Necessary first-order conditions for optimal crossing of a given region

We consider a nonlinear optimal control problem with an integral functional in which the integrand is the characteristic function of a closed set in the phase space. An approximation method is applied to prove the necessary conditions of optimality in the form of a Pontryagin maximum principle without any prior assumptions on the behavior of the optimal trajectory. Similarly to phase-constrained problems, we derive conditions of nondegeneracy and pointwise nontriviality of the maximum principle. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 179–204, 2004.     

Maximum principle for the optimal control of a hyperbolic equation in one space dimension,part 1: Theory

A maximum principle is developed for a class of problems involving the optimal control of a damped-parameter system governed by a linear hyperbolic equation in one space dimension that is not necessarily separable. A convex index of performance is formulated, which consists of functionals of the state variable, its first- and second-order space derivatives, its first-order time derivative, and a penalty functional involving the open-loop control force. The solution of the optimal control problem is shown to be unique. The adjoint operator is determined, and a maximum principle relating the control function to the adjoint variable is stated. The proof of the maximum principle is given with the help of convexity arguments. The maximum principle can be used to compute the optimal control function and is particularly suitable for problems involving the active control of structural elements for vibration suppression.     

Smooth and Nonsmooth Lipschitz Controls for a Class of Vector Differential Equations

This paper is devoted to the study of a class of control problems associated to a nonlinear second-order vector differential equation with pointwise state constraints. The control is realized via a function of the state. We extend the results of Akkouchi, Bounabat, and Goebel to vector differential equations and furthermore consider the more general case. Under proper conditions, we prove the existence of optimal controls in the class of Lipschitz functions and obtain an optimality condition which looks somehow like the Pontryagin maximum principle for a smooth optimal control function. For a nonsmooth optimal control function, we derive a suboptimality condition by means of the Ekeland variational principle.Communicated by M. J. BalasThis work was supported by 985 Project of Jilin University. The author thanks Professor Yong Li for valuable suggestions. He also thanks Professor M. J. Balas and the anonymous referees for their comments.     

Necessary optimality conditions for a class of optimal control problems with discontinuous integrand

We consider a nonlinear optimal control problem with an integral functional in which the integrand contains the characteristic function of a given closed subset of the phase space. Using an approximation method, we prove necessary optimality conditions in the form of the Pontryagin maximum principle without any a priori assumptions about the behavior of an optimal trajectory.     

Properties of extremals in optimal control problems with state constraints

An optimal control problem with state constraints is considered. Some properties of extremals to the Pontryagin maximum principle are studied. It is shown that, from the conditions of the maximum principle, it follows that the extended Hamiltonian is a Lipschitz function along the extremal and its total time derivative coincides with its partial derivative with respect to time.     

Properties of the Lagrange multipliers in the Pontryagin maximum principle for optimal control problems with state constraints

We study the Pontryagin maximum principle for an optimal control problem with state constraints. We analyze the continuity of a vector function µ (which is one of the Lagrange multipliers corresponding to an extremal by virtue of the maximum principle) at the points where the extremal trajectory meets the boundary of the set given by the state constraints. We obtain sufficient conditions for the continuity of µ in terms of the smoothness of the extremal trajectory.     

Resource allocation problem in a two-sector economic model of special form

In the present paper, we study the resource allocation problem for a two-sector economic model of special form, which is of interest in applications. The optimization problem is considered on a given finite time interval. We show that, under certain conditions on the model parameters, the optimal solution contains a singular mode. We construct optimal solutions in closed form. The theoretical basis for the obtained results is provided by necessary optimality conditions (the Pontryagin maximum principle) and sufficient optimality conditions in terms of constructions of the Pontryagin maximum principle.     

Maximum principle for optimal control of distributed-parameter systems with singular mass matrix

A maximum principle for the open-loop optimal control of a vibrating system relative to a given convex index of performance is investigated. Though maximum principles have been studied by many people (see, e.g., Refs. 1–5), the principle derived in this paper is of particular use for control problems involving mechanical structures. The state variable satisfies general initial conditions as well as a self-adjoint system of partial differential equations together with a homogeneous system of boundary conditions. The mass matrix is diagonal, constant, and singular, and the viscous damping matrix is diagonal. The maximum principle relates the optimal control with the solution of the homogeneous adjoint equation in which terminal conditions are prescribed in terms of the terminal values of the optimal state variable. An application of this theory to a structural vibrating system is given in a companion paper (Ref. 6).     

Sufficient Optimality Conditions for Optimal Control Problems with State Constraints

It is well-known in optimal control theory that the maximum principle, in general, furnishes only necessary optimality conditions for an admissible process to be an optimal one. It is also well-known that if a process satisfies the maximum principle in a problem with convex data, the maximum principle turns to be likewise a sufficient condition. Here an invexity type condition for state constrained optimal control problems is defined and shown to be a sufficient optimality condition. Further, it is demonstrated that all optimal control problems where all extremal processes are optimal necessarily obey this invexity condition. Thus optimal control problems which satisfy such a condition constitute the most general class of problems where the maximum principle becomes automatically a set of sufficient optimality conditions.     

Optimal processes in the model of two-sector economy with an integral utility function

An infinite-horizon two-sector economy model with a Cobb–Douglas production function is studied for different depreciation rates, the utility function being an integral functional with discounting and a logarithmic integrand. The application of the Pontryagin maximum principle leads to a boundary value problem with special conditions at infinity. The presence of singular modes in the optimal solution complicates the search for a solution to the boundary value problem of the maximum principle. To construct the solution to the boundary value problem, the singular modes are written in an analytical form; in addition, a special version of the sweep algorithm in continuous form is proposed. The optimality of the extremal solution is proved.     

向量值测度与线性系统最优控制问题的必要条件

The vector-valued measure defined by the well-posed linear boundary value problems is discussed. The maximum principle of the optimal control problem with non-convex constraintis proved by using the vector-valued measure. Especially, the necessary conditions of the optimal control of elliptic systems is derived without the convexity of the control domain and the cost function. optimal control, maximum principle, distributed parameter system, linear system,vector-valued measure.     

Maximum principle for the optimal control of a hyperbolic equation in one space dimension,part 2: Application

The optimal open-loop control of a beam subject to initial disturbances is studied by means of a maximum principle developed for hyperbolic partial differential equations in one space dimension. The cost functional representing the dynamic response of the beam is taken as quadratic in the displacement and its space and time derivatives. The objective of the control is to minimize a performance index consisting of the cost functional and a penalty term involving the control function. Application of the maximum principle leads to boundary-value problems for hyperbolic partial differential equations subject to initial and terminal conditions. The explicit solution of this system is obtained yielding the expressions for the state and optimal control functions. The behavior of the controlled and uncontrolled beam is studied numerically, and the effectiveness of the proposed control is illustrated.     

抑制剂作用下肿瘤生长模型的参数识别

该文考虑抑制剂作用下肿瘤生长的模型. 假设肿瘤是球对称的, 其表面为运动边界, 用函数r=R(t)表示. 既然多细胞肿瘤扁球体(MTS)通常作为肿瘤生长的体外模型, 在实验室能够被观察和控制, 因此研究如下反问题: 根据观察到的MTS动态增长(即给定R(t)), 来确定抑制剂的参数. 运用极大值原理, 作者证明了该抛物反问题解的唯一性. 进一步, 用最优控制框架来重构模型中的抑制剂参数, 证明了最优控制问题解的存在性, 并推导了最优控制满足的最优性必要条件.     

An optimal control model and algorithm for the deviated well’s trajectory planning

This paper presents a nonlinear piecewise smooth dynamical system of the trajectory of deviated wells according to engineering background. An optimal control model is established and the necessary conditions for optimality are proved via maximum principle. The optimal control problem is solved by a revised Hooke–Jeeves algorithm. The uniform design technique has been incorporated into the revised Hooke–Jeeves algorithm to handle the multimodal function. Computer simulation is used for this paper, and the numerical example illustrates the validity and efficiency of the algorithm. The procedure demonstrates its advantages in practical applications in Liaohe Oil Field.     

Normality of the maximum principle for nonconvex constrained Bolza problems

We consider a Bolza optimal control problem with state constraints. It is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle. In general, the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls. In the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate, cf. [A.V. Arutyanov, S.M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim. 35 (1997) 930–952; F. Rampazzo, R.B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control, IMA 16 (4) (1999) 335–351; F. Rampazzo, R.B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim. 39 (4) (2000) 989–1007]. Our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle. In particular we allow the initial condition to be fixed and the state constraints to be nonsmooth. To prove normality we use J. Yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined, in turn, by linearization of the original set of constraints along an extremal trajectory.     

A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems

In this paper, we discuss a new general formulation of fractional optimal control problems whose performance index is in the fractional integral form and the dynamics are given by a set of fractional differential equations in the Caputo sense. The approach we use to prove necessary conditions of optimality in the form of Pontryagin maximum principle for fractional nonlinear optimal control problems is new in this context. Moreover, a new method based on a generalization of the Mittag–Leffler function is used to solving this class of fractional optimal control problems. A simple example is provided to illustrate the effectiveness of our main result. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal boundary control of dynamics responses of piezo actuating micro-beams

Optimal control theory is formulated and applied to damp out the vibrations of micro-beams where the control action is implemented using piezoceramic actuators. The use of piezoceramic actuators such as PZT in vibration control is preferable because of their large bandwidth, their mechanical simplicity and their mechanical power to produce controlling forces. The objective function is specified as a weighted quadratic functional of the dynamic responses of the micro-beam which is to be minimized at a specified terminal time using continuous piezoelectric actuators. The expenditure of the control forces is included in the objective function as a penalty term. The optimal control law for the micro-beam is derived using a maximum principle developed by Sloss et al. [J.M. Sloss, J.C. Bruch Jr., I.S. Sadek, S. Adali, Maximum principle for optimal boundary control of vibrating structures with applications to beams, Dynamics and Control: An International Journal 8 (1998) 355–375; J.M. Sloss, I.S. Sadek, J.C. Bruch Jr., S. Adali, Optimal control of structural dynamic systems in one space dimension using a maximum principle, Journal of Vibration and Control 11 (2005) 245–261] for one-dimensional structures where the control functions appear in the boundary conditions in the form of moments. The derived maximum principle involves a Hamiltonian expressed in terms of an adjoint variable as well as admissible control functions. The state and adjoint variables are linked by terminal conditions leading to a boundary-initial-terminal value problem. The explicit solution of the problem is developed for the micro-beam using eigenfunction expansions of the state and adjoint variables. The numerical results are given to assess the effectiveness and the capabilities of piezo actuation by means of moments to damp out the vibration of the micro-beam with a minimum level of voltage applied on the piezo actuators.     

Hölder Continuity of Adjoint States and Optimal Controls for State Constrained Problems

We investigate Hölder regularity of adjoint states and optimal controls for a Bolza problem under state constraints. We start by considering any optimal solution satisfying the constrained maximum principle in its normal form and we show that whenever the associated Hamiltonian function is smooth enough and has some monotonicity properties in the directions normal to the constraints, then both the adjoint state and optimal trajectory enjoy Hölder type regularity. More precisely, we prove that if the state constraints are smooth, then the adjoint state and the derivative of the optimal trajectory are Hölder continuous, while they have the two sided lower Hölder continuity property for less regular constraints. Finally, we provide sufficient conditions for Hölder type regularity of optimal controls.     

Analysis of a gas field development model with an infinite planning horizon

We consider a nonlinear optimal control problem with an infinite planning horizon, which extends a dynamic gas field development model by taking into account a gas price forecast. (The prices varies in time.) The solution is constructed on the basis of the Pontryagin maximum principle. To prove the optimality of the extremal solution, we use the theorem on sufficient optimality conditions in terms of constructions of the Pontryaginmaximum principle. We discuss the problem of constructing an optimal solution by dynamic programming.