We consider an optimal control problem subject to the terminal state equality constraint and continuous inequality constraints on the control and the state. By using the control parametrization method used in conjunction with a time scaling transform, the constrained optimal control problem is approximated by an optimal parameter selection problem with the terminal state equality constraint and continuous inequality constraints on the control and the state. On this basis, a simple exact penalty function method is used to transform the constrained optimal parameter selection problem into a sequence of approximate unconstrained optimal control problems. It is shown that, if the penalty parameter is sufficiently large, the locally optimal solutions of these approximate unconstrained optimal control problems converge to the solution of the original optimal control problem. Finally, numerical simulations on two examples demonstrate the effectiveness of the proposed method.

Constrained optimal control problems often arise in a wide range of practical applications, including the swing minimization of transferring container [

Most constrained optimal control problems are much too complex to obtain analytical solutions. Thus, they can only be solved by some numerical methods, such as the discretization method [

In this paper, we develop a new computational approach based on the control parametrization method [

The rest of the paper is organized as follows. In Section

Consider the following nonlinear dynamical system:

Define

Consider the following continuous inequality constraints on the control and the state:

Now, the optimal control problem considered in this paper is stated formally as follows.

Let the interval

In order to further improve the accuracy of the approximate optimal control problem, the switching points are also taken as decision variables. We will employ the time scaling transform originally proposed in [

Let

After the control parametrization and a time scaling transformation, the approximate parameter selection problem corresponding to Problem (

Problem (

Define the following constraint violation function on

From the definition of

Problem (

Let

The gradients of the objective function

In particular, when

In the following, we will turn our attention to the new objective function given by (

Suppose that

First, we will show that (

Following arguments similar to those given for the proof of Lemma

Next, from the proof process given for (

Furthermore, we will move on to show that (

From (

In this section, we will show that if (

Suppose that

On the contrary, we assume that the conclusion is false. Then, there exists a subsequence of

Suppose that

On the contrary, we assume that the conclusion is false. Then,

Suppose that

By Lemma

Applying (

Suppose that

Since (

Once the optimal parameter

Let

The proof is similar to that given for Theorem

Based on the results of the convergence analysis, we are in a position to present the following numerical algorithm for solving problem (

The steps are as follows.

Set

Solve problem

If

In Step

Although we have proven that a local minimizer of the exact penalty function optimization problem

In this example, we consider a realistic and complex problem of transferring containers from a ship to a cargo truck at the port of Kobe in [

By utilizing the control parametrization method used conjunction with a time scaling transform and the exact penalty function method, the constrained optimal control problem is transformed into the following unconstrained optimal control problem (

Optimal controls.

Optimal states under optimal controls.

Optimal states under optimal controls.

Constraint functions under optimal controls.

Optimal control.

The following problem is taken from [

In this problem, we set

Optimal state under the optimal control.

Optimal state under the optimal control.

Constraint function under the optimal control.

In this paper, we have presented optimal control problems subject to the terminal state equality constraint and continuous inequality constraints on the control and the state. Our aim is to design an optimal control that minimizes total system cost and ensures satisfaction of all constraints. After the control parametrization, together with the time scaling transformation, the constrained optimal control problem is transformed into a constrained approximate optimal parameter selection problem. A simple exact penalty function method is then used to design a computational method to solve the constrained optimal parameter selection problem. Its main idea is to augment the constraint violation function constructed from the terminal state equality constraint and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimal control problems, which are easily solved by a numerical algorithm. From numerical simulation results, we observe that our proposed method can find a high quality approximate optimal control such that the objective function is minimized, while the terminal state constraint and the constraints on the control and state are both satisfied.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grant no. 11371006.