Turnpike solutions of optimal control problems

Minimizing sequences for degenerate optimal control problems are constructed from turnpike solutions. Two variational approximation schemes for the turnpike solution are described: the first is a direct improvement of the simple approximation of piecewise-continuous turnpikes by the solutions of the initial differential system and the second consists of constructing an approximate optimal control in the neighborhood of a turnpike by a parametric curve in the state space. Since a sequence of state-linear feedback controls with variable coefficients is generated in both variants, turnpikes can be easily realized in practice.     

Numerical solutions for constrained time-delayed optimal control problems

In this paper, time-delayed optimal control problems governed by delayed differential equation are solved. Two different techniques based on integration and differentiation matrices are considered. The time-delayed term of the problem has been approximated by Chebyshev interpolating polynomials. On this basis, the optimal control problem can be solved as a mathematical programming problem. The example illustrates the robustness, accuracy and efficiency of the proposed numerical techniques.     

Singular solutions in problems of optimal control

A particular class of optimization problems, in which the system equations and index of performance are linear in the control variable, is examined in detail. Pontryagin's Maximum Principle seems to indicate an optimal control of the bang-bang type for this class of problems. However, it is shown that the optimal control may actually consist of intervals of variable control effort (called "singular control") combined with intervals of bang-bang control. The conditions which characterize singular control are derived. Some techniques are given which may be helpful in detecting and calculating singular controls in this class of problems. In general, it cannot be stated a priori that singular control will necessarily constitute part of the optimal control. Two examples are worked out in detail to illustrate application of the techniques given in the paper.     

Invertibility of quantum-mechanical control systems

This is the first of two papers concerned with the formulation of a continuous-time quantum-mechanical filter. Efforts focus on a quantum system with Hamiltonian of the formH ₀+u(t)H ₁, whereH ₀ is the Hamiltonian of the undisturbed system,H ₁ is a system observable which couples to an external classical field, andu(t) represents the time-varying signal impressed by this field. An important problem is to determine when and how the signalu(t) can be extracted from the time-development of the measured value of a suitable system observableC (invertibility problem). There exist certain quasiclassical observables such that the expected value and the measured value can be made to coincide. These are called quantum nondemolition observables. The invertibility problem is posed and solved for such observables. Since the physical quantum-mechanical system must be modelled as aninfinite-dimensional bilinear system, the domain issue for the operatorsH ₀,H ₁, andC becomes nontrivial. This technical matter is dealt with by invoking the concept of an analytic domain. An additional complication is that the output observableC is in general time-dependent.Research supported in part by the National Science Foundation under Grant Nos. ECS-8017184, INT-7902976 and DMR 8008229 and by the Department of Energy under Contract No. DE-AC01-79ET-29367.     

最优控制问题弱间断解的一个自适应算法

针对最优解有弱间断的最优控制问题提出一个自适应算法.时间区间被划分为若干子区间,使用分段多项式逼近最优控制问题的解,在每个子区间内,最优控制问题被拟谱方法离散,使用的配置点是Chebyshev-Gauss-Lobatto点.根据计算出的数值解提供的后验信息,该自适应算法既能剖分产生新的子区间,又能在子区间内增加逼近多项式的次数.最后通过若干例子表明了所提出算法的高精度和有效性.     

Stochastic maximum principle for optimal control problems involving delayed systems

Dear editor, The main objective of this study is to investigate one type of stochastic optimal control problem for a delayed system using the maximum principle ...     

Trigonometric approximation of optimal periodic control problems for distributed-parameter systems

The optimal periodic control problem for a system described by first order partial differential equations is approximated by a sequence of discretized optimization problems. Trigonometric polynomials in two variables are used in the latter problems to approximate the state trajectory, the control and functions appearing in differential equations and in the criterion of the basic problem. The state equations and the instantaneous constraints on the state and the control are taken into account by the mixed exterior-interior penalty function. Sufficient conditions are given for the convergence of solutions of discretized problems to the optimal solution of the basic problem. The possibility of applying the method to a class of optimal periodic control problems in chemical engineering is emphasized.     

Analytic solutions to optimal control problems in crystal growth processes

A unified approach to optimal control of univariate and multivariate crystallization particulate processes with size-independent or/and size-dependent growth rate kinetics is developed by utilizing the minimum principle and the method of characteristics in conjunction with novel approximate integro-differential and ordinary differential equation computational schemes. The proposed theoretical approach leads to simple analytic solutions involving numerical boundary value problems with few unknown parameters.     

On one class of optimal control problems for quantum systems

We present a new modification of the global control improvement method based on a known Krotov’s method for optimal control in quantum systems from a certain class. The algorithm is implemented for high-dimensional systems as a parallel program. We give computations for the control in a quantum dynamical system that represents a well-known model of communicating the quantum state in spin chains.     

Viscosity solutions and optimal control problems with integral constraints

We discuss optimal control problems with integral state-control constraints. We rewrite the problem in an equivalent form as an optimal control problem with state constraints for an extended system, and prove that the value function, although possibly discontinuous, is the unique viscosity solution of the constrained boundary value problem for the corresponding Hamilton–Jacobi equation. The state constraint is the epigraph of the minimal solution of a second Hamilton–Jacobi equation. Our framework applies, for instance, to systems with design uncertainties.     

Turnpike properties of a class of aquifer control problems

In this paper we are concerned with the long-run behavior of solutions to infinite horizon optimal control systems, where the state equation is given by a non-linear parabolic equation. The goal is to establish convergence to steady state of solutions of the optimal control problems under consideration. Our motivation to consider this problem is the study of agricultural economics models concerned with the optimal management of groundwater. The main object of the study towards achieving this goal is the continuity of solutions of the state equation with respect to the initial condition and the non-homogeneous term. We establish this continuity by employing well-known norm estimates for linear parabolic equations.     

Finite-dimensional regularizers for optimal control problems on solutions of ill-posed variational inequalities

A method is proposed for construction of finite-dimensional regularizers for optimal control on solutions of ill-posed variational inequalities with pseudomonotone operators. It is assumed that all the input data of the problems, including the control sets, are known approximately.Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 84–88, July–August, 1991.     

Positional solutions of Hamilton-Jacobi equations in control problems for discrete-continuous systems

We develop a canonical global optimality theory based on operating with the set of solutions for the Hamilton-Jacobi inequalities that parametrically depend on the initial (or final) position. These solutions, called positional L-functions (of Lyapunov type), naturally arise in the studies of control problems for discrete-continuous (hybrid, impulse) systems; an important prototype of such problems are classical optimal control problems with general end constraints on the trajectory. We analyze sufficient optimality conditions with this new class of L-functions and invert the maximum principle into a sufficient condition for nonlinear problems of optimal impulse control.     

Degenerate problems of optimal control for discrete-continuous (hybrid) systems

The notion of the degenerate problem of optimal control for the discrete-continuous systems was formulated. The main approaches to the problems of this class that were developed for the uniform continuous and discrete systems such as the transformations to the derivative systems and the method of multiple maxima, a special technique to define the Krotov functions under the like sufficient conditions, were extended to the discrete-continuous systems. The fields of possible efficient applications were indicated, and an example was presented.     

Iterative methods for solution of problems of optimal control of logic-dynamic systems

Optimal control problem for logic-dynamic systems is considered. Based on sufficient optimality conditions, similar to Krotov’s conditions, a method for successive control improvement is developed. First- and second-order computational algorithms are constructed.     

Trigonometric approximation of optimal periodic control problems for systems with cascade structure

A cascade-structured system controlled by the input of the first subsystem is considered. Trigonometric polynomials are used to approximate the state trajectory, the control and the functions in differential equations and in the criterion for the constrained optimal periodic control of the above system. The formulation of approximating problems is adapted to the system structure to obtain particularly good approximating properties. Sufficient conditions for the convergence of solutions of discretized problems to the optimal solution of the basic problem are given, and the convergence rate is estimated. An application of the proposed approximation to multi-stage chemical processes is discussed.     

Time reparameterization in problems of optimal control of impulsive hybrid systems

We consider an optimal control problem for a hybrid dynamic system, where jumps of a trajectory may occur only at the moments of hitting a given closed subset of an extended phase space. A time reparameterization technique is developed to reduce the original problem to the one with bounded controls. We show that the reparameterized problem is equivalent to optimization in a class of generalized solutions to the hybrid system.     

Some nonlinear optimal control problems with closed‐form solutions

Optimal controllers guarantee many desirable properties including stability and robustness of the closed‐loop system. Unfortunately, the design of optimal controllers is generally very difficult because it requires solving an associated Hamilton–Jacobi–Bellman equation. In this paper we develop a new approach that allows the formulation of some nonlinear optimal control problems whose solution can be stated explicitly as a state‐feedback controller. The approach is based on using Young's inequality to derive explicit conditions by which the solution of the associated Hamilton–Jacobi–Bellman equation is simplified. This allows us to formulate large families of nonlinear optimal control problems with closed‐form solutions. We demonstrate this by developing optimal controllers for a Lotka–Volterra system. Copyright © 2001 John Wiley & Sons, Ltd.     

Realization of sliding modes as generalized solutions to optimal control problems

Consideration is given to the problem of realization of generalized solutions of different nature in classes of admissible models traditional for applications for optimal control problems. The main attention is focused on sliding modes in systems with a bounded set of velocities (velocity hodograph). Constructive correction schemes for elements of the sequence converging to the sliding mode are specified so that they are strictly admissible. This defines the procedure for constructing a minimizing sequence the elements of which are approximate optimal practically realizable solutions to estimates of accuracy.     

Discontinuous solutions of the optimal control problems. Iterative optimization method

Consideration was given to the class of optimal control problems for the differential systems with unbounded linear control and, in particular, the classes of problems for the bilinear systems. Such problems are distinguished for the lack of minimum (maximum) on the ordinary class of permissible processes (continuous trajectories, sectionally continuous controls) and for reaching it on some closure of this class including processes with discontinuous trajectories. A direct iterative method for optimization of controls was developed.