Existence of Solutions for Optimal Control Problems on Time Scales whose States are Absolutely Continuous

ABSTRACT

This paper considers a class of optimal control problems on time scales described by dynamic equations on time scales. For this class, the states are absolutely continuous. We have established sufficient conditions for the existence of optimal controls.

Keywords:
Time scales; control systems; optimal control

RESUMO

Este artigo considera uma classe de problemas de controle ótimo em escalas temporais descritos por equações dinâmicas em escalas temporais. Para essa classe, os estados são absolutamente contínuos. Nós estabelecemos condições suficientes para a existência de controles ótimos.

Palavras-chave:
escalas temporais; sistemas de controle; controle ótimo

1 INTRODUCTION

The existence of solutions to optimal control problems on time scales can be found in ^7,11,13.

In this report, we have studied control systems covered in ⁶ to the study of necessary optimality conditions. However, unlike ⁶ here the states are absolutely continuous functions and the controls are Δ-measurable functions. To the class of control systems studied here, we establish an extension of Filippov's selection theorem ⁴.

Using standards arguments to the obtaining of optimal controls, see for instance ^4,8,12, we get the existence of solutions to optimal control problems on time scales. The result of existence obtained here relies on [¹¹, Theorem 5.2], the difference is in the classes of control systems studied.

The paper is organized as follows. In Section 2, we provided basic concepts and results on time scales theory. Section 3 presents the result of existence of optimal processes to optimal control problems on time scales.

2 PRELIMINARIES

2.1 Time scales

A time scale is a nonempty closed subset of real numbers. Here we use a bounded time scale 𝕋, where a = min 𝕋 and b = max 𝕋 are such that a < b.

Define the forward jump operator σ : 𝕋 → 𝕋 by

σ (t) = inf{s ∈ 𝕋 : s > t}

and the backward jump operator ρ: 𝕋 → 𝕋 by

ρ(t) = sup{s ∈ 𝕋 : s < t}.

We assume that inf Ø= sup 𝕋 and sup Ø = inf 𝕋.

Let μ: 𝕋 → [0, +∞) be defined as

μ(t) = σ (t) − t.

Lemma 2.1 (⁽³⁾ ). There exist I ⊂ ℕ and {t_i }_i∈I ⊂ 𝕋 such that

RS: = {t ∈ 𝕋 : t < σ(t)}= {ti }i∈I,

where RS stands for right scattered points of the time scale 𝕋.

If A ⊂ ℝ, define the set A _𝕋 as A _𝕋 = A ∩ 𝕋. We set 𝕋^κ = 𝕋 \ (ρ(sup 𝕋), sup 𝕋]_𝕋.

Take a function f: 𝕋 → ℝ and t ∈ 𝕋^κ . If ξ ∈ ℝ is such that, for all ε > 0 there exists δ > 0 satisfying

| f (σ(t)) − f (s) − ξ(σ(t) − s)| ≤ ε|σ (t) − s|

for all s ∈ (t − δ, t + δ)_𝕋, we say that ξ is the delta derivative of f at t and we denote it by

ξ := f ^Δ(t).

Now, consider a function f: 𝕋 → ℝⁿ and t ∈ 𝕋^κ . We say that f is Δ-differentiable at t if each component fi: 𝕋 → ℝ of f is Δ-differentiable at t. In this case f ^Δ(t) = f ₁ ^Δ(t), ... , f_n ^Δ(t)).

1n

The next result is an easy consequence of [¹, Theorem 1.16].

Theorem 2.1 (⁽¹⁾ ). Consider a function f: 𝕋 → ℝⁿ and t ∈ 𝕋^κ . Then the following statements hold:

If f is Δ-differentiable at t then f is continuous at t .

If f is continuous at t and σ (t) > t, then f is Δ-differentiable at t. Furthermore,

If σ (t) = t, then f is Δ-differentiable at t if and only if the following limit exists

as an element of ℝ^{_{n. In this case}}

If f is Δ-differentiable at t, then

f (σ (t)) = f (t) + μ(t) f ^Δ(t).

2.2 Δ-measurable functions and Δ-integrability

We denote the family of Δ-measurable sets of 𝕋 by Δ. We recall that Δ is a σ-algebra of 𝕋.

Given a function f: 𝕋 → ℝⁿ we define a, b] → ℝⁿ as: [

where I ⊂ ℕ and {t_i }_i∈I ⊂ 𝕋 are such that {t ∈ 𝕋 : t < σ(t)}= {t_i }_i∈I .

Proposition 2.1 given below can be easily obtained from [³, Proposition 4.1].

Proposition 2.1 (⁽³⁾ ). Consider a function f: 𝕋 → ℝⁿ . Then f is Δ-measurable if and only ifis Lebesgue measurable.

Take a function f: 𝕋 → E ∈ Δ. We indicate by and

the Lebesgue Δ-integral of f over E. We denote the set of functions f: 𝕋 → ℝ which are Lebesgue Δ-integrable over E by L ₁(E). If f: 𝕋 → ℝⁿ is Δ-measurable and E ∈ Δ, we indicate by L ₁(E, ℝⁿ ) the set of functions f: 𝕋 → ℝⁿ which are Lebesgue Δ-integrable over E.

More in the subject on Lebesgue integration on time scales, can be found in ^3,5,10.

Proposition 2.2 (⁽¹⁰⁾ ). Let f: 𝕋 → [0, +∞) be a function in L ₁([a, b)_𝕋). Given ε > 0 there exists δ > 0 such that, if A ∈ Δ and μ _Δ(A) < δ then

Proposition 2.3 (⁽¹⁰⁾ ). Let g ∈ L ₁([a, b)_𝕋). Suppose that

for each c, d ∈ 𝕋 such that c < d. Then g(t) ≥ 0 Δ-a.e. t ∈ [a, b)_𝕋.

2.3 Absolutely continuous functions on time scales

A function f: 𝕋 → ℝⁿ is absolutely continuous if given ε > 0 there exists δ > 0 such that

whenever a_i ≤ b_i and {[a_i, b_i )_𝕋} are disjoint intervals satisfying

The next theorem is an immediate consequence of [², Theorem 4.1].

Theorem 2.2. A function f: 𝕋 → ℝⁿ is absolutely continuous if and only if the following assertions are valid:

f is 11-differentiable Δ-a.e. on [a, b)_𝕋 and f ^Δ ∈ L ₁([a, b)𝕋, ℝⁿ );

for each t ∈ 𝕋 we have

A function f: 𝕋 → ℝⁿ is said to be an arc if it is absolutely continuous. We denote the set of all arcs with domain 𝕋 and taking values on ℝⁿ by AC(𝕋, ℝⁿ ).

Proposition 2.4 (⁽¹⁰⁾ ). Consider the sequence x_i : 𝕋 → ℝⁿ of arcs and a function ϕ: 𝕋 → [0, +∞) in L ₁([a, b)𝕋). Suppose that for each i, we have

If {x_i (a)} is a bounded sequence, then there exists a subsequencex_i } and an arc x: 𝕋 → ℝⁿ such thatx. Furthermore, if c, d ∈ 𝕋 and c < d then ⊂ {

Proposition 2.5 (⁽¹⁰⁾ ). Let x: 𝕋 → ℝⁿ be an arc and let k: 𝕋 → [0, +∞) be a function in L ₁([a, b)𝕋). Suppose

with γ₁ ≥ 0. Then

for all t ∊ 𝕋.

2.4 Set-valued functions properties

Here we consider basic results of measurable set-valued functions.

Let (Ω, ℱ) be a measurable space. A set-valued function E: Ω ⁿ is said to be ℱ-measurable if the set ℝ

E⁻¹(V) = {x ∈ Ω : E(x) ∩ V ≠ Ø}

is ℱ-measurable for all compact sets V ⊂ ℝⁿ .

A set-valued function E is said to be closed, compact, convex or nonempty when its image E(x) obeys the required property, each point x ∈ Ω.

Definition 2.1. Take U: 𝕋 ^m . We define the set-valued functiona, b] ^m by ℝ ℝ: [

Lemma 2.2 (⁽¹¹⁾ ). Let U: 𝕋 ^m be a Δ-measurable set-valued function. Then the set-valued functiona, b] ^m is Lebesgue measurable. ℝ ℝ: [

Similarly to [¹²,Theorem 2.3.14], we have the proposition below.

Proposition 2.6. Consider a function g: [a, b] × ℝ^m → ℝ and a closed nonempty set-valued function Γ : [a, b] ^m . Assume that ℝ

g is a Carathéodory function;

Γ is a Lebesgue measurable set-valued function.

Let η: [a, b]→ be defined by

Then η is a Lebesgue measurable function.

Using Proposition 2.1 and Lemma 2.2, we get the following consequence of the Proposition 2.6.

Lemma 2.3. Consider a function g: 𝕋 × ℝ^m → ℝ and a closed nonempty set-valued function U: 𝕋 ^m . Assume that ℝ

g is Δ-measurable at t , for each fixed (x, u), and continuous at (x, u), for each fixed t ;

U is a Δ-measurable set-valued function.

Then the function η: 𝕋 → given by

is a Δ-measurable function.

Take a function f: 𝕋 × ℝⁿ × ℝ^m → ℝⁿ and a set-valued function U: 𝕋 ^m . Consider the following hypotheses: ℝ

(H1) f is continuous at (x, u), for each fixed t , and Δ-measurable at t , for each fixed (x, u).

(H2) U is a compact, nonempty and Δ-measurable set-valued function.

We define the function H: 𝕋 × ℝⁿ × ℝⁿ → ℝ as

If x: 𝕋 → ℝⁿ is a 11-measurable function and p ∈ ℝⁿ , it is a consequence of Lemma 2.3 that

is a Δ-measurable function.

2.5 Filippov's selection theorem

Consider the control system of equations

(2.1)

If the control process pair (x, u) obeys (2.1) then x obeys the following dynamical inclusion

(2.2)

Similar to [¹¹,Theorem 4.8] we have the following extension of Filippov's selection theorem.

Theorem 2.3. Consider a function f: 𝕋 × ℝⁿ × ℝ^m → ℝⁿ and a set-valued function U: 𝕋 ^m . Assume that the function f and the set-valued function U satisfy the hypothesis (H1) and (H2). ℝ

If x ∈ AC(𝕋, ℝⁿ ) obeys(2.2)then there is a Δ-measurable selection u of U such that the process (x, u) obeys(2.1).

3 OPTIMAL CONTROL PROBLEMS IN TIME SCALES

Here we get the existence of optimal controls to optimal control problems on time scales. Consider the optimal control problem on time scales

where A, C ⊂ ℝⁿ , x ∈ AC(𝕋, ℝⁿ ), u: 𝕋 → ℝ^m is a Δ-measurable function, g: ℝⁿ × ℝⁿ → ℝ, f: 𝕋 × ℝⁿ × ℝ^m → ℝⁿ and U: 𝕋 ^m is a set-valued function. ℝ

Consider the following hypotheses:

(H3) A is a compact set and C is a closed set.

(H4) g is a lower semicontinuous function.

(H5) There exists K > 0 such that for any t ∈ 𝕋, one has

ǁf (t, x, u) − f (t, y, u)ǁ ≤ K ǁx − yǁ

for any u ∈ U(t), and any x, y ∈ ℝⁿ .

(H6) There are positive constant γ > 0 and a function c: 𝕋 → [0, ∞) in L ₁([a, b)_𝕋) such that

for all x ∈ ℝⁿ and all u ∈ U(t).

(H7) The set f (t, x, U(t)) is convex for each t ∈ 𝕋 and x ∈ ℝⁿ .

(H8) Let j > 0. Suppose that

and

Let u: 𝕋 → ℝ^m be a Δ-measurable function and x ∈ AC(𝕋, ℝⁿ ). We say that (x, u) is an admissible process for (P), if the pair (x, u) satisfies (2.1) and x obeys the condition (x(a), x(b)) ∈ A × C. A process P), if it is an admissible process for (P) that satisfies is called an optimal process for (

for all admissible process (x, u) of (P).

Theorem 3.4 ensures the existence of optimal processes for (P). We get the Theorem 3.4 using the next two lemmas.

Lemma 3.4. Assume (H8). Let f: 𝕋 × ℝⁿ × ℝ^m → ℝⁿ be a function obeying (H6). If (x, u) satisfies(2.1)then

Proof. Let t ∈ [a, b)_𝕋 be such that

x^Δ(t) = f (t, x(σ(t)), u(t)).

If σ (t) = t we have

and if σ (t)> t we get

since

and

and then

Therefore

□

Lemma 3.5. Assume (H8) and consider a function f: 𝕋 × ℝⁿ × ℝ^m → ℝⁿ obeying (H6). Take a sequence (x_i, u_i ) satisfying(2.1). If {x_i (a)} is a bounded sequence, then there exists a subsequencex_i } and an arc x: 𝕋 → ℝⁿ such thatx. ⊂ {

Proof. Since

it follows from Lemma 3.4 that

Let γ ₁: = jγ and let k(t) := (1 + γ (b − a)j)c(t). Consider a real number L > 0 obeying

From Proposition 2.5 we get

for all t ∈ 𝕋. Then

Now, using Proposition 2.4 we conclude the proof. □

Theorem 3.4. Assume that (H 1) − (H 8) are satisfied for the optimal control problem (P). If the problem (P) has an admissible process, then there exists an optimal process (x, u) for (P).

Proof. Let inf{P} denote the greatest lower bound on g(x(a), x(b)) over admissible processes (x, u) of (P). Thence there exists a sequence of admissible processes (x_i, u_i ) of (P) obeying

From Lemma 3.5 there exists a subsequence of {x_i }, we do not relabel, such that {x_i } converges uniformly for an arc x, that is, x_i x.

Now we prove that there is a Δ-measurable selection u of U such that (x, u) is an admissible process for (P).

Let t, s ∈ 𝕋 be such that t ≤ s. We have

By (H5) we get

and thus

Note that

and that

because of the dominated convergence theorem ⁽⁽⁹⁾⁾. Using Proposition 2.4, we obtain

Then

(3.1)

Let p ∈ ℝⁿ be arbitrarily fixed. Then by (3.1) we have

For any i we also have

and therefore

From Proposition 2.3 we deduce that

that is,

As p is arbitrary, we can use standard procedures (e.g., ¹², p. 91) which involves the separability of ℝⁿ and the geometric Hahn-Banach separation theorem for convex sets, to obtain

Theorem 2.3 ensures the existence of a Δ-measurable selection u of U such that the process (x, u) satisfies (2.1).

Since A × C is a closed set we deduce that (x(a), x(b)) ∈ A × C. Hence (x, u) is an admissible process for (P).

Finally, since

we may conclude that (x, u) is an optimal process for (P). □

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Publication Dates

• Publication in this collection
  Jan-Apr 2016

History

• Received
  21 June 2015
• Accepted
  17 Feb 2016