A maximum principle for optimal control system with endpoint constraints
© Wang and Liu; licensee Springer 2012
Received: 30 May 2012
Accepted: 28 September 2012
Published: 12 October 2012
Pontryagin’s maximum principle for an optimal control system governed by an ordinary differential equation with endpoint constraints is proved under the assumption that the control domain has no linear structure. We also obtain the variational equation, adjoint equation and Hamilton system for our problem.
MSC:65K10, 34H05, 93C15.
Optimal control problems have been studied for a long time and have a lot of practical applications in the fields such as physics, biology and economics, etc. Hamilton systems are derived from Pontryagin’s maximum principle, which is known as a necessary condition for optimality. Many results have been obtained both for finite and infinite dimensional control systems such as [1–5]. Regarding the state constraint problems, lots of results are also obtained. For example, readers can refer to [6, 7] and the references therein.
To our best knowledge, to derive the necessary conditions of Pontryagin’s maximum principle type for optimal control problems, there are two main perturbation methods. When the control domain is convex, we often use the convex perturbation. When the control domain is non-convex and does not have any linear structure, we usually use the spike perturbation. Many relevant results have been obtained; see [3, 6, 8–10] and the references therein. The two methods have their advantages and disadvantages. The convex variational needs the control domain being convex, but in reality it is not always satisfied. And the spike variational needs more regularity for the coefficients and the solutions to the state equations, especially in the stochastic case.
In 2010, Lou  introduced a new method to study the necessary and sufficient conditions of optimal control problems in the absence of linear structure for the deterministic case. The author gave a local linearization of the optimal control problem along the optimal control, and transformed the original problem into a new relaxed control problem. Moreover, he proved the equivalence of the two problems in some sense. Being directly inspired by , we are also interested in applying this method to an endpoint constraints optimal control system, which is also in the absence of linear structure. Also, Pontryagin’s maximum principle is obtained for our problem.
The rest of this paper is organized as follows. Section 2 begins with a general formulation of our state constraints optimal control problem and the local linearization of the problem is given. In Section 3, we give our main result and its proof. Moreover, we obtain the variational equation, adjoint equation and Hamilton system for our optimal control system.
where V is a non-convex set in .
where and are closed convex subsets of . Let .
Then the optimal control problem can be stated as follows.
Any satisfying the above identity is called an optimal control, and the corresponding state is called an optimal trajectory; is called an optimal pair.
Let the following hypotheses hold:
() The metric space is separable, and d is the usual metric in .
where denotes the usual Euclidean norm.
From the above conditions, it is easy to know that the state equation (2.1) has a unique solution for any .
We can easily find that and coincide with and respectively. Thus, can be viewed as a subset of in the sense of identifying and . Because the elements of are very simple, we need neither pose additional assumptions like that the control domain is compact as Warga  did nor introduce the relaxed control defined by finite-additive probability measure as Fattorini  did. Now, we can see that already has a linear structure at . First, we give some lemmas to show that is a minimizer of over .
Lemma 2.2 (Lou )
Lemma 2.3 (Lou )
Let - hold and be a minimizer of over . Then is a minimizer of over .
Remark 2.4 We can see that Lemma 2.3 shows the equivalence of the above two problems. And this lemma is essential for our following maximum principle.
3 Pontryagin’s maximum principle
In this section, we give our main result first and then prove it. Let denote the derivative of f on x, and others can be defined in the same way. denotes the inner product in .
Theorem 3.1 (Pontryagin’s maximum principle)
where will be defined in the following part.
We recall that under the conditions -, for any , the state equation (2.1) admits a unique solution with . So, the cost functional is uniquely determined by . In the sequel, we denote the unique solution of (2.1) with by . Now, let be an optimal control. We denote . Without loss of generality, we may assume that ; otherwise, we may consider the optimal control problem with a cost functional of .
Now, we give some definitions and lemmas for Theorem 3.1.
First, we define a penalty functional, via which, for convenience, we can transform the original problem to another one called the approximate problem, which has no endpoint constraint.
where denote the measure of (it obviously is a metric).
where denotes the inner product in . For more properties of subdifferential , one can see p.146 in .
Taking supremum in the above inequality, the first inequality in (3.11) is obtained. The second inequality can be proved similarly. □
By the definition of and Lemma 3.2, we can easily obtain the following result.
Corollary 3.3 The functional is continuous on the space .
The above implies that if we let , then is an optimal pair for the problem where the state equation is (2.1) and the cost functional is .
Now, we derive the necessary conditions for .
where η, , will be defined in the following proof.
Proof Similar to Section 2, we linearize along and denote , , where ( denotes the ball whose center is 0 and radius is 1). Recall that , then we derive the variational equation.
In order to pass to the limit as , we give the following lemma first, which is necessary for the derivation of our maximum principle.
Thus, the proof of this lemma is completed. □
Based on the above preparation, now we start to prove Theorem 3.1 by the duality relations.
Note that in this case, so this gives a contradiction to (3.22). The Hamilton system (3.6) is obvious. Then the proof of the maximum principle is completed. □
- (i)The control problem with fixed endpoints. In this case, the constraint set is and the endpoint constraint becomes of the following form:
- (ii)The control problem with a terminal state constraint, i.e.,
We would like to thank the referees for the useful suggestions which helped to improve the previous version of this paper. This work was partially supported by NNSF of China (Grant No. 11171122).
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