Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 45, 1-10; http://www.math.u-szeged.hu/ejqtde/

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

EQUATIONS ON BANACH SPACES

JinRong Wang^{1,2,3}, X. Xiang^{1} and W. Wei^{1}

1. College of Science of Guizhou University, Guiyang, Guizhou 550025, P. R. China 2. College of Science of Kaili University, Kaili, Guizhou 556000, P. R. China 3. Policy Research Room of Qiandongnan State Committee of Chinese Communist Party, Kaili, Guizhou 556000, P. R. China

ABSTRACT. 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. Keywords. Time optimal control, C_{0}-semigroup, Existence, Transformation, Meyer approximation.

1. 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

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

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)  \{ \begin{matrix}z^{.} (t) = Az(t) + f(t, z(t), B(t)v(t)), t ∈ (0, τ), \\ z(0) = z_{0} ∈ X, v ∈ V_{ad}, \end{matrix}

where A is the infinitesimal generator of a C_{0}-semigroup {T (t) , t ≥ 0} on Banach space X and V_{ad} is the admissible control set.

Then, we construct the Meyer approximation (P_{ε_{n}}) to Problem (P). Our new control system is

(2.1)  \{ \begin{matrix}ẋ(s) = kAx(s) + kf(ks, x(s), B(ks)u(s)), s ∈ (0, 1], \\ x(0) = z(0) = z_{0} ∈ X, w = (u, k) ∈ W, \end{matrix}

whose controls are taken from a product space. A chosen subsequence {P_{ε_{n}}} is the Meyer approximation to Problem (P).

By applying the family of C_{0}-semigroups with parameters, the existence of optimal controls for Meyer problem (P_{ε}) is proved. Then, we show that there exists a subsequence of Meyer Problems (P_{ε_{n}}) whose corresponding sequence of optimal controls {w_{ε_{n}}} ∈ W converges to a time optimal control of Problem (P) in some sense. In other words, in a limiting process, the sequence {w_{ε_{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 (P_{ε}). In Section 3, existence of optimal controls for Meyer problems (P_{ε}) 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.

2. Time Optimal Control Problem (P) and Meyer problem (P_{ε})

For each τ < +∞, let I_{τ} ≡ [0, τ] and let C(I_{τ} , X) be the Banach space of continuous functions from I_{τ} to X with the usual supremum norm.

Consider the following nonlinear control system

(1.1)  \{ \begin{matrix}z^{.} (t) = Az(t) + f(t, z(t), B(t)v(t)), t ∈ (0, τ), \\ z(0) = z_{0} ∈ X, v ∈ V_{ad}. \end{matrix}

We make the following assumptions:

[A] A is the infinitesimal generator of a C_{0}-semigroup {T (t) , t ≥ 0} on X with domain D(A).

[F] f : I_{τ} × X × X → X is measurable in t on I_{τ} and for each ρ > 0, there exists a constant L (ρ) > 0 such that for almost all t ∈ I_{τ} and all z_{1}, z_{2}, y_{1}, y_{2} ∈ X, satisfying ∥z_{1}∥ , ∥z_{2}∥ , ∥y_{1}∥, ∥y_{2}∥ ≤ ρ, we have

∥f (t, z_{1}, y_{1}) − f (t, z_{2}, y_{2})∥ ≤ L (ρ) (∥z_{1} − z_{2}∥ + ∥y_{1} − y_{2}∥) .

For arbitrary (t, y) ∈ I_{τ} × X, there exists a positive constant M > 0 such that

∥f (t, z, y)∥ ≤ M(1 + ∥z∥).

[B] Let E be a reflexive Banach space. Operator B ∈ L_{∞}(I_{τ} , L(E, X)), ∥B∥_{∞} stands for the norm of operator B on Banach space L_{∞}(I_{τ} , L(E, X)). B : L_{p}(I_{τ} , E) → L_{p}(I_{τ} , X)(1 < p < +∞) is strongly continuous.

EJQTDE, 2009 No. 45, p. 2

[U] Multivalued maps Γ (·) : I_{τ} → 2^{E}\{Ø} has closed, convex and bounded values. Γ (·) is graph measurable and Γ (·) ⊆ Ω where Ω is a bounded set of E.

Set

V_{ad} = {v (·) | I_{τ} → E measurable, v(t) ∈ Γ(t) a.e.} .

Obviously, V_{ad} ≠ Ø (see Theorem 2.1 of [12]) and V_{ad} ⊂ L_{p}(I_{τ} , E)(1 < p < +∞) is bounded, closed and convex.

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(I_{τ} , X) which satisfies the following integral equation

z(t) = T (t)z_{0} + ∫_{0}^{t} T (t − θ) f(θ, z (θ) , B(θ)v(θ))dθ.

Definition 2.1: (Admissible trajectory) Take two points z_{0}, z_{1} in the state space X. Let z_{0} be the initial state and let z_{1} be the desired terminal state with z_{0} ≠ z_{1}. Denote z (v) ≡ {z (t, v) ∈ X | t ≥ 0} be the state trajectory corresponding to the control v ∈ V_{ad}. A trajectory z (v) is said to be admissible if z (0, v) = z_{0} and z (t, v) = z_{1} for some finite t > 0.

Set

V_{0} = {v ∈ V_{ad} | z(v) is an admissible trajectory} ⊂ V_{ad}.

For given z_{0}, z_{1} ∈ X and z_{0} ≠ z_{1}, if V_{0} ≠ Ø (i.e., There exists at least one control from the admissible class that takes the system from the given initial state z_{0} to the desired target state z_{1} in the finite time.), we say the system (1.1) can be controlled. Let

τ(v) ≡ inf {t ≥ 0 | z (t, v) = z_{1}}

denote the transition time corresponding to the control v ∈ V_{0} ≠ Ø and define

τ^{∗} = inf {τ (v) ≥ 0 | v ∈ V_{0}} .

Then, the time optimal control problem can be stated as follows:

Problem (P): Take two points z_{0}, z_{1} in the state space X. Let z_{0} be the initial state and let z_{1} be the desired terminal state with z_{0} ≠ z_{1}. Suppose that there exists at least one control from the admissible class that takes the system from the given initial state z_{0} to the desired target state z_{1} in the finite time. The time optimal control problem is to find a control v^{∗} ∈ V_{0} such that

τ (v^{∗}) = τ^{∗} = inf {τ (v) ≥ 0 | v ∈ V_{0}} .

For fixed v̂ ∈ V_{ad}, T̂ = τ(v̂) > 0. Now we introduce the following linear transformation

t = ks, 0 ≤ s ≤ 1 and k ∈ [0, T̂].

Through this transformation system (1.1) can be replaced by

(2.1)  \{ \begin{matrix}ẋ(s) = kAx(s) + kf(ks, x(s), B(ks)u(s)), s ∈ (0, 1] \\ x(0) = z(0) = z_{0} ∈ X, w = (u, k) ∈ W, \end{matrix}

where x(·) = z(k·), u(·) = v(k·) and define

W = {(u, k) | u(s) = v (ks) , 0 ≤ s ≤ 1, v ∈ V_{ad}, k ∈ [0, T̂]} .

By Theorem 2.1, one can verify that

EJQTDE, 2009 No. 45, p. 3

Theorem 2.2: Under the assumptions [A], [B], [F] and [U], for every w ∈ W, system (2.1) has a unique mild solution x ∈ C([0, 1], X) which satisfies the following integral equation

x(s) = T_{k}(s)z_{0} + ∫_{0}^{s} T_{k} (s − θ) kf(kθ, x (θ) , B(kθ)u(θ))dθ,

where kA is the generator of a C_{0}-semigroup {T_{k}(t), t ≥ 0} (see Lemma 3.1).

For the controlled system (2.1), we consider

Meyer problem (P_{ε}): Minimize the cost functional given by

J_{ε} (w) = \frac{1}{2ε} ∥x (w) (1) − z_{1}∥^{2} + k

over W, where x (w) is the mild solution of (2.1) corresponding to control w. i.e., Find a control w_{ε} = (u_{ε}, k_{ε}) such that the cost functional J_{ε} (w) attains its minimum on W at w_{ε}.

3. Existence of optimal controls for Meyer Problem (P_{ε})

In this section, we discuss the existence of optimal controls for Meyer Problem (P_{ε}). First, in order to study system (2.1), we have to deal with family of C_{0}-semigroups with parameters which are widely used in this paper.

Lemma 3.1: If the assumption [A] holds, then (1) For given k ∈ [0, T̂], kA is the infinitesimal generator of C_{0}-semigroup {T_{k}(t), t ≥ 0} on X.

(2) There exist constants C ≥ 1 and ω ∈ (−∞, +∞) such that

∥T_{k}(t)∥ ≤ Ce^{ωkt} for all t ≥ 0.

(3) If k_{n} → k_{ε} in [0, T̂] as n → ∞, then for arbitrary x ∈ X and t ≥ 0,

T_{k_{n}}(t) ⟶^{τ_{s}} T_{k_{ε}}(t) as n → ∞ (τ_{s} denotes strong operator topology)

uniformly in t on some closed interval of [0, T̂] in the strong operator topology sense.

Proof. (1) By the famous Hille-Yosida theorem (see Theorem 2.2.8 of [2]),

\{ \begin{matrix}(i) A is closed and \overline{D(A)} = X; \\ (ii) ρ(A) ⊃ (ω, +∞) and ∥R(λ, A)∥ ≤ (λ − ω)^{−1}, for λ > ω. \end{matrix}

It is obvious that for fixed k ∈ [0, T̂],

\{ \begin{matrix}(i̇) kA is also closed and \overline{D(kA)} = X; \\ (i̇i) ρ(kA) ⊃ (kω, +∞) and ∥R(λ, kA)∥ = k^{−1}∥R(k^{−1}λ, A)∥ ≤ (λ − kω)^{−1}, for λ > kω. \end{matrix}

Using Hille-Yosida theorem again, one can complete it.

(2) By virtue of (1) and Theorem 1.3.1 of [2], one can verify it easily.

(3) Since {k_{n}} is a bounded sequence of [0, T̂] and k_{n} > 0, due to continuity theorem of real number, there exists a subsequence of {k_{n}}, denoted by {k_{n}} again such that k_{n} → k_{ε} in [0, T̂] as n → ∞. For arbitrary x ∈ X and λ > k_{n}ω, we have

R(λ, k_{n}A)x = (λI − k_{n}A)^{−1}x → (λI − k_{ε}A)^{−1}x = R(λ, k_{ε}A)x, as n → ∞.

Using Theorem 4.5.4 of [2], then

T_{k_{n}}(t)x → T_{k_{ε}}(t)x as n → ∞.

Further,

T_{k_{n}}(t) ⟶^{τ_{s}} T_{k_{ε}}(t) as n → ∞

EJQTDE, 2009 No. 45, p. 4

uniformly in t on some closed interval in the strong operator topology sense. □

Lemma 3.2: For each g ∈ L_{p}([0, T_{0}], X) with 1 ≤ p < +∞,

lim_{h→0} ∫_{0}^{T_{0}} ∥g(t + h) − g(t)∥^{p}dt = 0

where g(s) = 0 for s does not belong to [0, T_{0}].

Proof. See details on problem 23.9 of [12]. □

We show that Meyer problem (P_{ε}) has a solution w_{ε} = (u_{ε}, k_{ε}) for fixed ε > 0.

Theorem 3.A: Assumptions [A], [B], [F] and [U] hold. Meyer problem (P_{ε}) has a solution.

Proof. Let ε > 0 be fixed. Since J_{ε}(w) ≥ 0, there exists inf{J_{ε}(w), w ∈ W }. Denote m_{ε} ≡ inf{J_{ε}(w), w ∈ W } and choose {w_{n}} ⊆ W such that

J_{ε}(w_{n}) → m_{ε}

where

w_{n} = (u_{n}, k_{n}) ∈ W = V_{ad} × [0, T̂].

By assumption [U], there exists a subsequence {u_{n}} ⊆ V_{ad} such that

u_{n} ⟶^{w} u_{ε} in V_{ad} as n → ∞,

and V_{ad} is closed and convex, thanks to Mazur Lemma, u_{ε} ∈ V_{ad}. By assumption [B], we have

Bu_{n} ⟶^{s} Bu_{ε} in L_{p}([0, 1], X) as n → ∞.

Since k_{n} is bounded and k_{n} > 0, there also exists a subsequence {k_{n}} denoted by {k_{n}} ⊆ [0, T̂] again, such that

k_{n} → k_{ε} in [0, T̂] as n → ∞.

Let x_{n} and x_{ε} be the mild solutions of system (2.1) corresponding to w_{n} = (u_{n}, k_{n}) ∈ W and w_{ε} = (u_{ε}, k_{ε}) ∈ W respectively. Then we have

x_{n}(s) = T_{n}(s)z_{0} + ∫_{0}^{s} T_{n}(s − θ)k_{n}F_{n}(θ)dθ,
x_{ε}(s) = T_{ε}(s)z_{0} + ∫_{0}^{s} T_{ε}(s − θ)k_{ε}F_{ε}(θ)dθ,

where

T_{n}(·) ≡ T_{k_{n}}(·); F_{n}(·) ≡ f(k_{n}·, x_{n}(·), B(k_{n}·)u_{n}(·));
T_{ε}(·) ≡ T_{k_{ε}}(·); F_{ε}(·) ≡ f(k_{ε}·, x_{ε}(·), B(k_{ε}·)u_{ε}(·)).

By assumptions [F], [B], [U] and Gronwall Lemma, it is easy to verify that there exists a constant ρ > 0 such that

∥x_{ε}∥_{C([0,1],X)} ≤ ρ and ∥x_{n}∥_{C([0,1],X)} ≤ ρ.

Further, there exists a constant M_{ε} > 0 such that

∥F_{ε}∥_{C([0,1],X)} ≤ M_{ε}(1 + ρ).

EJQTDE, 2009 No. 45, p. 5

Denote

R_{1} = ∥T_{n}(s)z_{0} − T_{ε}(s)z_{0}∥,
R_{2} = ∥ ∫_{0}^{s} T_{n}(s − θ)k_{n}F_{n}(θ)dθ − ∫_{0}^{s} T_{n}(s − θ)k_{n}F_{n}^{ε}(θ)dθ∥,
R_{3} = ∥ ∫_{0}^{s} T_{n}(s − θ)k_{n}F_{n}^{ε}(θ)dθ − ∫_{0}^{s} T_{ε}(s − θ)k_{ε}F_{ε}(θ)dθ∥,

where

F_{n}^{ε}(θ) ≡ f(k_{n}θ, x_{ε}(θ), B(k_{ε}θ)u_{ε}(θ)).

By assumption [F],

R_{2} ≤ C_{k_{n}}k_{n} ∫_{0}^{s} ∥F_{n}(θ) − F_{n}^{ε}(θ)∥dθ
≤ C_{k_{n}}k_{n}L(ρ) ∫_{0}^{s} ∥x_{n}(θ) − x_{ε}(θ)∥dθ
+ C_{k_{n}}k_{n}L(ρ) ∫_{0}^{s} ∥B(k_{n}θ)u_{n}(θ) − B(k_{ε}θ)u_{ε}(θ)∥dθ
≤ R_{21} + R_{22} + R_{23}

where

C_{k_{n}} ≡ Ce^{ωk_{n}},
R_{21} ≡ C_{k_{n}}k_{n}L(ρ) ∫_{0}^{s} ∥x_{n}(θ) − x_{ε}(θ)∥dθ,
R_{22} ≡ C_{k_{n}}k_{n}L(ρ) ∫_{0}^{s} ∥B(k_{n}θ)u_{ε}(θ) − B(k_{ε}θ)u_{ε}(θ)∥dθ,
R_{23} ≡ C_{k_{n}}k_{n}L(ρ) ∫_{0}^{s} ∥B(k_{n}θ)u_{n}(θ) − B(k_{n}θ)u_{ε}(θ)∥dθ.

and

R_{3} ≤ ∫_{0}^{s} ∥k_{n}T_{n}(s − θ)F_{n}^{ε}(θ) − k_{ε}T_{n}(s − θ)F_{ε}(θ)∥dθ
+ k_{ε} ∫_{0}^{s} ∥T_{n}(s − θ)F_{ε}(θ) − T_{ε}(s − θ)F_{ε}(θ)∥ dθ
≤ R_{31} + R_{32} + R_{33}

where

R_{31} ≡ C_{k_{n}} ∫_{0}^{s} ∥k_{n}F_{n}^{ε}(θ) − k_{n}F_{ε}(θ)∥dθ,
R_{32} ≡ C_{k_{n}} ∫_{0}^{s} ∥k_{n}F_{ε}(θ) − k_{ε}F_{ε}(θ)∥dθ,
R_{33} ≡ k_{ε}M_{ε}(1 + ρ) ∫_{0}^{s} ∥T_{n}(s − θ) − T_{ε}(s − θ)∥dθ.

By Lemma 3.1, one can obtain R_{1} → 0 and R_{33} → 0 as n → ∞ immediately. It follows from

R_{22} ≤ C_{k_{n}}k_{n}L(ρ)( ∫_{0}^{1} ∥B(k_{n}θ) − B(k_{ε}θ)∥^{p}dθ)^{\frac{1}{p}} ( ∫_{0}^{1} ∥u_{ε}(θ)∥^{q}dθ)^{\frac{1}{q}} ,
R_{23} ≤ C_{k_{n}}k_{n}L(ρ) ∫_{0}^{s} ∥B(k_{ε}θ)u_{n}(θ) − B(k_{ε}θ)u_{ε}(θ)∥dθ
+ C_{k_{n}}k_{n}L(ρ)( ∫_{0}^{1} ∥B(k_{n}θ) − B(k_{ε}θ)∥^{p}dθ)^{\frac{1}{p}} ( ∫_{0}^{1} ∥u_{n}(θ) − u_{ε}(θ)∥^{q}dθ)^{\frac{1}{q}} ,
R_{31} ≤ C_{k_{n}}k_{n} ∫_{0}^{s} ∥F_{n}^{ε}(θ) − F_{ε}(θ)∥dθ,

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Lemma 3.2 and assumption [B] that R_{22} → 0, R_{23} → 0 and R_{31} → 0, as n → ∞.

Since k_{n} → k_{ε} as n → ∞,

R_{32} → 0 as n → ∞.

Then, we obtain that

∥x_{n}(s) − x_{ε}(s)∥ ≤ R_{1} + R_{2} + R_{3}
≤ σ_{ε} + C_{k_{n}}k_{n}L(ρ) ∫_{0}^{s} ∥x_{n}(θ) − x_{ε}(θ)∥dθ

where

σ_{ε} = R_{1} + R_{22} + R_{23} + R_{31} + R_{32} → 0 as n → ∞.

By Gronwall Lemma, we obtain

x_{n} ⟶^{s} x_{ε} in C ([0, 1] , X) as n → ∞.

Thus, there exists a unique control w_{ε} = (u_{ε}, k_{ε}) ∈ W such that

m_{ε} = lim_{n→∞} J_{ε}(w_{n}) = J_{ε}(w_{ε}) ≥ m_{ε}.

This shows that J_{ε}(w) attains its minimum at w_{ε} ∈ W, and hence x_{ε} is the solution of system (2.1) corresponding to control w_{ε}. □

4. 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.

Step 1: By Theorem 3.A, there exists w_{ε} = (u_{ε}, k_{ε}) ∈ W such that J_{ε}(w) attains its minimum at w_{ε} ∈ W, i.e.,

J_{ε}(w_{ε}) = \frac{1}{2ε} ∥x (w_{ε}) (1) − z_{1}∥^{2} + k_{ε} = inf_{w∈W} J_{ε}(w).

By controllability of problem (P), V_{0} ≠ Ø. Take ṽ ∈ V_{0} and let τ \binom{}{ṽ} = τ̃ < +∞ then z \binom{}{ṽ} \binom{}{τ̃} = z_{1}. Define ũ(s) = ṽ (τ̃s), 0 ≤ s ≤ 1 and w̃ = (ũ, τ̃) ∈ W. Then x̃(·) = z \binom{}{ṽ} (τ̃·) is the mild solution of system (2.1) corresponding to control w̃ = (ũ, τ̃) ∈ W. Of course we have x̃(1) = z_{1}.

For any ε > 0 submitting w̃ to J_{ε}, we have

J_{ε} \binom{}{w̃} = τ̃ ≥ J_{ε}(w_{ε}) = \frac{1}{2ε} ∥x (w_{ε}) (1) − z_{1}∥^{2} + k_{ε}.

This inequality implies that

\{ \begin{matrix}0 ≤ k_{ε} ≤ τ̃, \\ ∥x (w_{ε}) (1) − z_{1}∥^{2} ≤ 2ετ̃ hold for all ε > 0. \end{matrix}

We can choose a subsequence {ε_{n}} such that ε_{n} → 0 as n → ∞ and

k_{ε_{n}} → k^{0} in [0, T̂],

x (w_{ε_{n}}) (1) ≡ x_{ε_{n}}(1) → z_{1} in X, as n → ∞,

u_{ε_{n}} ⟶^{w} u^{0} in V_{ad}, w_{ε_{n}} = (u_{ε_{n}}, k_{ε_{n}}) ∈ W.

Since V_{ad} is closed and convex, thanks to Mazur Lemma again, u^{0} ∈ V_{ad}. Further, by assumption [B], we obtain

k_{ε_{n}} → k^{0} in [0, T̂],

x (w_{ε_{n}}) (1) ≡ x_{ε_{n}}(1) → z_{1} in X, as n → ∞,

Bu_{ε_{n}} ⟶^{s} Bu^{0} in L_{p}([0, 1], X).

EJQTDE, 2009 No. 45, p. 7

Step 2: Let x_{ε_{n}} and x^{0} be the mild solutions of system (2.1) corresponding to w_{ε_{n}}=(u_{ε_{n}}, k_{ε_{n}}) ∈ W and w^{0} = (u^{0}, k^{0}) ∈ W respectively. Then we have

x_{ε_{n}}(s) = T_{ε_{n}}(s)z_{0} + ∫_{0}^{s} T_{ε_{n}}(s − θ)k_{ε_{n}}F_{ε_{n}}(θ)dθ,
x^{0}(s) = T_{0}(s)z_{0} + ∫_{0}^{s} T_{0}(s − θ)k^{0}F^{0}(θ)dθ,

where

T_{ε_{n}}(s) ≡ T_{k_{εn}} (s) ; F_{ε_{n}}(θ) ≡ f(k_{ε_{n}}θ, x_{ε_{n}}(θ), B(k_{ε_{n}}θ)u_{ε_{n}}(θ));
T_{0}(s) ≡ T_{k^{0}}(s);  F^{0}(θ) ≡ f(k^{0}θ, x^{0}(θ), B(k^{0}θ)u^{0}(θ)).

By assumptions [F], [B], [U] and Gronwall Lemma, it is easy to verify that there exists a constant ρ > 0 such that

∥x_{ε_{n}}∥_{C([0,1],X)} ≤ ρ and ∥x^{0}∥_{C([0,1],X)} ≤ ρ.

Further, there exists a constant M^{0} > 0 such that

∥F^{0}∥_{C([0,1],X)} ≤ M^{0}(1 + ρ).

Denote

L_{1} = ∥T_{ε_{n}}(s)z_{0} − T_{0}(s)z_{0}∥,
L_{2} = ∥ ∫_{0}^{s} T_{ε_{n}}(s − θ)k_{ε_{n}}F_{ε_{n}}(θ)dθ − ∫_{0}^{s} T_{ε_{n}}(s − θ)k_{ε_{n}}F_{ε_{n}}^{0}(θ)dθ∥,
L_{3} = ∥ ∫_{0}^{s} T_{ε_{n}}(s − θ)k_{ε_{n}}F_{ε_{n}}^{0}(θ)dθ − ∫_{0}^{s} T_{0}(s − θ)k^{0}F^{0}(θ)dθ∥.

where

F_{ε_{n}}^{0}(θ) ≡ f(k_{ε_{n}}θ, x^{0}(θ), B(k^{0}θ)u^{0}(θ)).

By assumption [F],

L_{2} ≤ C_{k_{εn}} k_{ε_{n}} ∫_{0}^{s} ∥F_{ε_{n}}(θ) − F_{ε_{n}}^{0}(θ)∥dθ,
≤ C_{k_{εn}} k_{ε_{n}}L(ρ) ∫_{0}^{s} ∥x_{ε_{n}}(θ) − x^{0}(θ)∥dθ
+ C_{k_{εn}} k_{ε_{n}}L(ρ) ∫_{0}^{s} ∥B(k_{ε_{n}}θ)u_{ε_{n}}(θ) − B(k^{0}θ)u^{0}(θ)∥dθ,
≤ L_{21} + L_{22} + L_{23},

where

C_{k_{εn}} ≡ Ce^{ωk_{εn}} ,
L_{21} ≡ C_{k_{εn}} k_{ε_{n}}L(ρ) ∫_{0}^{s} ∥x_{ε_{n}}(θ) − x^{0}(θ)∥dθ,
L_{22} ≡ C_{k_{εn}} k_{ε_{n}}L(ρ) ∫_{0}^{s} ∥B(k_{ε_{n}}θ)u^{0}(θ) − B(k^{0}θ)u^{0}(θ)∥dθ,
L_{23} ≡ C_{k_{εn}} k_{ε_{n}}L(ρ) ∫_{0}^{s} ∥B(k_{ε_{n}}θ)u_{ε_{n}}(θ) − B(k_{ε_{n}}θ)u^{0}(θ)∥dθ.

and

L_{3} ≤ ∫_{0}^{s} ∥T_{ε_{n}}(s − θ)k_{ε_{n}}F_{ε_{n}}^{0}(θ) − T_{ε_{n}}(s − θ)k^{0}F^{0}(θ)∥dθ
+ ∫_{0}^{s} ∥T_{ε_{n}}(s − θ)k^{0}F^{0}(θ) − T_{0}(s − θ)k^{0}F^{0}(θ)∥ dθ
≤ L_{31} + L_{32} + L_{33}

EJQTDE, 2009 No. 45, p. 8

where

L_{31} ≡ C_{k_{εn}} ∫_{0}^{s} ∥k_{ε_{n}}F_{ε_{n}}^{0}(θ) − k_{ε_{n}}F^{0}(θ)∥dθ,
L_{32} ≡ C_{k_{εn}} ∫_{0}^{s} ∥k_{ε_{n}}F^{0}(θ) − k^{0}F^{0}(θ)∥dθ,
L_{33} ≡ k^{0}M^{0}(1 + ρ) ∫_{0}^{s} ∥T_{ε_{n}}(s − θ) − T_{0}(s − θ)∥dθ.

Similar to the proof in Theorem 3.A, one can obtain

∥x_{ε_{n}}(s) − x^{0}(s)∥ ≤ L_{1} + L_{2} + L_{3}
≤ σ^{0} + C_{k_{εn}} k_{ε_{n}}L(ρ) ∫_{0}^{s} ∥x_{ε_{n}}(θ) − x^{0}(θ)∥dθ

where

σ^{0} = L_{1} + L_{22} + L_{23} + L_{31} + L_{32} → 0 as n → ∞.

Using Gronwall Lemma again, we obtain

x_{ε_{n}} ⟶^{s} x^{0} in C ([0, 1] , X) as n → ∞.

Step 3: It follows from Step 1 and Step 2,

\{ \begin{matrix}∥x_{ε_{n}} (1) − z_{1}∥ ≤\sqrt{2ε_{n}τ̃} ⟶ 0, as n → ∞, \\ ∥x_{ε_{n}}(1) − x^{0}(1)∥ ⟶ 0, as n → ∞, \end{matrix}

and

∥x^{0}(1) − z_{1}∥ ≤ ∥x_{ε_{n}} (1) − z_{1}∥ + ∥x_{ε_{n}}(1) − x^{0}(1)∥ ⟶ 0, as n → ∞,

that

x^{0}(1) = z_{1}.

It is very clear that k^{0} ≠ 0 unless z_{0} = z_{1}. This implies that

k^{0} > 0.

Define v^{0}(·) = u^{0} (·/k^{0}) . In fact, z^{0}(·) = x^{0} (·/k^{0}) is the mild solution of system (1.1) corresponding to

control v^{0} ∈ V_{0}, then

z^{0}(k^{0}) = x^{0}(1) = z_{1} and τ(v^{0}) = k^{0} > 0.

By the definition of τ^{∗} = inf {τ (v) ≥ 0 | v ∈ V_{0}},

k^{0} ≥ τ^{∗}.
For any v ∈ V_{0},

τ(v) ≥ J_{ε}(w_{ε}) = \frac{1}{2ε} ∥x (w_{ε}) (1) − z_{1}∥^{2} + k_{ε}.

Thus,

τ(v) ≥ k_{ε}.

Further,

τ(v) ≥ k_{ε_{n}} for all ε_{n} > 0.

Since k^{0} is the limit of k_{ε_{n}} as n → ∞,

τ(v) ≥ τ(v^{0}) = k^{0} for all v ∈ V_{0}.

Hence,

k^{0} ≤ τ^{∗}.

Thus,

0 < τ(v^{0}) = k^{0} = τ^{∗}.

EJQTDE, 2009 No. 45, p. 9

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 (P_{ε_{n}}) whose corresponding sequence of optimal controls {w_{ε_{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 {w_{ε_{n}}} ∈ W can be used to find the solution of time optimal control problem (P).

Theorem 4.A: Assumptions [A], [B], [F] and [U] hold. Problem (P) has a solution.

Remark 1: If B(t) ≡ B, then Theorem 3.A and Theorem 4.A also hold.

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.

Acknowledgment: Special thanks go to Professor N.U. Ahmed and referees for their useful suggestion.

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(Received January 13, 2009)

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