Nonlinear optimal control problems with continuous state inequality constraints

In this paper, we consider a class of optimal switching control problems with multiple time-delays and a cost on changing control and subject to terminal state constraints. A computational method involving three stages is developed to solve this class of optimal control problems. First, by parameterizing the control function with piecewise-constant functions, the optimal switching control problem is approximated by a sequence of finite-dimensional optimization problems, where the original switching times, the control heights and the control switching times are decision variables. Second, by introducing new variables, the total variation of the control variables is transformed into an equivalently smooth function. Third, we convert the constrained optimization problem into one only with box constraints by an exact penalty function method. The gradients of the cost functional are then derived, which can be combined with any gradient-based optimization method to determine the optimal solution. Finally, a numerical example is given to illustrate the effectiveness of the proposed algorithm.

This paper introduces a numerical technique for solving a class of optimal control problems containing nonlinear dynamical system and functional of state variables. This numerical method consists of two major parts. In the first part, using linear combination property of intervals, we convert the nonlinear dynamical system into an equivalent linear system. And in the second part, which we are dealing with a linear dynamical system, using Legendre expansions for approximating both the state and associated control together with discretizing the constraints over the Chebyshev-Gauss-Lobatto points, the optimal control problem is transformed into a corresponding NLP problem which is diretly solved. The proposed idea is illustrated by several numerical examples.

In this paper, we consider a class of optimal control problems involving continuous control and state inequality constraints where the control is almost smooth. We first employ the control parametrization technique via approximating the control signal by a piecewise linear function. Then, we develop a time scaling transformation procedure for transforming the approximate problem into an equivalent problem that can be solved readily using conventional methods. On this basis, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. The gradient formulas and convergent properties ensure that the transformed unconstrained optimal parameter selection problems can be solved by existing optimization algorithms or software packages. Finally, an example is solved showing the effectiveness and applicability of the approach proposed.

Most direct methods solve finite time horizon optimal control problems with nonlinear programming solver. In this paper, we propose a numerical method for solving nonlinear optimal control problem with state and control inequality constraints. This method used quasilinearization technique and Haar wavelet operational matrix to convert the nonlinear optimal control problem into a quadratic programming problem. The linear inequality constraints for trajectories variables are converted to quadratic programming constraint by using Haar wavelet collocation method. The proposed method has been applied to solve Optimal Control of Multi-Item Inventory Model. The accuracy of the states, controls and cost can be improved by increasing the Haar wavelet resolution.

In this paper, we consider a class of optimal control problems in which the system is to be determined in an optimal way. This class of problems involves the choice of a fixed number of switching time points which divide the system’s time horizon into a number of time periods. For each of these time periods, a subsystem is selected, from a finite number of given candidate subsystems, to run during that time periods. The choice of the switching points and the selection of the subsystems are carried out in such a way that a given cost functional is minimized. We consider only problems involving ordinary differential equations over a finite time horizon. A computational method is developed for solving these problems. In our method, the candidate subsystems are combined into a single system by considering their ‘linear combination’. By introducing suitable constraints on the coefficients of the linear combination and using a time rescaling technique, the original problem is transformed into an equivalent optimal control problem with system parameters. An algorithm is proposed for solving this transformed problem and the required gradient formulae are derived. To show the effectiveness of the method, a numerical example is solved.

For Meyer type optimal control problems the principle of optimality states that every subarc of an optimal solution that extends to the final time furnishes again a solution to the same optimal control problem, only with different initial conditions. Through a simple counter example it is shown in this paper that this principle is not satisfied, in general, for solutions obtained from a standard collocation approach. Introduction The principle of optimality [1] is one of the most intuitive and yet most fundamental theorems in the theory of optimal control. Its basic statement is that for Meyer type optimal control problems every subarc of an optimal solution that extends to the final time furnishes again a solution to the same optimal control problem, only with different initial conditions. This theorem has significant implications. For example, it is the starting point for the derivation of the Hamilton-Jacobi-Bellman equations [2] and it is the underlying principle in Bellman's dynamic programming approach [1], [3]. The present paper shows that the principle of optimality is not satisfied, in general, for solutions obtained from a standard collocation approach[4]. For a simple example whose collocation solution can be generated analytically it is shown that further improvements are possible along an optimal collocation solution if the design parameters along a number of leading nodes are frozen and the remainder of the trajectory is re-optimized. I. Continuous Optimal Control Problem In this section of the paper we introduce a simple class of optimal control problems without state and control constraints. In Section 1.2 we present the associated necessary conditions for optimality [2], [5]. For ease of notation we assume that the right-hand side of the 'Supervising Engineer, Member AIAA, t Senior Engineer, Member AIAA, *17 Research Drive, Hampton, VA 23666. ^Graduate Student, Member AIAA, Department, 304 Town Engineering, Ames, IA 50011 equations of motion does not depend explicitly on time t. This does not represent a loss of generality, as it is always possible to transform explicit time dependence away through the introduction of an additional, timelike state variable. 1. Problem Formulation Let us consider the following class of optimal control problems: min «e(Pwc[*o,*/]), (1) subject to the conditions ±(t) = /(*(*),«(*)), tf.(z(*oMo) = 0, (2)

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.

In this paper, an iterative computational method is proposed to handle constrained nonlinear quadratic optimal control problems. with terminal state constraints, saturation state and saturation control variables constraints. The method is based on replacing the original constrained nonlinear optimal control problem by a sequence of constrained linear time-varying quadratic optimal control problems. To this end, an iterative technique is used to replace the original constrained nonlinear dynamic system by a sequence of constrained linear time-varying systems. Then each of constrained linear time-varying quadratic optimal control problems is approximated by a quadratic programming problem by parameterizing each of the state variable by a finite length Legendre polynomials with unknown parameters. To show the effectiveness of the proposed method, simulation results of a constrained nonlinear optimal control problem are presented.

The paper is concerned with a nonlinear optimal impulsive control problem with trajectories of bounded variation. Vector Lebesgue-Stieltjes measures play the part of controls. Studied were state and mixed constraints in the conventional and fast times, as well as joint conditions for trajectory and impulsive control. A method of reduction to the classical problem of optimal control was developed relying on the discontinuous time reparameterization. The original and reduced problems were established to be equivalent.

In this paper, we consider a class of optimal control problems with control and terminal inequality constraints, where the system dynamics is governed by a linear second-order parabolic partial differential equation with first boundary condition. A feasible direction algorithm for solving this class of optimal control problems has already been obtained in the literature. The aim of this paper is to improve the convergence result by using a topology arising in the study of relaxed controls.

A class of optimal control problems for semilinear elliptic equations with mixed control-state constraints is considered. The existence of bounded and measurable Lagrange multipliers is proven. As a particular application, the Lavrentiev-type regularization of pointwise state constraints is discussed. Here, the existence of associated regular multipliers is shown, too.

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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. In this paper, we announce new sensitivity relations for state constrained optimal control problems. For the class of optimal control problems considered there is no guarantee that the co-state arc is unique; a key feature of the results is that they assert `some? choice of co-state arc can be made, for which the sensitivity relations are valid. The proof technique is to introduce a new 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 obtained is precisely the desired set of sensitivity relations.

Optimal minimal variation control with quality constraint for fed-batch fermentation processes involving multiple feeds

A new approach is proposed for constructing a relaxation sequence of admissible controls in the class of optimal control problems with constraints. The approach is based on the construction of a system of non-local conditions for improving the admissible control in the form of a fixed point problem of the control operator. To build the conditions for improving the admissible control, we apply the transition to an auxiliary optimization problem based on the well-known principle of extension. Sufficient conditions for the optimality of admissible control and the existence of a minimizing sequence of admissible controls in the considered class of problems with constraints are substantiated. A comparative analysis of the computational efficiency of the proposed iterative method of fixed points with the exact implementation of constraints in model and test optimal control problems is carried out.