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
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,
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