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In this paper, the analytic solutions to constrained optimal control problems are considered. A novel approach based on canonical duality theory is developed to derive the analytic solution of this problem by reformulating a constrained optimal control problem into a global optimization problem. A differential flow is presented to deduce some optimality conditions for solving global optimizations, which can be considered as an extension and a supplement of the previous results in canonical duality theory. Some examples are given to illustrate the applicability of our results.

In this paper, we consider the following linear-quadratic optimal control problem involving control constraints:

where

In recent years, significant advances have been made in efficiently tackling optimal control problems [

In this paper, a different way, canonical dual approach is used to study the problem

Now, we shall give the Pontryagin maximum principle and an important Lemma.

Pontryagin Maximum Principle If

then we have,

and

Lemma 1. An admissible pair

Proof. Denote

Let

Moreover, it is easy to see that the minimizer

Taking into account of the convexity of the integrand in the cost functional as well as the set U, the function

which leads to

Thus, we have

This means that J attains its minimum at

The above Lemma reformulates the optimal control problem

The rest of the paper is organized as follows. In Section 2, we consider the optimal control problem with a sphere constraint. By introducing the differential flow and canonical dual function for solving global optimizations, we derive the analytic solution expressed by the co-state via canonical dual variables. Based on the similar canonical dual strategy, the box constrained optimal control problem is studied and the corresponding analytic expression of optimal control is obtained in Section 3. Meanwhile, some examples are given to demonstration.

In this section, we let

solution for the problem

Consider the following general optimization problem

where

The original idea of this section is from the paper [

Assume that a

We focus on the differential flow

Based on the classical theory of ODE, we can obtain the solution

and the canonical dual problem associated with the problem (10) can be proposed as follows

Notice that

dual function

Theorem 1. If the flow

that

Detailed proof of Theorem 1 can be referred to [

In what follows, we show that

Lemma 2. Let

Since U is bounded and

by Brown fixed-point theorem, which means that the pair

Let

between

So, the canonical dual function can be formulated as, for each

Next, we have the following properties.

Lemma 3. Let

Proof. Since

Lemma 4. Let

1)

2) if there exists

Proof. By (21), it follows that

decreasing on

If there exists one point

Theorem 2. For the sphere constrained optimal control problem

where

and

Proof. We first consider

For any

Case 1: Suppose that

where

and

Further, it follows from Lemma 4 that

Thus, for every

Case 2: Suppose that

On the other hand, If there exists one point

Define

where

If consider

Theorem 3. Let R be an identity matrix I in (1). Then the analytic solution to problem

Proof. Suppose that

solution can be expressed as, a.e.

In this section, we consider

Similarly, consider the general box constrained problem

where

Denote

where

Assumed that

we focus on the flow

where

Based on the extension theory, the solution

and the canonical dual problem associated with the problem (32) can be formulated as follows

Lemma 5. Let

Proof. Since

By (35), it follows that

Form (34), we have

By the definition of

Lemma 5 shows that the canonical dual function

Theorem 4. (Perfect duality theorem) The canonical dual problem

Proof. By the KKT theory,

where

The proof is completed.

Theorem 5. (Triality theorem) Consider

Proof. By Lemma 5 and the fact that

In the following deducing, we need to note the fact that since

can show that

Thus, we have

By (43), (44) and the canonical duality theory, it leads to the conclusion we desired.

Now, let

and the canonical dual function

Set

Lemma 6. Let

Proof. Notice that

^{th} diagonal element of H.

By properties of the positive definite matrix, it follows that the diagonal element

In the rest part of this section, we suppose that

Theorem 6. For the box constrained optimal control problem

Proof. Set

Consider complementarity conditions

Lemma 6, it is easy to verify that there must exist one point

which can be rewritten as

In what follows, parallel to the proof of Theorem 2, we shall show that

By statements as the above and Lemma 6, we have

where

By Lemma 5 and (46), we have

Thus, ^{opt} as a function with respect to the co-state

We will give an example to illustrate our results.

Example 2: For the box constrained optimal control problem

Following idea of Lemma 1 and Theorem 2 as above, we need to solve a system on the state and co-state for deriving the optimal solution

and

By solving Equations (52)-(54) in MATLAB, we can obtain the optimal optimal feedback control

We thank the Editor and the referee for their comments. Research of D. Wu is supported by the National Science Foundation of China under grants No.11426091, 11471102.

DanWu, (2015) Analytic Solutions to Optimal Control Problems with Constraints. Applied Mathematics,06,2326-2339. doi: 10.4236/am.2015.614205