THE CONSTRUCTIVE APPROACH ON EXISTENCE OF TIME OPTIMAL CONTROLS OF SYSTEM GOVERNED BY NONLINEAR EQUATIONS ON BANACH SPACES

In this paper, a new approach to the existence of time optimal controls of system governed by nonlinear equations on Banach spaces is provided. A sequence of Meyer problems is constructed to approach a class of time optimal control problems. A deep relationship between time optimal control problems and Meyer problems is presented. The method is much different from standard methods.


Introduction
The research on time optimal control problems dates back to the 1960's.Issues such as existence, necessary conditions for optimality and controllability have been discussed.We refer the reader to [6] for the finite dimensional case, and to [1,4,7,9] for the infinite dimensional case.The cost functional for a time optimal control problem is the infimum of a number set.On the other hand, the cost functional for a Lagrange, Meyer or Bolza problem contains an integral term.This difference leads investigators to consider a time optimal control problem as another class of optimal control problems and use different studying framework.
Recently, computation of optimal control for Meyer problems has been extensively developed.Meyer problems can be solved numerically using methods such as dynamic programming (see [10]) and control parameterization(see [11]).However, the computation of time optimal controls is very difficult.For finite dimensional problems, one can solve a two point boundary value problems using a shooting method.However, this method is far from ideal since solving such two point boundary value problems numerically is a nontrivial task.
In this paper, we provide a new constructive approach to the existence of time optimal controls.The method is called Meyer approximation.Essentially, a sequence of Meyer problems is constructed to approximate the time optimal control problem.That is, time optimal control problem can be approximated by a sequence of optimal controls from an associated Meyer problem.Although the existence of time optimal control can be proved using other methods, the method presented here is constructive.Hence the algorithm based on Meyer approximation can be used to actually compute the time optimal control.This is in contrast to previously desired results.
We consider the time optimal control problem (P) of a system governed by (1.1) .
where A is the infinitesimal generator of a C0-semigroup {T (t) , t ≥ 0} on Banach space X and V ad is the admissible control set.
By applying the family of C0-semigroups with parameters, the existence of optimal controls for Meyer problem (Pε) is proved.Then, we show that there exists a subsequence of Meyer Problems (Pε n ) whose corresponding sequence of optimal controls {wε n } ∈ W converges to a time optimal control of Problem (P) in some sense.In other words, in a limiting process, the sequence {wε n } ∈ W can be used to find the solution of time optimal control problem (P).The existence of time optimal controls for problem (P) is proved by this constructive approach which offers a new way to compute the time optimal control.
The rest of the paper is organized as follows.In Section 2, we formulate the time optimal control Problem (P) and Meyer problem (Pε).In Section 3, existence of optimal controls for Meyer problems (Pε) is proved.
The last section contributes to the main result of this paper.Time optimal control can be approximated by a sequence of Meyer problems.

Time Optimal Control Problem (P) and Meyer problem (Pε)
For each τ < +∞, let Iτ ≡ [0, τ ] and let C(Iτ , X) be the Banach space of continuous functions from Iτ to X with the usual supremum norm.
Consider the following nonlinear control system (1.1) .
We make the following assumptions: [A] A is the infinitesimal generator of a C0-semigroup {T (t) , t ≥ 0} on X with domain D(A).
By standard process (see Theorem 5.3.3 of [2]), one can easily prove the following existence of mild solutions for system (1.1).

Theorem 2.1: Under the assumptions [A], [B], [F] and [U]
, for every v ∈ V ad , system (1.1) has a unique mild solution z ∈ C(Iτ , X) which satisfies the following integral equation Definition 2.1: (Admissible trajectory) Take two points z0, z1 in the state space X.Let z0 be the initial state and let z1 be the desired terminal state with For given z0, z1 ∈ X and z0 = z1, if V0 = Ø (i.e., There exists at least one control from the admissible class that takes the system from the given initial state z0 to the desired target state z1 in the finite time.),we say the system (1.1) can be controlled.Let denote the transition time corresponding to the control v ∈ V0 = Ø and define Then, the time optimal control problem can be stated as follows: Problem (P): Take two points z0, z1 in the state space X.Let z0 be the initial state and let z1 be the desired terminal state with z0 = z1.Suppose that there exists at least one control from the admissible class that takes the system from the given initial state z0 to the desired target state z1 in the finite time.The time optimal control problem is to find a control v * ∈ V0 such that For fixed v ∈ V ad , T = τ (v) > 0. Now we introduce the following linear transformation Through this transformation system (1.1) can be replaced by (2.1) ẋ(s) = kAx(s) + kf ks, x(s), B(ks)u(s) , s ∈ (0, 1] By Theorem 2. where kA is the generator of a C0-semigroup {T k (t), t ≥ 0} (see Lemma 3.1).
For the controlled system (2.1), we consider Meyer problem (Pε): Minimize the cost functional given by over W , where x (w) is the mild solution of (2.1) corresponding to control w.
i.e., Find a control wε = (uε, kε) such that the cost functional Jε (w) attains its minimum on W at wε.

Existence of optimal controls for Meyer Problem (Pε)
In this section, we discuss the existence of optimal controls for Meyer Problem (Pε).First, in order to study system (2.1), we have to deal with family of C0-semigroups with parameters which are widely used in this paper.
It is obvious that for fixed k ∈ [0, T ], ( i) kA is also closed and D(kA) = X; Using Hille-Yosida theorem again, one can complete it.
(3) Since {kn} is a bounded sequence of [0, T ] and kn > 0, due to continuity theorem of real number, there exists a subsequence of {kn}, denoted by {kn} again such that kn → kε in [0, T ] as n → ∞.For arbitrary x ∈ X and λ > knω, we have Using Theorem 4.5.4 of [2], then Proof.See details on problem 23.9 of [12].
We show that Meyer problem (Pε) has a solution wε = (uε, kε) for fixed ε > 0. By assumption [U], there exists a subsequence {un} ⊆ V ad such that and V ad is closed and convex, thanks to Mazur Lemma, uε ∈ V ad .
Let xn and xε be the mild solutions of system (2.1) corresponding to wn = (un, kn) ∈ W and wε = (uε, kε) ∈ W respectively.Then we have where
Since kn → kε as n → ∞, Then, we obtain that where By Gronwall Lemma, we obtain Thus, there exists a unique control wε = (uε, kε) ∈ W such that This shows that Jε(w) attains its minimum at wε ∈ W , and hence xε is the solution of system (2.1) corresponding to control wε.

Meyer Approximation
In this section, we will show the main result of Meyer approximation of the time optimal control problem (P).In order to make the process clear we divide it into three steps.
We can choose a subsequence {εn} such that εn → 0 as n → ∞ and Since V ad is closed and convex, thanks to Mazur Lemma again, u 0 ∈ V ad .
By assumption [F], where Similar to the proof in Theorem 3.A, one can obtain where Using Gronwall Lemma again, we obtain Step The equality implies that v 0 is an optimal control of Problem (P) and k 0 > 0 is just optimal time.
The following conclusion can be seen from the discussion above.

Conclusion:
Under the above assumptions, there exists a sequence of Meyer problems (Pε n ) whose corresponding sequence of optimal controls {wε n } ∈ W can approximate the time optimal control problem (P) in some sense.In other words, by limiting process, the sequence of the optimal controls {wε n } ∈ W can be used to find the solution of time optimal control problem (P).Remark 2: If B(t) does not have strong continuity and semigroup is compact, we will carry out the full details as well as some related problems in a forthcoming paper.
This work is supported by National Natural Science Foundation of China, Key Projects of Science and Technology Research in the Ministry of Education (No.207104), International Cooperate Item of Guizhou Province (No.(2006) 400102) and Undergraduate Carve Out Project of Department of Guiyang Science and Technology([2008] No.15-2).E-mail: wjr9668@126.com.EJQTDE, 2009 No. 45, p. 1 ) If kn → kε in [0, T ] as n → ∞, then for arbitrary x ∈ X and t ≥ 0, T kn (t) τs −→ T kε (t) as n → ∞ (τs denotes strong operator topology) uniformly in t on some closed interval of [0, T ] in the strong operator topology sense.Proof.(1) By the famous Hille-Yosida theorem (see Theorem 2.2.8 of [2]), (i) A is closed and D(A) = X;