Minimum-fuel rocket trajectories involving intermediate-thrust arcs

The optimal trajectories in the neighborhood of an optimal intermediate-thrust arc are investigated for the minimumfuel orbit rendezvous problem with fixed specific impulse. Since such an arc is singular, the thrust acceleration magnitude being the singular control component, a second-variation analysis leads to the identification of a field of neighboring, singular arcs in a state space of dimension four rather than six, provided that a suitable Jacobi condition is met. A given neighboring initial six-dimensional state vector does not generally lie on a neighboring singular arc, and junction onto the appropriate singular arc must be accomplished by a short period of strong variations in the acceleration. This contributes an addition to the fuel expenditure which is of order 5/2 rather than 2 in the initial state displacement. The minimization of this higher-order cost, in the case of bounded acceleration, leads to an unsymmetrical version of Fuller's problem, whose solution requires an infinite number of switches between maximum and zero thrust during the short period. For unbounded thrust, the junction simplifies to either coast-impulse-singular trajectories or impulse-coast-impulse-singular trajectories. The neighboring singular arc meets the final condition in 4 dimensions, rather than 6 dimensions, and rendezvous must be completed by another, terminal short period of strong variations in the acceleration. Implications for midcourse guidance near a singular arc are discussed.     

Singular optimal controls of stochastic recursive systems and Hamilton–Jacobi–Bellman inequality

In this paper, we study the optimal singular controls for stochastic recursive systems, in which the control has two components: the regular control, and the singular control. Under certain assumptions, we establish the dynamic programming principle for this kind of optimal singular controls problem, and prove that the value function is a unique viscosity solution of the corresponding Hamilton–Jacobi–Bellman inequality, in a given class of bounded and continuous functions. At last, an example is given for illustration.     

Location of paths on trees with minimal eccentricity and superior section

Network location theory has traditionally been concerned with the optimal location of a single-point facility at either a vertex or along an arc in the network. Recently, some authors have departed from this traditional problem and have considered the location of extensive facilities, such as paths, trees or cycles. In this paper, we consider the optimal location of paths on trees with regard to two objective functions: the eccentricity and the superior section. We first present methods to find paths with minimal eccentricity and minimal superior section on trees with arbitrary positive lengths. Then, we analyse the biobjective optimization problem and propose an algorithm, based on a progressive reduction of the initial tree, to obtain all efficient paths. Modifications of the proposed algorithm to solve the problem when a general objective function is used instead of the eccentricity function are also given. This work has been supported by Fundación Séneca under grant PB/11/FS/97     

Optimality necessary conditions in singular stochastic control problems with nonsmooth data

The present paper studies the stochastic maximum principle in singular optimal control, where the state is governed by a stochastic differential equation with nonsmooth coefficients, allowing both classical control and singular control. The proof of the main result is based on the approximation of the initial problem, by a sequence of control problems with smooth coefficients. We, then apply Ekeland's variational principle for this approximating sequence of control problems, in order to establish necessary conditions satisfied by a sequence of near optimal controls. Finally, we prove the convergence of the scheme, using Krylov's inequality in the nondegenerate case and the Bouleau-Hirsch flow property in the degenerate one. The adjoint process obtained is given by means of distributional derivatives of the coefficients.     

Second-order nonlinear impulsive evolution equations with time-varying generating operators and optimal controls

In this article, the second-order nonlinear impulsive evolution differential equations with time-varying generating operators is considered. Constructing evolution systems generated by time-varying operator matrix, we introduce suitable mild solution of the second-order nonlinear impulsive evolution differential equations. The existence and uniqueness of the mild solutions and the continuous dependence on initial value are proved. The existence of the optimal controls for a Lagrange problem of the systems governed by the second-order nonlinear impulsive evolution equations is also presented. An example is given for demonstration.     

A Twofold Spline Chebyshev Linearization Approach for a Class of Singular Second-Order Nonlinear Differential Equations

The aim of this paper is to present a twofold approach for the numerical solution of a class of singular second-order nonlinear differential equations. The first is based on a modified version of an adaptive spline collocation method (ASCM). The second is a patching approach (PASCM) that splits the problem domain into two subintervals: Chebyshev economization procedure is implemented in the vicinity of the singular point and outside this domain the resulting initial or boundary value problem is handled by the (ASCM). The second strategy is based on the linearization of the nonlinear term about the given initial condition at the singular point. The choice of either technique relies on the specified boundary or initial conditions. Performance of the approach is investigated numerically through a number of application examples that demonstrate the efficiency of the approach and that it has O(h ⁴) rate of convergence. Results confirm that the scheme yields highly accurate results when compared with the exact and/or numerical solutions that exist in the literature.     

Existence of solutions for impulsive Dirichlet problems with the parameter inequality reverse

In this paper, we consider the existence and multiplicity for second‐order nonlinear impulsive differential equations with Dirichlet boundary condition and a parameter. By using critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution or infinitely many solutions, assuming that the impulsive functions satisfy the superlinear growth condition and the parameter inequality is reverse. Our results extend and improve some recent results. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on the approximation of Dirichlet boundary control problems for the wave equation on curved domains

We consider an exact boundary control problem for the wave equation with given initial and terminal data and Dirichlet boundary control. The aim is to steer the state of the system that is defined on a given domain to a position of rest in finite time. The optimal control that is obtained as the solution of the problem depends on the data that define the problem, in particular on the domain. Often for the numerical solution of the control problem, this given domain is replaced by a polygon. This is the motivation to study the convergence of the optimal controls for the polygon to the optimal controls for the given domain. To study the convergence, the values of the optimal controls that are defined on the boundaries of the approximating polygons are mapped in the normal directions of the polygon to control functions defined on the boundary of the original domain. This map has already been used by Bramble and King, Deckelnick, Guenther and Hinze and by Casas and Sokolowski. Using this map, we can show the strong convergence of the transformed controls as the polygons approach the given domain. An essential tool to obtain the convergence is a regularization term in the objective functions to increase the regularity of the state.     

The structure of optimal solutions for harvesting a renewable resource

We consider the problem of optimal harvesting of a renewable resource whose dynamics are governed by logistic growth and whose payoff is proportional to the harvest. We consider both the case of a finite and an infinite time horizon and analyse the structure of the optimal solutions and their dependence on the parameters of the model. We show that the optimal policy can only have one of three structures: (1) maximal harvesting effort until the resource is depleted, (2) zero harvesting during an initial time interval followed by a subsequent switch to maximal harvesting effort, or (3) a singular solution, which corresponds to an intermediate level of harvesting, accompanied by the most rapid approach path. All three scenarios emerge, with minor variations, with finite and infinite time horizons, depending on the particular combination of parameters of the system. We characterize the conditions under which the singular solution is optimal and present suggestions for designing an optimal and sustainable harvesting strategy. Recommendations for Resource Managers :

• We have rigorously explored a standard optimal harvesting model and its steady states.
• We show that three different types of solutions may emerge: (i) maximal harvesting eventually leading to a complete depletion of the stock; (ii) maximal harvesting with a potential period of idleness leading to a positive stock; (iii) an initial phase of either no or full harvesting followed by a period of intermediate harvesting intensity leading to a positive stock (singular solution).
• With some modifications, similar results hold for a finite planning horizon.
• Which of these three scenarios emerges in the finite horizon case depends not only on the parameter values but also on the length of the planning horizon.

     

Evolution of the Scattering Coefficients of the Camassa–Holm Equation, for General Initial Data

We consider the Camassa–Holm equation for general initial data, particularly when the potential in the scattering problem of the Lax pair, m +κ, becomes negative over a finite region. We show that the direct scattering problem of the eigenvalue problem of the Lax pair for this equation may be solved by dividing the spatial infinite interval into a union of separate intervals. Inside each of these intervals, the initial potential is uniformly either positive or negative. Due to this, one can define Jost functions inside each interval, each of which will have a uniform asymptotic form. We then demonstrate that one can obtain the t -evolution of the scattering coefficients of the scattering matrix of each interval. In the process, we also demonstrate that the evolution of the zeros of m +κ can be given entirely in terms of limits of the scattering coefficients at singular points.     

Direct methods with maximal lower bound for mixed-integer optimal control problems

Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed-integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed-integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed-integer optimal control problems, with a focus on discrete-valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.      

Optimizing venture capital investments in a jump diffusion model

We study two practical optimization problems in relation to venture capital investments and/or Research and Development (R&D) investments. In the first problem, given the amount of the initial investment and the cash flow structure at the initial public offering (IPO), the venture capitalist wants to maximize overall discounted cash flows after subtracting subsequent investments, which keep the invested company solvent. We describe this problem as a mixture of singular stochastic control and optimal stopping problems. The second problem is concerned with optimal dividend policy. Rather than selling the company at an IPO, the investor may want to harvest technological achievements in the form of dividend when it is appropriate. The optimal control policy in this problem is a mixture of singular and impulse controls. E. Bayraktar was supported in part by the National Science Foundation, under grant DMS-0604491.     

On the solution of boundary value problems for the laplace equation on the plane with a three-layer film inclusion

We derive analytic formulas that, on the basis of given harmonic functions, permit one to construct solutions of boundary value problems on a plane containing an infinitely thin three-layer film by simple quadratures with the preservation of singular points of the harmonic functions. The three-layer film consists of alternating strongly and weakly permeable layers and has the form of a ray, a segment, or a circle arc.     

Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations

In this paper, we study optimal relaxed controls and relaxation of nonlinear fractional impulsive evolution equations. Firstly, existence of piecewise continuous mild solutions for the original fractional impulsive control system is presented. Secondly, fractional impulsive relaxed control system is constructed by using a regular countably additive measure and making the original control system convexified. Thirdly, optimal relaxed controls and relaxation theorems are obtained. Finally, application to initial-boundary value problem of fractional impulsive parabolic control system is considered.     

Comparison results for initial value problems of second-order impulsive integro-differential inclusions

In this paper, we present new maximum and minimum principles for the initial value problem for a class of second-order nonlinear impulsive integro-differential inclusions with discontinuous multivalued functions on the right hand side in a real Banach space. This allows us to lead on to the development of the comparison principle and monotone iterative technique for a nonlinear second-order impulsive integro-differential inclusion.     

一类具漂移的“跳-停”奇异型随机控制

以随机分析和最优控制理论为基础,讨论了一类带停时的奇异型随机控制问题.在原模型状态过程的基础上添加了漂移因子,并将原模型中的控制费用函数推广为一般的费用函数.在某些条件下,得到"跳一停"策略是其最优控制策略,并给出了"跳一停"策略存在的条件以及控制方法,所得的结论在实际中有较深的应用背景.     

Optimal control problem described by impulsive differential equations with nonlocal boundary conditions

We study an optimal control problem in which the plant state is described by impulsive differential equations with nonlocal boundary conditions. By using the contraction mapping principle, we prove the existence and uniqueness of a solution of the nonlocal impulsive boundary value problem for given feasible controls. We compute the first and second variations of the performance functional and use them to obtain various necessary second-order optimality conditions.     

Structural Stability Investigation of Bang-Singular-Bang Optimal Controls

The paper is devoted to parametric optimal control problems with a scalar, partially singular optimal control function. In contrast to the case of pure bang-bang behavior, the investigation of structural stability properties for partially singular controls so far has been rarely addressed in literature. The central result of the paper deals with the case of one first order singular arc under regular concatenation to bang-arcs. Conditions will be provided which ensure the Lipschitz stability of bang-singular junction times positions with respect to small parameter changes. Three examples illustrate the main theorem.     

Farkas-type results for fractional programming problems

Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results.