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The optimal control problem constrained by a fractional diffusion equation arises in a great deal of applications. Fast and efficient numerical methods for solving such kinds of problems have attracted much attention in recent years. In this paper, we consider an optimal control problem constrained by a fractional diffusion equation (FDE). After the state and costate equations are derived, the closed form of the optimal control variable is then given. The decoupled gradient projection method is applied to solve the coupled system of the state and costate equations to obtain the solution of the optimal control problem. The second-order Crank-Nicolson method as well as the weighted and shifted Grunwald difference (CN-WSGD) methods are utilized to discretize these two equations. We get the discretized state and costate equations as systems of linear equations with both coefficient matrices having the structure of the sum of a diagonal and a Toeplitz matrix. A diagonal and a R.Chan’s circulant splitting (DRCS) preconditioner is developed and combined in the Krylov subspace methods to solve the resulting discretized linear systems. Theoretical analysis of spectral distributions of the preconditioned matrix is also given. Numerical results exhibit that the proposed preconditioner can significantly improve the convergence of the Krylov subspace iteration methods.

We have developed a new method for solving low-thrust fuel-optimal orbit transfer problems in the vicinity of a large body (planet or asteroid), considering a high-fidelity spherical harmonic gravity model. The algorithm is formulated via the indirect optimization method, leading to a two-point boundary value problem (TPBVP). We make use of a hyperbolic tangent smoothing law for performing continuation on the thrust magnitude to reduce the sharpness of the control switches in early iterations and thus promote convergence. The TPBVP is solved using the method of particular solutions (MPS) shooting method and Picard-Chebyshev numerical integration. Application of Picard-Chebyshev integration affords an avenue for increased efficiency that is not available with step-by-step integrators. We demonstrate that computing the particular solutions with only a low-fidelity force model greatly increases the efficiency of the algorithm while ultimately achieving near machine precision accuracy. A salient feature of the MPS is that it is parallelizable, and thus further speedups are available. It is also shown that, for near-Earth orbits and over a small number of en-route revolutions around the Earth, only the zonal perturbation terms are required in the costate equations to obtain a solution that is accurate to machine precision and optimal to engineering precision. The proposed framework can be used for trajectory design around small asteroids and also for orbit debris rendezvous and removal tasks.

Velocity at landing should be within a safe limit, to make soft landing possible. For this the thrust steering angle should be controlled to optimize the fuel consumption. This paper deals with designing soft landing trajectory, from lunar parking orbit to the surface of Moon. To ensure vertical landing using minimum fuel, the landing trajectory design is done in four phases: Hohmann transfer phase, power descend phase, attitude maneuver and vertical descend phase. The fuel optimization is done in the power descend phase. Then the attitude of the probe is adjusted and the probe is made vertical during the attitude maneuver. In final vertical descend phase the steering angle is kept as 90 degree to land the probe vertically. Optimal soft landing trajectory is designed using indirect approach of optimization. The indirect approach uses, Pontryagin's minimum principle to transfer optimal control problem to Two Point Boundary Value (TPBV) problem, and to express the control variable as a function of costate variables. Then the state and costate equations are integrated at optimum control angle, with the initial costate found by the different optimization techniques, to get the optimal trajectory. Two optimization techniques are used in this paper: Control Random Search (CRS) and Differential Evolution (DE). The former optimization techniques are compared with optimal trajectory designed by indirect approach. The simulation results show that CRS has taken 10000 iterations in a simulation time of 120 minutes to converge while DE converged with 2000 iteration in 30 minutes of CPU time. Even though the velocity obtained after power descent phase and optimum landing mass are almost same in both technique, the time of convergence of CRS is four times that of DE. i.e., DE is found superior to CRS, since it converges faster and load taken by system is less. The problem of soft landing is extended to design a fuel optimal three dimensional trajectory using DE technique.

In this article, we describe a discontinuous finite volume method with interpolated coefficients for the numerical approximation of the distributed optimal control problem governed by a class of semilinear elliptic equations with control constraints. The proposed distributed control problem involves three unknown variable: control, state and costate. For the approximation of control, we have adopted three different methodologies: variational discretization, piecewise constant and piecewise linear discretization, while the approximation of state and costate variables is based on discontinuous piecewise linear polynomials. As the resulted scheme is non-symmetric, optimize-then-discretize approach is used to approximate the control problem. Optimal a priori error estimates in suitable natural norms for state, costate and control variables are derived. Moreover, numerical experiments are presented to support the derived theoretical results. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

This paper deals with the Pontryagin maximum principle for optimal control problems governed by 3D Navier---Stokes equations with pointwise control constraint. The obtained result is proved by using some results on regularity of solutions of the Navier---Stokes equations and techniques of optimal control theory.

This paper studies a time optimal control problem for a class of ordinary differential equations. The control systems may have multiple solutions. Based on the properties fulfilled by the solutions of the concerned equations, we get both the existence and the Pontryagin maximum principle for optimal controls.

Preliminary design of low-thrust trajectories generally benefits from broad searches over the feasible space. Despite convergence issues, indirect methods are generally faster than direct methods and are therefore well-suited for such searches. Nonetheless, indirect solutions typically require expensive numerical integration of at least the state and costate equations. Here, based on the physical interpretation of the primer vector, a fast model that approximates solutions to all the dynamics is introduced. If the ballistic dynamics have a closed-form solution, then the costate equations can be approximated without resorting to numerical integration. Analogous to the Sims–Flanagan model for direct optimization, the new model approximates thrust arcs with a series of ballistic arcs and impulsive maneuvers. The closed-form solution is obtained using a Sundman-transformed independent variable, which also provides an efficient discretization. Furthermore, a low-order series solution is used for the ballistic ...

We present models of differential terror queue games, wherein terrorists seek to determine optimal attack rates over time, while simultaneously the government develops optimal counterterror staffing levels. The number of successful and interdicted terror attacks is determined via an underlying dynamic terror queue model. Different information structures and commitment abilities derive from different assumptions regarding what the players in the game can and cannot deduce about the underlying model. We compare and explain the impact of different information structures, i.e., open loop, closed loop, and asymmetric. We characterize the optimal controls for both the terrorists and the government in terms of the associated state and costate variables and deduce the costate equations that must be solved numerically to yield solutions to the game for the different cases. Using recently assembled data describing both terror attack and staffing levels, we compare the differential game models to each other as well as to the optimal control model of Seidl et al. (Eur J Oper Res 248:246–256, 2016). The paper concludes with a discussion of the lessons learned from the entire modeling exercise.

A pseudospectral method is presented for direct trajectory optimization and costate estimation of infinite-horizon optimal control problems using global collocation at flipped Legendre-Gauss-Radau points which include the end point +1. A distinctive feature of the method is that it uses a new smooth, strictly monotonically decreasing transformation to map the scaled left half-open interval τ ϵ (-1, +1] to the descending time interval t ϵ (+∞, 0]. As a result, the singularity of collocation at point +1 associated with the commonly used transformation, which maps the scaled right half-open interval τ ϵ [-1, +1) to the increasing time interval [0,+∞), is avoided. The costate and constraint multiplier estimates for the proposed method are rigorously derived by comparing the discretized necessary optimality conditions of a finite-horizon optimal control problem with the Karush-Kuhn-Tucker conditions of the resulting nonlinear programming problem from collocation. Another key feature of the proposed method is that it provides highly accurate approximation to the state and costate on the entire horizon, including approximation at t = +∞, with good numerical stability. Numerical results show that the method presented in this paper leads to the ability to determine highly accurate solutions to infinite-horizon optimal control problems.

The problem of determining the optimum shape of a homogeneous Euler–Bernoulli beam of a circular cross-section, in which the coupled axial and bending vibrations arose due to complex boundary conditions, is considered. The beam mass is minimized at prescribed fundamental frequency. The problem is solved applying Pontryagin’s maximum principle, with the beam cross-sectional diameter derivative with respect to longitudinal coordinate taken for control variable. This problem involves first-order singular optimal control, the calculations of which allowed the application of the Poisson bracket formalism and the fulfillment of the Kelley necessary condition on singular segments. Numerical solution of the two-point boundary value problem is obtained by the shooting method. An inequality constraint is imposed to the beam diameter derivative. Depending on the size of the diameter derivative boundaries, the obtained solutions are singular along the entire beam or consist of singular and non-singular segments, where the diameter derivative is at one of its boundaries. It is shown that such system is self-adjoint, so that only one differential equation of the costate equations system was integrated and the rest costate variables were expressed via the state variables. Also, the paper shows the fulfillment of necessary conditions for the optimality of junctions between singular and non-singular segments, as well as the percent saving of the beam mass compared to the beams of constant diameter at identical value of the fundamental frequency.

The control of an epidemic disease consists in introducing the strategies able to reduce the number of infected subjects by means of medication/quarantine actions, and the number of the subjects that could catch the disease through an informative campaign and, when available, a vaccination strategy. Some diseases, like the influenza, do not guarantee immunity; therefore, the subjects could get ill again by different strain of the same viral subtype. The epidemic model adopted in this paper introduces the cross-immune individuals; it is known in literature as SIRC model, since the classes of susceptible (S), infected (I), removed (R) and cross-immune (C) subjects are considered. Its control is herein determined in the framework of the linear quadratic regulator, by applying to the original nonlinear model the optimal control found on the linearized system. The results appear satisfactory, and the drawback of using a control law based on the linear approximation of the system is compensated by the advantages arising from such a solution: no costate equations to be solved and a solution depending on the current state evolution which allows a feedback implementation.

An optimal control problem for a system of stochastic partial differential equations with nonlocal terms is considered. The control of this system is motivated by a model for optimal harvesting of a population on a spatial grassland habitat. Existence of optimal controls and necessary conditions for the optimal control of such systems are established.

We formulate and solve the conjugate problem for Lagrange multipliers connected with designing a Laval nozzle optimal contour, including its subsonic part. In approximation of an ideal (inviscid and nonheat-conducting) gas, the sought contour provides a thrust maximum under a number of constraints; in particular, given total nozzle length and gas mass flow. For a contour of a contracting (subsonic) part of the nozzle (which is suspected to be optimal) we take an abrupt contraction. Because of the constraint on the nozzle length, the abrupt contraction can be a region of boundary extremum with positive permissible variations of the longitudinal (“axial”) coordinate of the contour. To clarify whether this is true, we use the method of Lagrange multipliers and formulate the conjugate problem for finding the Lagrange multipliers. As in the case of quasilinear Euler equations governing a flow, the linear equations in the conjugate problem are elliptic (hyperbolic) in the subsonic (supersonic) flow domain. The requirement that the conjugate problem be solvable for any contour highlights some features that can be of interest not only for this special problem, but, possibly for the general theory of mixed type partial differential equations.

The problem of designing low-energy transfers between the Earth and the Moon has attracted recently a major interest from the scientific community. In this paper, an indirect optimal control approach is used to determine minimum-fuel low-thrust transfers between a low Earth orbit and a Lunar orbit in the Sun–Earth–Moon Bicircular Restricted Four-Body Problem. First, the optimal control problem is formulated and its necessary optimality conditions are derived from Pontryagin’s Maximum Principle. Then, two different solution methods are proposed to overcome the numerical difficulties arising from the huge sensitivity of the problem’s state and costate equations. The first one consists in the use of continuation techniques. The second one is based on a massive exploration of the set of unknown variables appearing in the optimality conditions. The dimension of the search space is reduced by considering adapted variables leading to a reduction of the computational time. The trajectories found are classified in several families according to their shape, transfer duration and fuel expenditure. Finally, an analysis based on the dynamical structure provided by the invariant manifolds of the two underlying Circular Restricted Three-Body Problems, Earth–Moon and Sun–Earth is presented leading to a physical interpretation of the different families of trajectories.

The finite element method is used in the field of solid and structural mechanics. Various types of problems solved by the finite element method in this field include the elastic, elastoplastic, and viscoelastic analysis of trusses, frames, plates, shells, and solid bodies. This chapter presents general equations of solid and structural mechanics and their derivation and solution. The primary aim of any stress analysis or solid mechanics problem is to find the distribution of displacements and stresses under the stated loading and boundary conditions. If an analytical solution of the problem is to be found, one has to satisfy the basic equations of solid mechanics. The number of unknown quantities is equal to the number of equations available. The finite element equations can also be derived by using either a differential equation formulation method (for example, Galerkin approach) or variational formulation method (for example, Rayleigh-Ritz approach). In the case of solid and structural mechanics problems, each of the differential equation and variational formulation methods is classified into three categories, which are displacement, force, and displacement-force method. The chapter describes the displacement method (or equivalently the principle of minimum potential energy) for deriving the finite element equations. Variational formulation methods are based on the principle of minimum potential energy, principle of minimum complementary energy, and the principle of stationary Reissner Energy.

We propose an “optimize-then-discretize” approach for the numerical solution of optimal control problems for systems with delays in both state and control. We first derive the optimality conditions and an explicit representation of the gradient of the cost functional. Then, we use explicit discretizations of the state/costate equations and employ general-purpose Non-Linear Programming (NLP) solvers, in particular Conjugate Gradient or Quasi-Newton schemes, to easily implement a descent method. Finally, we prove convergence of the algorithm to stationary points of the cost, and present some numerical simulations on model problems, including performance evaluation.

This paper presents a system representation that can be applied to the description of the interaction between systems connected through common boundaries. The systems consist of partial differential equations that are first order with respect to time, but spatially higher order. The representation is derived from the instantaneous multisymplectic Hamiltonian formalism; therefore, it possesses the physical consistency with respect to energy. In the interconnection, particular pairs of control inputs and observing outputs, called port variables, defined on the boundaries are used. The port variables are systematically introduced from the representation.

In this paper, the time and norm optimal control problems of controlled heat equations with a weight function are considered. For the time optimal problems, we study the following two cases: one is for equations with multi-domain control under null controllability, and the other is for equations under approximate null controllability. We prove the solvability, and obtain the bang-bang principle of the time optimal controls for aforementioned both cases. For the norm optimal control problems, we focus on equations with multi-time and multi-domain control, and present the solvability of these problems.

We develop a partial Hamiltonian framework to obtain reductions and closed-form solutions via first integrals of current value Hamiltonian systems of ordinary differential equations (ODEs). The approach is algorithmic and applies to many state and costate variables of the current value Hamiltonian. However, we apply the method to models with one control, one state and one costate variable to illustrate its effectiveness. The current value Hamiltonian systems arise in economic growth theory and other economic models. We explain our approach with the help of a simple illustrative example and then apply it to two widely used economic growth models: the Ramsey model with a constant relative risk aversion (CRRA) utility function and Cobb Douglas technology and a one-sector AK model of endogenous growth are considered. We show that our newly developed systematic approach can be used to deduce results given in the literature and also to find new solutions.

Pontryagin's maximum principle reduces optimal control problems to the investigation of Hamiltonian systems of ordinary differential equations with discontinuous right-hand side. An optimal synthesis is the totality of solutions to this system with a fixed terminal (or initial) condition, which fill a region in the phase space one-to-one. In the construction of optimal synthesis, singular trajectories that go along the discontinuity surface N of the right-hand side of the Hamiltonian system of ordinary differential equations, are crucial. The aim of the paper is to prove that the system of singular trajectories makes up a Hamiltonian flow on a submanifold of N. In particular, it is proved that the flow of singular trajectories in the problem of control of the magnetized Lagrange top in a variable magnetic field is completely Liouville integrable and can be embedded in the flow of a smooth superintegrable Hamiltonian system in the ambient space. Bibliography: 17 titles.

A pseudospectral (PS) method based on Hermite interpolation and collocation at the Legendre-Gauss-Lobatto (LGL) points is presented for direct trajectory optimization and costate estimation of optimal control problems. A major characteristic of this method is that the state is approximated by the Hermite interpolation instead of the commonly used Lagrange interpolation. The derivatives of the state and its approximation at the terminal time are set to match up by using a Hermite interpolation. Since the terminal state derivative is determined from the dynamic, the state approximation can automatically satisfy the dynamic at the terminal time. When collocating the dynamic at the LGL points, the collocation equation for the terminal point can be omitted because it is constantly satisfied. By this approach, the proposed method avoids the issue of the Legendre PS method where the discrete state variables are over-constrained by the collocation equations, hence achieving the same level of solution accuracy as the Gauss PS method and the Radau PS method, while retaining the ability to explicitly generate the control solution at the endpoints. A mapping relationship between the Karush-Kuhn-Tucker multipliers of the nonlinear programming problem and the costate of the optimal control problem is developed for this method. The numerical example illustrates that the use of the Hermite interpolation as described leads to the ability to produce both highly accurate primal and dual solutions for optimal control problems.

Summary Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) points. It is shown that the derivative of the costate of the continuous-time optimal control problem is equal to the negative of the costate of the integral form of the continuous-time optimal control problem. Using this continuous-time relationship between the differential and integral costate, it is shown that the discrete approximations of the differential costate using LG and LGR collocation are related to the corresponding discrete approximations of the integral costate via integration matrices. The approach developed in this paper provides a way to approximate the costate of the original optimal control problem using the Lagrange multipliers of the integral form of the LG and LGR collocation methods. The methods are demonstrated on two examples where it is shown that both the differential and integral costate converge exponentially as a function of the number of LG or LGR points. Copyright © 2014 John Wiley & Sons, Ltd.

SUMMARY This article describes the application of optimal control to a solar furnace that is used to perform temperature stress cycling tests in material samples. This process is characterized by having nonlinearities that depend on the sample properties and relate the temperature of the sample with the solar energy fluctuations and the position of the furnace shutter. An optimal control problem with fix terminal time and free terminal state and control constraints is addressed in continuous time domain. The solution is approximated using discretized state and costate equations and applied to the furnace according to a receding horizon strategy. The performance of the overall system is evaluated from computer simulations which show that the controller is able to tolerate, up to some degree, the presence of parameter uncertainty. Copyright © 2014 John Wiley & Sons, Ltd.

This paper describes flying multi-vehicle control strategies and its benefit for saving fuel. Exposition starts from inspiration of flying multi-vehicle in daily life. Furthermore, from model of single flying vehicle, we construct the model of multi-vehicle and cost functional model that describe the state of the cost to be met the flying vehicle. The flying multi-vehicle control designed with optimal control strategy. The design of optimal control is done through the Pontryagin Maximum Principle, brings the model to a system of equations consisting of state equations and costate equations. In the system of states equations, each having initial and final condition, in the costate equations system has no requirements at all. The next problem is converted to the initial value problem and search for the approximate initial condition equation of costate equations system which has no requirements using a modified method of steepest descent. Thus, the control of multi-vehicle successfully performed and the simulation results presented on the results and discussion section. In addition, we also calcute the fuel which used by multi-vehicle, compared by the fuel which used by each vehicle in solo flying. The result can be conclude that the fuel more efficient if the flying vehicles in formation flying.

This paper presents an algorithm that provides a regularization for the costate dynamics of state constrained optimal control problems with a scalar constraint under the assumption that the Hamiltonian is convex in the control and the state dynamics equation of the constrained state is monotonically increasing in the control variable. The algorithm is demonstrated with a classical optimal control problem.

In this chapter, we present an introduction to the optimal control of partial differential equations. After explaining what an optimal control problem is and the goals of the analysis of these problems, we focus the study on a model example. We consider an optimal control problem governed by a semilinear elliptic equation, the control being subject to bound constraints. Then we explain the methods to prove the existence of a solution; to derive the first and second order optimality conditions; to approximate the control problem by discrete problems; to prove the convergence of the discretization and to get some error estimates. Finally we present a numerical algorithm to solve the discrete problem and we provide some numerical results. Though the whole analysis is done for an elliptic control problem, with distributed controls, some other control problems are formulated, which show the scope of the field of control theory and the variety of mathematical methods necessary for the analysis. Among these problems, we consider the case of evolution equations, Neumann or Dirichlet boundary controls, and state constraints.

This chapter introduces an approach for calculating pressure-volume-temperature equations of state with nested sampling.

A complete list of third order vector Hamiltonian equations with the Hamiltonian operator D x having an infinite series of higher conservation laws is presented. A new vector integrable equation on the sphere is found.

Parturient paresis(milk fever) is a common disease associated with the onset of parturition in dairy cows. The disease is considered due to a large increased demand for calcium. Several work has mathematically and biologically modelled this process. Based on the existing models on calcium dynamics in diary cows, an optimal impulse treatment is proposed in this paper. The treatment is executed at a fixed time interval and lasts a relatively very small time duration, which is termed as a "fixed time impulse" control. For the optimization, with a selected objective function, a series of equations for optimality are to be satisfied, including control equations, costate equations and state equations. Those impulsive differential equations form a two point boundary value problem and are difficult to solve. A numerical scheme, SNAC(Single Network Adaptive Critic), is then proposed. The algorithm key is to use one neural network to capture the optimal relation between the pre-impulse state and the after-impluse costate. After the neural network is trained and the relation is captured, the optimal impulse dosage of medicine can be provided when a parturient paresis is detected, and the cow's calcium level can be brought back to the normal status. Simulations are presented for illustrative purposes.

The dynamic principle is proposed that a thermodynamic system evolves in time so that the total energy balance including the energy drawn from the environment becomes stationary for all admissible variations of thermodynamic states. It is shown that this principle allows to obtain variational characterizations of contact Hamiltonian equations (even in presence of ports), reaction equations and doubly nonlinear reaction–diffusion equations. Further, examples are discussed which support this principle.

In this paper, we consider a numerical method for designing optimal controlon Linear Quadratic Regulator (LQR) problem. In the optimal control design process through Pontryagin Maximum Principle (PMP), we obtain a system of diferential equations in state and costate variables. This system lacks of initial condition on the adjoint variables, and this situation creates classic dificulty for solving optimal control problems.This paper proposes a constructive method to approximate the initial condition of the adjoint system.DOI : http://dx.doi.org/10.22342/jims.15.1.40.13-20

This paper deals with the Pontryagin's principle of optimal control problems governed by the 2D Navier-Stokes equations with integral state constraints and coupled integral control--state constraints. As an application, the necessary conditions for the local solution in the sense of $L^r(0,T;L^2(\Omega))$ ($2 < r < \infty$) are also obtained.

WOS: 000293829300016, In this paper, a new approach to the variational iteration method is introduced to solve systems of first-order differential equations. Since higher-order differential equations can almost always be converted into a first-order system of equations, the proposed method is still applicable to a wide range of differential equations. This generalised approach, unlike the classical method, uses restricted variations only for nonlinear terms by generalising the Lagrange multipliers. Consequently, this allows us to use the well known, but ignored, theory of linear ODEs for computing the matrix-valued Lagrange multipliers. In order to validate the newly proposed approach in solving linear and nonlinear systems of differential equations, illustrative examples are presented: it turns out that the use of the generalised Lagrange multipliers is more reliable and efficient. (C) 2011 Elsevier Ltd. All rights reserved.

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.

In all the applications of pseudo-time-stepping method mentioned in Chapters 7- 11, the state equations have been the Euler equations. In this chapter we extend the method to viscous compressible flow modeled by the Reynolds Averaged Navier- Stokes equations together with algebraic turbulence model of Baldwin and Lomax. While inviscid formulations are useful for the design in transonic cruise conditions, inclusion of viscous effects are essential for optimal design encompassing off-design conditions and high-lift configurations. The computational complexity in viscous design is at least an order of magnitude greater than that in inviscid design since the number of mesh points are to be increased by a factor of two or more to resolve the boundary layer. The convergence of the Navier-Stokes solver is much more slower than the Euler solver due to discrete stiffness and directional decoupling arising from the highly stretched boundary layer cells. Since we use inaccurate state and costate solutions, and hence inaccurate gradients, in our one-shot pseudotime- stepping method, therefore slow convergence of the Navier-Stokes (forward and adjoint) solver may affect the convergence of this method. We investigate that numerically in this chapter.

The relations between the partial differential equations of multivariable vector functions and the integral equations of multivariable vector functions which are correspond to them are discussed. The partial differential linear equations of multivariable vector functions can be transformed into the integral linear equations of multivariable vector functions by using the continuous wavelet transforms of multivariable vector function spaces. The result that in the weak topology the partial differential equations of multivariable vector functions are equivalent to the integral equations of multivariable vector functions which are correspond to them is obtained.

By converting an optimal control problem for nonlinear systems to a Hamiltonian system, a symplecitc-preserving method is proposed. The state and costate variables are approximated by the Lagrange polynomial. The state variables at two ends of the time interval are taken as independent variables. Based on the dual variable principle, nonlinear optimal control problems are replaced with nonlinear equations. Furthermore, in the implementation of the symplectic algorithm, based on the 2 N algorithm, a multilevel method is proposed. When the time grid is refined from low level to high level, the initial state and costate variables of the nonlinear equations can be obtained from the Lagrange interpolation at the low level grid to improve efficiency. Numerical simulations show the precision and the efficiency of the proposed algorithm in this paper.

A novel unified approach to two-degrees-of-freedom control is devised and applied to a classical chemical reactor model. The scheme is constructed from the optimal control point of view and along the lines of the Hamiltonian formalism for nonlinear processes. The proposed scheme optimizes both the feedforward and the feedback components of the control variable with respect to the same cost objective. The original Hamiltonian function governs the feedforward dynamics, and its derivatives are part of the gain for the feedback component. The optimal state trajectory is generated online, and is tracked by a combination of deterministic and stochastic optimal tools. The relevant numerical data to manipulate all stages come from a unique off-line calculation, which provides design information for a whole family of related control problems. This is possible because a new set of PDEs (the variational equations) allow to recover the initial value of the costate variable, and the Hamilton equations can then be solved as an initial-value problem. Perturbations from the optimal trajectory are abated through an optimal state estimator and a deterministic regulator with a generalized Riccati gain. Both gains are updated online, starting with initial values extracted from the solution to the variational equations. The control strategy is particularly useful in driving nonlinear processes from an equilibrium point to an arbitrary target in a finite-horizon optimization context. Copyright © 2010 John Wiley & Sons, Ltd.

In optimal control theory, it is well known that the costate arc and the associated maximized Hamiltonian function can be interpreted in terms of gradients of the value function, evaluated along the optimal state trajectory. Such relations have been referred to as “sensitivity relations” in the literature. We provide in this paper new sensitivity relations for state constrained optimal control problems. For the class of optimal control problems considered, there is no guarantee that the costate arc is unique; a key feature of the results is that they assert some choice of costate arc can be made for which the sensitivity relations are valid. The proof technique is to introduce an auxiliary optimal control problem that possesses a richer set of control variables than the original problem. The introduction of the additional control variables in effect enlarges the class of variations with respect to which the state trajectory under consideration is a minimizer; the extra information thereby obtained yields the desired set of sensitivity relations.

This work focuses on the optimal control of a quantum system composed of harmonic oscillators under linear control agents (dipole function, objective operator, and the penalty operator whose expectation value is to be suppressed). The main purpose of the work is to determine the temporal external field amplitude function. Paper recalls the formulation of the optimal control equations first. Then a set of ordinary differential equations over the expectation values of certain unknown entities is constructed. These temporal differential equations have time varying coefficients unless the weight functions appearing in the cost functional are constant. Certain accompanying conditions are needed to get unique solutions. Investigations show that one half of the conditions should be given at the initial instant and the other half should be specified at the final moment. Since the differential equations contain another unknown entity, deviation parameter, solutions must satisfy an algebraic equation derived from the definition of this parameter. Results do not involve the explicit structure of the wave function and costate function. Only the external field amplitude and the deviation parameter are determined here. The evaluation of the wave function and costate function needs additional treatments to the control equations. We report certain analytical results for external field amplitude and the deviation parameter and give certain illustrative implementations to finalize the paper.

We study in this article Pontryagin’s maximum principle for a class of control problems associated with the stationary primitive equations (PEs) of the ocean. These optimal problems involve a state constraint similar to that considered in [G. Wang, Pontryagin’s maximum principle for optimal control of the stationary Navier–Stokes equations, Nonlinear Anal. 52 (2003) 1853–1866] in the case of the three-dimensional Navier–Stokes (NS) equations. The main difference between this work and the one cited above is that the nonlinearity in PEs is stronger than that in the three-dimensional NS equations.

A modification of the classical needle variation, namely, the so-called two-parameter variation of controls is proposed. The first variation of a functional is understood as a repeated limit. It is shown that the modified needle variation can be effectively used to derive necessary optimality conditions for a rather wide class of optimal control problems involving partial differential equations with weak solutions. Specifically, the two-parameter variation is used to obtain necessary optimality conditions in the form of a maximum principle for the optimal control of divergent hyperbolic equations.

The dynamics of a mechanical system in the Lagrange space yields a set of differential equations of the second order and involves much less variables and constraints than that described in the state space. This paper presents a so-called Legendre pseudo-spectral (PS) approach for directly estimating the costates of the Bolza problem of optimal control of a set of dynamic equations of the second order. Under a set of closure conditions, it is proved that the Karush-Kuhn-Tucker (KKT) multipliers satisfy the same conditions as those determined by collocating the costate equations of the second order. Hence, the KKT multipliers can be used to estimate the costates of the Bolza problem via a simple linear mapping. The proposed approach can be used to check the optimality of the direct solution for a trajectory optimization problem involving the dynamic equations of the second order and to remove any conversion of the dynamic system from the second order to the first order. The new approach is demonstrated via two classical benchmark problems.

The analytic solution to an optimal control problem is investigated using the canonical dual method. By means of the Pontryagin principle and a transformation of the cost functional, the optimal control of a nonconvex problem is obtained. It turns out that the optimal control can be expressed by the costate via canonical dual variables. Some examples are illustrated.