Time optimal controls for fractional differential systems with Riemann-Liouville derivatives

Definition 2.1

A function y ∈ C_1−γ([0, c], X) is called the mild solution of (1.1) if

for each t ∈ (0, c] and u ∈ U_ad, where

Remark 2.1

The function ω_γ(⋅) is an one-side stable probability density defined on (0, ∞) (see [22]), whose Laplace transformation satisfies

The function h_γ is a probability density function defined on (0, ∞) satisfying

We now list all the assumptions which will be applied in the whole paper.

(HA) The linear closed and densely defined operator A on X generates a compact C₀ semigroup {T(t)}_{t > 0}, and set M:=supt∈[0,c]‖T(t)‖<+∞.

(Hg) (1) g(t, y) is measurable in t on [0, c] for all y ∈ X, and continuous in y on X for a.e. t ∈ [0, c].

(2) For all y ∈ X and a.e. t ∈ [0, c], there exist a function η ∈ L^p([0, c], ℝ⁺) and a constant ρ with p > 1γ and 0<ρ<Γ(1+γ)cM such that

(HB) B ∈ L^∞([0, c], 𝓛(Y, X)).

The set U_ad for control functions is defined as

where p > 1γ, and the multivalued map U : [0, c] → P_f(Y) (the set of all nonempty closed and convex subset of Y) satisfies the following condition (HU).

(HU) (1) U(⋅) is graph measurable.

(2) For a.e. t ∈ [0, c], there exists a function m ∈ L^p([0, c], ℝ⁺) with p > 1γ such that

A noteworthy fact in [9] is that (HU) implies that U_ad ≠ ∅, and obviously U_ad is bounded, closed and convex. Moreover, ∥u∥_L^p ≤ ∥m∥_L^p and Bu ∈ L^p([0, c], X) for p > 1γ and u ∈ U_ad.

The following properties play an important role in this paper.

Lemma 2.1

Let 0 < γ < 1 and (HA) be satisfied. Then, the operator 𝓢_γ(t) (t ≥ 0) defined in Definition 2.1 satisfies:

1. {𝓢_γ(t)}_{t ≥ 0} ⊆ 𝓛(X), and for each x ∈ X and t ≥ 0, there holds:

   ‖Sγ(t)x‖≤γMΓ(1+γ)‖x‖.(2.2)

2. for each x ∈ X, 𝓢_γ(⋅)x ∈ C([0, c], X).

3. for each t > 0, 𝓢_γ(t) is a compact operator on X.

4. the operator 𝓢_γ(t) is continuous in the uniform operator topology for t > 0.

5. limh→0+‖1Γ(γ)Sγ(t+h)−Sγ(t)Sγ(h)‖=0,    t>0,        limh→0+‖1Γ(γ)Sγ(t)−Sγ(h)Sγ(t−h)‖=0,    t>0.

Proof

For the properties (1) (2) and (3), we refer to [33] for details. We now verify property (4). For 0 < t₁ < t₂, there exist positive numbers ϱ and N such that

The compactness of T(t) (t > 0) yields ‖T(t2γθ)−T(t1γθ)‖→0 as t₁ → t₂ and θ ∈ [ϱ, N]. This together with the arbitrariness of ϱ and N as well as the fact ∫0∞θhγ(θ)dθ=1Γ(1+γ) gives that ∥𝓢_γ(t₂) − 𝓢_γ(t₁)∥ → 0 as t₁ → t₂.

For property (5), a similar manner as did in [6] gives the conclusion. This completes the proof. □

Lemma 2.2

If (HA) holds, then the operator F : L^p([0, c], X) → C_1−γ([0, c], X) given by

is compact for p > 1γ.

Proof

For fixed r > 0, let B_L^p(r) = {h ∈ L^p([0, c], X) : ∥h∥_L^p ≤ r}. We will show that the set {Fh : h ∈ B_L^p(r)} ⊆ C_1−γ([0, c], X) is precompact, that is {⋅^1−γFh(⋅) : h ∈ B_L^p(r)} ⊆ C([0, c], X) is precompact.

Firstly, we verify that {⋅^1−γFh(⋅) : h ∈ B_L^p(r)} is equicontinuous. Let 0 ≤ t₁} < t₂} ≤ c and h ∈ B_L^p(r). If t₁ = 0,

as t₂ → 0. If t₁ > 0, for ϱ > 0 small enough with t₁ − ϱ > 0, one has

as t₂ → t₁, due to the uniform continuity of 𝓢_γ(t), t > 0 and the arbitrariness of ϱ.

Secondly, for each t ∈ [0, c], we prove that {t^1−γFh(t) : h ∈ B_L^p(r)} is precompact in X. If t = 0, the conclusion is obvious. If t ∈ (0, c], for each ε > 0 with t − 2ε > 0, define the set {t^1−γF^εh(t) : h ∈ B_L^p(r)} in X, where

Taking into account the compactness of 𝓢_γ(ε), we obtain that the set {t^1−γF^εh(t) : h ∈ B_L^p(r)} is precompact in X. Moreover, for each h ∈ B_L^p(r), we have

as ε → 0 by using the property (5) of Lemma 2.1. Then, the set {t^1−γFh(t) : h ∈ B_L^p(r)} is precompact in X owning to the fact that the precompact set {t^1−γF^εh(t) : h ∈ B_L^p(r)} in X is close arbitrarily to it.

Finally, applying Ascoli-Arzela theorem, one gets that {⋅^1−γFh(⋅) : h ∈ B_L^p(r)} is precompact in C([0, c], X), which means that {Fh(⋅) : h ∈ B_L^p(r)} is precompact in C_1−γ([0, c], X). Then, we can come to the conclusion that F is compact. This completes the proof. □

Theorem 3.1

Let all the hypotheses listed in Section 2 be fulfilled. Then, for each u ∈ U_ad, system(1.1)possesses at least one mild solution in fC_1−γ([0, c], X).

Proof

For each y₀ ∈ X and u ∈ U_ad, consider the operator 𝓠 : C_1−γ([0, c], X) → C_1−γ([0, c], X) as follows:

It is not difficult to verify that 𝓠 is well defined. Then, the solvability of system (1.1) will be transformed into a fixed point problem of 𝓠. For clarity, we proceed into the following steps.

Step 1. Let r > 0 and B_C1−γ(r) = {y ∈ C_1−γ([0, c], X) : ∥y∥_C1−γ ≤ r}. We show that 𝓠B_C1−γ(r) ⊆ B_C1−γ(r) provided that

In fact, for each y ∈ B_C1−γ(r), one has

that is,

Step 2. We show that 𝓠 is continuous on B_C1−γ(r). To this end, let {y_n}_{n ≥1} ⊆ B_C1−γ(r) with limn→∞yn=y∈BC1−γ(r), that is,

as n → ∞. This yields t^1−γy_n(t) → t^1−γy(t) as n → ∞ uniformly for t ∈ [0, c]. Note that, for a.e. τ ∈ [0, t],

as n → ∞, and

where η(⋅) ∈ L^p}([0, c], ℝ⁺). Then, by applying the Lebesgue dominated convergence theorem, one has

as n → ∞ uniformly for each t ∈ [0, c], which means that

Step 3. We check the compactness of 𝓠 on C_1−γ([0, c], X). The definition of 𝓠 yields that the compactness of 𝓠 is reduced to the compactness of 𝓠̃ on C_1−γ([0, c], X), where

for each y ∈ C_1−γ([0, c], X). It should be point out that (Hg)(2) implies that g(⋅, y(⋅)) ∈ L^p([0, c], X). A similar manner utilized in Lemma 2.2 gives the compactness of 𝓠̃.

Now, it is obvious that the conclusion of Theorem 3.1 holds by using the Schauder’s fixed point theorem. □

Remark 3.1

By virtue of Theorem 3.1, for each u ∈ U_ad, let y^u ∈ C_1−γ([0, c], X) be any one of the corresponding mild solutions of system (1.1). Then, ∥y^u}∥_C1−γ ≤ R, where

which is independent of u, and Eγ(z)=∑n=0∞znΓ(nγ+1) is the Mittag-Leffler function. In fact, for each t ∈ [0, c], one has

By using Corollary 2 of [31], one can obtain that

This means that

Remark 3.2

For simplicity, we denote by

Remark 3.3

A pair (y^u, u) is said to be feasible for the system (1.1) if and only if (y^u, u) ∈ 𝓐_d.

Remark 4.1

In general case, the subset AdWT is defined as:

However, since the solution is obtained in the space C_1−γ([0, c], X) and y^u(t) is indeed unbounded near the zero, a rescaling technique is necessary and a reasonable definition of AdWT in case of Riemann-Liouville derivatives is given as above.

Remark 4.2

For each (y^u, u) ∈ AdWT, the definition of transition time gives

where 𝓣(y^u, u) := {t : t ∈ [0, c], t^1−γy^u(t) ∈ W_T}. We claim that t_{(y^u, u)} is well defined. In fact, if the set 𝓣(y^u, u) contains finite elements, the proof is trivial. Otherwise, let t̃ = inf 𝓣(y^u, u). This gives that

for some decreasing {t_n}_{n ≥ 1} ⊆ 𝓣(y^u, u), that is, tn1−γyu(tn)∈WT. The fact ⋅^1−γy^u(⋅) ∈ C([0, c], X) yields

This together with the closeness of W_T gives

This means that t̃ = min 𝓣(y^u, u), and t_{(y^u, u)} = t̃.

Based on the above definitions and notations, now we consider the time optimal control problem (P): Find (y_*, u_*) ∈ AdWT such that

If the control u_*, the time t_{(y*, u*)} and the pair (y_*, u_*) exist solving problem (P), we call them time optimal control, optimal time and time optimal pair, respectively.

Theorem 4.1

Assume that all the hypotheses given in Section 2 are satisfied. Then, problem (P) possesses at least one time optimal pair.

Proof

In view of Theorem 3.1, there exists at least one y^u ∈ B_C1−γ(R) such that (y^u, u) ∈ 𝓐_d for each u ∈ U_ad. We will proceed in the following two steps to derive the main result.

Step 1. For each u ∈ U₀, set tu=infyu∈SuWTt(yu,u). We now need to check that tu1−γy^u(tu)∈WT for some y^u∈SuWT. It is trivial in situation in which the set SuWT has finite elements. Otherwise, there is a monotone decreasing sequence {t(ynu,u)}n≥1 such that

where (ynu,u)∈AdWT for each n ≥ 1. Moreover, the definition of t(ynu,u) gives

The fact ynu ∈ S(u) yields

for each n ≥ 1 and t ∈ (0, c]. Exploiting the compactness of 𝓢_γ(t), t > 0, a similar method used in Lemma 2.2, we can infer that {ynu}n≥1 is precompact in C_1−γ([0, c], X). Then, there is a subsequence of {ynu}n≥1, still relabled by it, and a function ŷ^u ∈ B_C1−γ(R), such that

as n → ∞. This together with (Hg) gives that

for a.e. τ ∈ [0, t], where η(⋅) ∈ L^p([0, c], ℝ⁺). Now, taking n → ∞ to both sides of (4.3) and using Lebesgue dominated convergence theorem yield

for each t ∈ (0, c]. This gives that

It is worth noticing that (4.1) and (4.4) lead to

as n → ∞. In fact,

It follows from (4.4) that ‖t(ynu,u)1−γynu(t(ynu,u))−t(ynu,u)1−γy^u(t(ynu,u))‖→0 as n → ∞. The fact ⋅^1−γŷ^u(⋅) ∈ C([0, c], X) and (4.1) give ‖t(ynu,u)1−γy^u(t(ynu,u))−tu1−γy^u(tu)‖→0 as n → ∞. (4.2) and (4.7), together with the closeness of W_T give rise to the fact that

Combining this with (4.6) yields y^u∈SuWT.

Step 2. Put t∗=infu∈U0tu, where t_u is the optimal time for fixed u in Step 1. Our task now is to seek an u_* ∈ U₀ and y∗∈Su∗WT such that t∗1−γy∗(t∗)∈WT. The proof is trivial provided that U₀ contains finite elements, or there exists a monotone decreasing sequence {t_un}_{n ≥ 1} such that

According to Step 1, for each n ≥ 1, we can deduce that there is a function y^un∈SunWT such that

and

for each n ≥ 1 and t ∈ (0, c]. Note that B(⋅)u_n(⋅) ∈ L^p([0, c], X). So, an argument similar with the one employed in Lemma 2.2 gives rise to the precompactness of {ŷ^u_n}_{n ≥ 1} in C_1−γ([0, c], X). Then, a subsequence of {ŷ^u_n}_{n ≥ 1} can be extracted, and still denoted by it, which satisfies