A computational method for a class of non-standard time optimal control problems involving multiple time horizons

In this paper, we consider the optimal control problem of discrete‐time switched systems. This problem is formulated as an optimization problem involving both continuous and discrete‐valued variables. It can be transformed into a discrete optimization problem. A metric in the space of switching sequences is introduced and an appropriate discrete filled function is constructed. Then, an algorithm that combines the discrete filled function method and a descent method is developed for solving this problem. For illustration, some numerical examples are solved. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper, an optimization problem is performed to obtain an approximate solution for a class of fractional optimal control problems (FOCPs) with the initial and final conditions. The main characteristic of our approximation is to reduce the FOCP into a system of Volterra integral equations. Then by solving this new problem, based on minimization and control the total error, we transform the original FOCP into a discrete optimization problem. By obtaining the optimal solutions of this problem, we obtain the numerical solution of the original problem. This procedure not only simplifies the problem but also speeds up the computations. The numerical solutions obtained from the proposed approximation indicate that this approach is easy to implement and accurate when applied to FOCPs.

A new discrete filled function method for finding global minimizer of the integer programming

The author summarizes the theoretical results in "A discrete optimal control problem for descriptor systems," [IEEE Trans. Automat. Control, vol.34, pp.177-181, 1989] and "Authors' reply to 'Comments on 'A discrete optimal control problem for descriptor systems"," [ibid., to appear], and uses them in the computer-aided optimization problem for descriptor systems. An efficient algorithm for computing the optimal solution sequence of the optimization problem is developed and an example of the macroeconomic systems is shown.

Time-discrete systems with a finite set of states are considered. Discrete optimal control problems with infinite time horizon for such systems are formulated. We introduce a certain graph-theoretic structure to model the transitions of the dynamical system. Algorithms for finding the optimal stationary control parameters are presented. Furthermore, we determine the optimal mean cost cycles. This approach can be used as a decision support strategy within such a class of problems; especially so-called multilayered decision problems which occur within environmental emission trading procedures can be modelled by such an approach.

In this paper we define ``embedded optimal control problems" which prescribe parametrized families of well defined associated optimal control problems. We show that the extremal generating Hamiltonian equations for an embedded optimal control problem and any associated optimal control problem are simply related by a projection. Furthermore normal extremals project to normal extremals and similarly for abnormal extremals. An interesting class of embedded optimal control problems consists of Clebsch optimal control problems. We provide necessary conditions for a Clebsch optimal control problem to describe a variational problem and thereby a mechanical system. There may be many advantages to analyzing an embedded optimal control problem instead of a particular associated optimal control problem, for example the former being defined on a linear space and the latter on a nonlinear space. The continuous analysis is paralleled by a similar discrete analysis. We define a discrete embedded/Clebsch optimal control problem along with associated discrete optimal control problems and we show results that are analogous to the continuous results. We apply the theory, both in the continuous and the discrete setting, to two example systems: mechanical systems on matrix Lie groups and mechanical systems on $n$-spheres.

This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear dynamic systems on time scales. Introducing the reasonable weak solution of nonlinear dynamic systems, the existence of the weak solution for the nonlinear dynamic systems on time scales and its properties are presented. Discussing L1-strong-weak lower semicontinuity of integral functional, we give sufficient conditions for the existence of optimal controls. Using integration by parts formula and Hamiltonian function on time scales, the necessary conditions of optimality are derived respectively. Some examples on continuous optimal control problems, discrete optimal control problems, mathematical programming and variational problems are also presented for demonstration.

This paper considers discrete global optimization problems. A new transformation function for finding global minimizer on discrete global optimization problems is given in this paper. We prove that under some general assumptions the new transformation function possesses the properties of both the tunnelling functions and the filled functions. New transformation function contains only one parameter and can be easily adjusted in the realization of the proposed algorithm. Moreover, the paper intends to obtain promising computational results comparing with filled function method, and shows the efficiency of the transformation function method.

In the design of digital control systems, the time delay due to computation time of a processor cannot be often neglected. In this paper, we consider the problem of synthesizing a type-1 optimal digital servosystem taking account of controller delay of one sampling period.It is first shown that the optimal digital servo problem in which a quadratic criterion function is defined for a given continuous-time plant can be reasonably reformulated to a discrete optimal regulator problem. To assure the solvability of this problem, some conditions for the preservation of stabilizability and detectability of the plant under sampling are given. In the discrete optimal regulator problem considered here, not only control input but also the initial state of the servocompensator are referred as free design parameters. The obtained optimal digital control law contains a feedforward term of reference signal in addition to a feedback term. It is also shown that the optimal control law has a function of estimating the state variables at the following sampling period in order to recover the controller delay.

Hierarchical two-level methods for solving state space constrained optimization problems are studied. The approach is a generalization of the methods suggested by the author and L. Hakkala in earlier papers. In this paper the original state space constrained problem is modified so that three control constrained problems result. The relationship between the original problem and the modified problems is studied by a dual approach which also gives a collection of algorithms for solving the original problem. The proposed approach is applied to discrete time optimal control problems and two numerical examples are given

Inverse identification of inputs of a random utility based model using optimal control

Three layer control structure is proposed for optimal control of continuous fermentation processes. The start-up optimization problems are solved as a first step for optimization layer building. A steady state optimization problem is solved by a decomposition method using prediction principle. A discrete minimum time optimal control problem with state delay is formulated and a decomposition method, based on an augmented Lagrange's function is proposed to solve it. The problem is decomposed in time domain by a new coordinating vector. The obtained algorithms are used for minimum time optimal control calculation of Baker's Yeast fermentation process.

A Survey of Lagrangean Techniques for Discrete Optimization

Nonlinear discrete time and continuous time optimal control problems with terminal constraints are solved using a new evolutionary approach which seeks the control history directly by evolutionary computation. Unlike methods that use the first order necessary conditions to determine the optimum, the main advantage of the present method is that it does not require the development of a Hamiltonian formulation and consequently, it eliminates the requirement to solve the adjoint problem which usually leads to a difficult two-point boundary value problem. The method is verified on two benchmark problems. The first problem is the discrete time velocity direction programming problem with the effects of gravity, thrust and drag and a terminal constraint on the final vertical position. The second problem is a continuous time optimal control problem in rocket dynamics, the Goddard’s problem. The solutions of both problems compared favorably with published results based on gradient methods.

In this paper we consider autonomous discrete time optimal control problems. We discuss the reduction to finite cost and the representation formula, the existence of optimal solutions on infinite horizon and their structure, and the structure of optimal solutions on finite intervals.