative, and a penalty function involving the open-loop control force vector. The solution of the optimal control problem can easily be shown to be unique using convexity arguments. The given maximum principle involves a Hamiltonian, which contains an adjoint vector function as well as an admissible control vector function. The maximum principle can be used to compute the optimal control vector function and is particularly suitable for problems involving the active control of structural elements for vibration suppression. A numerical example is given which studies the active control of a beam under going flexural and torsional vibrations. A comparison of the energies of controlled and uncontrolled beams indicates that the proposed control method is quite effective in damping out the vibrations of structural systems.

This work deals with the optimal control of one dimensional quantum harmonic oscillator under an external field characterized by a linear dipole function. The penalty term is taken as kinetic energy. The objective operator whose expectation value is desired to get a prescribed target value is taken as the square of the position operator. The dipole function hypothesis in the external field is valid only for weak fields otherwise hyperpolarizability terms which contain powers of the field amplitude higher than 1 should be considered. The weak field assumption enables us to develop a first order perturbation approach to get approximate solutions to the wave and costate equations. These solutions contain the field amplitude and another unknown, so-called deviation constant, through some certain integrals. By inserting these expressions into the connection equation which functionally relates the field amplitude to the wave and costate function it is possible to produce an integral equation. Same manipulations on the objective equation results in an algebraic equation to determine the deviation parameter. The algebraic deviation equation produces incompatibility which can be relaxed by including second order perturbative term of wave function. The integral and algebraic equations mentioned above are asymptotically solved. Their global solutions are left to future works since the main purpose of this work is to obtain the so-called field and deviation equations. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Variational method is applied to the state equations in order to derive the costate equations and their boundary conditions. Thereafter, the analyses of the eigenvalues of the state and costate equations are performed. It is shown that the eigenvalues of the Jacobean matrices of the state and the transposed Jacobean matrices of the costate equations are analytically and numerically the same. Based on the eigenvalue analysis, the costate equations with their boundary conditions are numerically integrated. Numerical results of the eigenvalues problems of the state and costate equations and of a maximization problem are finally presented.

In a recent article, certain underdetermined linear systems of partial differential equations connected with Lie–Poisson structures have been studied. They were constructed via power series solutions of the evolution equation for a given Hamiltonian. We extend the results to arbitrary Poisson manifolds, correct an error in the case of degenerate Poisson structures, and show that these linear systems simply characterize Hamiltonian vector fields. Our basic tool is the formal theory of differential equations with its central concept of an involutive system.

In this paper, we shall study two optimal control problems (P1) and (P2) governed by some semilinear elliptic di%erential equations which can admit more than one solution. We shall call such systems non-well posed systems. We obtain the Pontryagin maximum principle for problems (P1) and (P2). In the 6rst problem (P1), the cost functional may be non-smooth and the set of controls is convex while in the second problem (P2), the cost functional is smooth and the set of controls is non-convex. In both problem (P1) and problem (P2), we consider some state constraint of integral type. Due to some practical interests, many authors studied optimal control for elliptic di%erential equations. Here, we cite [1–10,12–15]. However, most of these works, except for [7,10,13], deal with state equations which are governed by monotone operators and admit a unique solution corresponding to each control. In such problems, the Pontryagin maximum principle was obtained by studying the variations of the state with respect to some perturbations of the control (cf. [1,2,12,15]). In the present work, the state equations are governed by non-monotone operators and can admit more than one solution. We cannot obtain the variations of the state with respect to the control with the same methods in [1,2,12,15]. When the state equations admit more than one solution, we even do not know how to make the sensitivity analysis of the state with respect to the control.

Based on the theory of calculus of variation, some sufficient conditions are given for some Euler-Lagrange equations to be equivalently represented by finite or even infinite many Hamiltonian canonical equations. Meanwhile, some further applications for equations such as the KdV equation, MKdV equation, the general linear Euler-Lagrange equation and the cylindric shell equations are given.

An optimization methodology which uses the conservative field variables, is developed to solve a design optimization problem in fluid dynamical distributed parameter systems. This approach which is completely based on the variational method, is employed to derive the costate partial differential equations (pdes) and their transversality (boundary) conditions from the continuous pdes of the fluid flow. The costate equations coupled with the flow field equations are solved iteratively to get the functional derivative coefficients. Then, these derivative coefficients combined with the flow field variables are used to find the boundary shape which extremizes the performance index (functional). To demonstrate the method through examples, the shape of the nozzle is optimized for the maximum thrust. For this maximization problem, inlet and outlet flow conditions that depend on the upstream and downstream Mach numbers respectively are considered. In order to build confidence in the optimization procedure, high convergences of the state and costate equations were sought and the numerical and analytical solutions (in the form of pressure distributions) of the state equations are compared. In the purely supersonic flow case, the gain in thrust is remarkably high. Even in the supersonic-inlet–subsonic-outlet and the purely subsonic cases, the improvement of the thrust is found to be substantial. As demonstrated through the cases investigated, a new achievement is that the present variational shape optimization approach is capable of resolving flows with shocks.

This paper is concerned with the maximum principle of optimal control problems governed by some parabolic differential equations which could be non-well-posed. Both an integral type state constraint and a two point boundary (time variable) state constraint are considered.

This paper proposes a method for optimal dynamic control of nonlinear time-invariant systems with a quadratic performance criterion by using the adjoint formulation of the system. From the Lagrangian the adjoint system is derived in terms of the Lagrange Multipliers, which are also called costate variables. Our approach makes the elimination of the costate possible and provides a dynamic control law based on state feedback which can be determined off-line.

A neural-network-based synthesis of an optimal midcourse guidance law is presented in this study. We use a set of two neural networks; the first network called a “critic” outputs the Lagrange's multipliers arising in an optimal control formulation and second network, called an “action” network, outputs the optimal guidance/control. The system equations, the optimality conditions, the costate equations are used in conjunction with the network outputs to provide the targets for the neural networks. When the critic and action network are mutually consistent, the output of the action network yields optimal guidance/control. Numerical results for a number of scenarios show that the network performance is excellent. Corroboration for optimality is provided by comparisons of the numerical solutions using a shooting method for a number of scenarios.

The aim of this paper is to explore the implementation of receding horizon optimal control based on the Pontryagin's maximum principle for the dynamical systems represented by stochastic differential equations. A brief review of stochastic Pontryagin's maximum principle is given, and then the issues of solving forward-backward stochastic differential equations are addressed which leads to the possibility of implementing the receding horizon optimal controller for a class of stochastic systems. Numerical examples are demonstrated to show the outline of the presented design procedure.

This contribution is concerned with model predictive control (MPC) of systems governed by partial differential equations (PDEs) and subject to state and input constraints. In particular, the numerically efficient handling of the underlying optimal control problem (OCP) is considered. It is shown that the state- and input-constrained OCP can be transformed by using saturation functions into an unconstrained OCP. This then can be solved numerically by means of well-established optimization methods. For a numerically efficient implementation of the MPC, a first discretize then optimize-approach is used in conjunction with a tailored gradient method. Both the transformation into an unconstrained OCP and its numerical solution are investigated for a simple heat conduction problem.

A continuous adjoint approach is developed to obtain the sensitivity derivatives for the Euler equations. The completederivationofthecostateequations andtheirtransversality (boundary)conditionsarepresented. Both the state and the costate equations are second-order e nite volume discretized for unstructured meshes, and they are coupled with a constrained optimization algorithm. Also integrated into the overall methodology are a geometry parameterization method for the shape optimization, and a dynamic unstructured mesh method for the shape evolution and the consequent volume mesh adaptations. For the proof of concept, three transonic airfoil optimization problems are presented. This method accepts general cost functionals, which are not necessarily functions of pressure only. It is also shown that a switch to the natural coordinate system in conjunction with the reduction of the governing state equation to the control surface results in sensitivity integrals that are only a function of the tangential derivativesof the state variables. Thisapproach eliminates the need fornormal derivativecomputations that can be erroneous.

A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a second-order accurate discretization method is described. The relationship between the continuous formulation and a discrete formulation is explored for inviscid, as well as for viscous flow. Several limitations in a strict adherence to the continuous approach are uncovered, and an approach that circumvents these difficulties is presented. The issue of grid sensitivities, which do not arise naturally in the continuous formulation, is investigated and is observed to be of importance when dealing with geometric singularities. A method is described for modifying inviscid and viscous meshes during the design cycle to accommodate changes in the surface shape. The accuracy of the sensitivity derivatives is established by comparing with finite-difference gradients and several design examples are presented.

In this paper, we present a method for solving nonlinear optimal control problems subject to terminal state constraints and control saturation constraints. This method is based on using the quasilinearization and the state variables parametrization by using Chebyshev polynomials. In this method, there is no need to integrate the system state equations or the costate equations. By virtue of the quasilinearization and the state parametrization, the difficult constrained nonlinear optimal control problem is approximated by a sequence of small quadratic programming problems which can be solved easily. To show the effectiveness of the proposed method, the simulation results of a numerical example are shown.

The stability of nonlinear dispersive-wave equations near the short wave limit is examined analytically and computationally. Equations of first and second order are analyzed based on their linear dispersion relations. Computational tests are performed on the nonlinear version of the equations, which confirm the theoretical estimates. Equations are derived based on approximations of the water-wave Hamiltonian, and are shown to possess different stability properties depending on the order of the Hamiltonian expansion in terms of a small parameter. It is shown that the smoothest solution is achieved by the regularized form of the equations, which, however, lead to excessively dissipative computational results. Consistent results are obtained by the second-order, Hamiltonian-based equations. All equations are solved by a Fourier pseudo-spectral method, which permits direct comparison of the various equation forms based on identical initial conditions that asymptotically reach the short wave limit.

A new approach for optimization of control problems defined by fully implicit differential-algebraic equations is described in the paper. The main feature of the approach is that system equations are substituted by discrete-time implicit equations resulting from the integration of the system equations by an implicit Runge–Kutta method. The optimization variables are parameters of piecewise constant approximations to control functions; thus, the control problem is reduced to the control space only. The method copes efficiently with problems defined by large-scale differential-algebraic equations.

This chapter is devoted to second order equations and is organized as follows. First we deal with general linear homogeneous equations, including linear independence of solutions and the reduction of order. Then we discuss general linear nonhomogeneous equations. Sections 5.5 and 5.6 deal with the constant coefficients case. Section 5.7 is devoted to the study of oscillation theory and the oscillatory behavior of solutions. Finally, in the last section we deal with some nonlinear equations.

International audience; Sensitivity relations in optimal control provide an interpretation of the costate trajectory and the Hamiltonian, evaluated along an optimal trajectory, in terms of gradients of the value function. While sensitivity relations are a straightforward consequence of standard transversality conditions for state constraint free optimal control problems formulated in terms of control-dependent differential equations with smooth data, their verification for problems with either pathwise state constraints, nonsmooth data, or for problems where the dynamic constraint takes the form of a differential inclusion, requires careful analysis. In this paper we establish validity of both 'full' and 'partial' sensitivity relations for an adjoint state of the maximum principle, for optimal control problems with pathwise state constraints, where the underlying control system is described by a differential inclusion. The partial sensitivity relation interprets the costate in terms of partial Clarke subgradients of the value function with respect to the state variable, while the full sensitivity relation interprets the couple, comprising the costate and Hamiltonian, as the Clarke subgradient of the value function with respect to both time and state variables. These relations are distinct because, for nonsmooth data, the partial Clarke subdifferential does not coincide with the projection of the (full) Clarke subdifferential on the relevant coordinate space. We show for the first time (even for problems without state constraints) that a costate trajectory can be chosen to satisfy the partial and full sensitivity relations simultaneously. The partial sensitivity relation in this paper is new for state constraint problems, while the full sensitivity relation improves on earlier results in the literature (for optimal control problems formulated in terms of Lipschitz continuous multifunctions), because a less restrictive inward pointing hypothesis is invoked in the proof, and because it is validated for a stronger set of necessary conditions.

The maximum principle for the time optimal control of the Navier-Stokes equations in 2-D is established.

International audience; Sensitivity relations in optimal control identify the costate trajectory and the Hamiltonian, evaluated along a minimizing trajectory, as gradients of the value function. Sensitivity relations for optimal control problems not involving state constraints and formulated in terms of controlled differential equation with smooth data follow easily from standard transversality conditions. In the presence of pathwise state constraints, if the data is nonsmooth or when the dynamic constraint takes the form of a differential inclusion, deriving the sensitivity relations is far from straightforward. We announce both 'full' and 'partial' sensitivity relations for differential inclusion problems with pathwise state constraints. The partial sensitivity relation identifies the costate with a partial subgradient of the value function with respect to the state, and the full sensitivity relation identifies the costate and the Hamiltonian with a subgradient of the value function with respect to time and state. The partial sensitivity relation is new for state constraint problems. The full sensitivity relation is valid under reduced hypotheses and for a stronger form of necessary conditions, as compared with earlier literature. It is shown for the first time also that the costate arc can be chosen to satisfy the two relations simultaneously. An example illustrates that a costate trajectory may be specially chosen to satisfy the sensitivity relations, and it is possible that some costate trajectories fail to do so.

This paper mainly works on an optimal control problem of mean-field forward-backward stochastic differential equations (MFFBSDEs) with partial information. But different from the general optimal control problems, this paper is concerned with the case of partial information and state equations are coupled at initial time. Meanwhile, we introduce the mean-field theory. By virtue of the classical convex variational technique, we establish a necessary maximum principle for the optimization problems.

The relationship between the Heun class of second-order linear equations and the Painleve second-order nonlinear equations is studied. The symbol of the Heun class equations is regarded as a quantum Hamiltonian. The independent variable and the differentiation operator correspond to the canonical variables and one of the parameters of the equation is assumed to be time. Painleve equations appear to be Euler - Lagrange equations related to corresponding classical motion.

A method for the automatic calculation of costates using only the results obtained from direct optimization techniques is presented. The approach exploits the relation between the time-varying costates and certain sensitivities of the variational cost function, a relation that also exists between the Lagrangian multipliers obtained from a direct optimization approach and the sensitivities of the associated nonlinear-programming cost function. The complete theory for treating free, control-constrained, interior-point-constrained, and state-constrained optimal control problems is presented. As a numerical example, a state-constrained version of the brachistochrone problem is solved and the results are compared to the optimal solution obtained from Pontryagin's minimum principle. The agreement is found to be excellent. Nomenclature / = right-hand side of state equations ge = control equality constraints gi = control inequality constraints he = state equality constraints hi = state inequality constraints J = cost function M = interior-point constraints m = dimension of control vector u N = total number of nodes minus 1 = total number of subintervals n — dimension of state vector x PWC = set of piecewise continuous functions t = time tf = final time ti = nodes along the time axis to = initial time u = control vector x = state vector Xf = final state Xi = state vector at node £/ Xo = initial state \(t) = costate A/ = Lagrangian multiplier associated with differential constraints along subinterval / Hi - Lagrangian multiplier associated with state constraints at node i (Ti = Lagrangian multiplier associated with control constraints along subinterval /

A variational approach to the Euler-Lagrange equations for the quasi-hydrostatic model is presented together with an explicit Hamiltonian (Poisson bracket) formulation. the bracket acts on functionals of the zonal and meridional angular momentum. Features of the relationships between the Hamiltonian formulations of the equations for an ideal fluid, the hydrostatic primitive equations and the quasi-hydrostatic equations are discussed.

A widely used approach to parameter identification is the output least-squares formulation. Numerical methods for solving the resulting minimization problem almost invariably require the computation of the gradient of the output least-squares functional. When the identification problem involves time-dependent distributed parameter systems (or approximations thereof), numerical evaluation of the gradient can be extremely time consuming. The costate method can greatly reduce the cost of computing these gradients. However, questions have been raised concerning the accuracy and convergence of costate approximations, even when the numerical methods being used are known to converge rapidly on the forward problem. In this paper it is shown that the use of time-marching schemes that yield high-order accuracy on the forward problem does not necessarily lead to high-order accurate costate approximations. In fact, in some cases these approximations do not converge at all. However, under certain circumstances, rapidly converging gradient approximations do result because of rapid weak-star-type convergence of the costate approximations. These issues are treated both theoretically and numerically.