In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order 1<r<2. Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems.
Citation: Marimuthu Mohan Raja, Velusamy Vijayakumar, Anurag Shukla, Kottakkaran Sooppy Nisar, Wedad Albalawi, Abdel-Haleem Abdel-Aty. A new discussion concerning to exact controllability for fractional mixed Volterra-Fredholm integrodifferential equations of order r∈(1,2) with impulses[J]. AIMS Mathematics, 2023, 8(5): 10802-10821. doi: 10.3934/math.2023548
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In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order 1<r<2. Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems.
Fractional derivatives have lately emerged as a very significant area of research as a result their steadily increasing numerous uses in several sectors of applied science and engineering. For more information, see the books [11,16,31,47,48]. Fractional differential systems produce a better and more accurate realistic scenario for understanding a wide variety of physical phenomena as compared to differential equations represented by regular integer order derivatives. Integer order definitions can be interpolated to non-integer order using a variety of techniques. Riemann-Liouville and Caputo derivatives are two of the most well-known. As a consequence, a large number of scholars have lately played an important role to fields like electromagnetic theory, rheology, image analysis, diffusion, data processing, porous materials, physiological engineering challenges, fluid mechanics, theology, and many others, see [4,9,10,34,35,40,41].
Recently, [5,39,45] the authors discussed some important methods for long-time anomalous heat flow study to the fractional derivatives, Laplace transformation, singular boundary approach, and dual reciprocity technique. Recently, the study of impulsive functional differential systems has offered a natural framework for the mathematical modelling of a variety of practical situations, particularly in the fields of control, biology, and medicine. The explanation for this applicability is that impulsive differential issues are a suitable model for explaining changes that occur their state rapidly at some points and can't be represented using traditional differential equations. For additional information on this theory as well as its implications, we suggest reading [7,37,38,42,43].
Mathematical control theory, a subfield of framework mathematics, focuses on the basic ideas that underlie the formulation and evaluation of control systems. While occasionally appearing to move in opposite directions, the two main study fields in control theory have typically been complementary. One of them assumes that a proper model of the object to be managed is offered and that the user wants to improve the behaviour of the object in some way. For instance, to design a spacecraft's trajectory that minimizes the overall trip time or fuel consumption, physical concepts and engineering standards are applied. The methods used here are strongly related to other branches of optimization theory as well as the classical calculus of variations; the result is typically a pre-programmed flight plan. The limitations imposed by uncertainty regarding the model or the environment in which the item operates provide the basis of the other key area of research. The use of feedback to correct for deviations from the anticipated behaviour in this case is the key strategy.
It is essential to carry out study on the consequences of such controllability of systems utilizing the resources at hand. As controllability is crucial to this theory, it makes sense to seek to generalize finite-dimensional control theory to infinite dimensions. Control system analysis and innovation have proven to benefit from the usage of controllability notation by employing fractional derivatives. It is used in a multitude of industries, as well as biochemistry, physicists, automation, electronics, transport, fields of study include economics, robotics, biology, physics, power systems, chemical outgrowth control, space technology and technology. As indicated by the researchers' papers [7,9,13,29,30,33,49], resolving these challenges has become a important undertaking for young researchers.
In addition, integrodifferential equations are used in many technological fields, including control theory, biology, medicine, and ecology, where a consequence or delay must be considered. In fact, it is always necessary to characterize a model with hereditary properties using integral-differential systems. Further, many researchers done the fruitful achievements in fractional Volterra-Fredholm integrodifferential systems with or without delay utilizing the mild solutions, semigroup theory, neutral systems, and fixed point theorems in [14,15,17,18,19,21,22,33]. In [1,25,26,27,28,46], the authors discussed the solution of a functional derivatives utilizing to weak and strong convergence, Chebyshev quadrature collocation algorithm, mixed Volterra-Fredholm type, almost contraction mapping, the iterative method, weak w2-stability, and faster iteration method.
In [35,36], the researchers established the existence and uniqueness for fractional differential equations of α∈(1,2) by applying the upper and lower solution methods, sectorial operators, and nonlocal conditions. The authors [6,49] established Caputo fractional derivative of 1<α<2 using nonlocal conditions, the Laplace transform, mild solutions, cosine families, measure of noncompactness(MNC), as well as other fixed point techniques. Additionally, using fractional derivatives, cosine functions, and Sobolev type, the authors discussed exact controllability outcomes for fractional differential systems of (1,2) with finite delay in [13].
In [18], the authors explored the approximate controllability of Caputo fractional differential systems of (1,2) by utilizing the impulsive system, sequence method, and cosine families. Furthermore, [40] proved fractional integrodifferential inclusions of (1,2) using Laplace transforms, Fredholm integro-differential systems, and the fixed point approach. Moreover, in [29,37], the researchers looked at Gronwall's inequality for the semilinear fractional derivatives of (1,2], stochastic systems, asymptotic stability, optimal control concerns, Lipschitz continuity, and impulsive systems. The researchers are currently investigating the optimal controls for fractional derivative of (1,2) with infinite delay in [19,20].
In [35], the authors looked into the existence and uniqueness outcomes of fractional differential equations of (1,2). In [12,32], the fixed point theorem, Gronwall's inequality, impulsive systems, and sectorial operators are used to analyze optimal control for fractional derivatives of order (1,2). To identify extremal solutions of fractional partial differential equations of order (1,2), the authors of [36] used upper and lower solution approaches, sectorial operators, the Mittag-Leffler function, and mild solutions. The existence of positive mild solutions for Caputo fractional differential systems of order r∈(1,2) was also addressed by the authors in [34].
Taking inspiration from the preceding information, let's investigate impulsive fractional integrodifferential systems of mixed type with order r∈(1,2) with the following form:
{CDrϱz(ϱ)=Az(ϱ)+f(ϱ,z(ϱ),(E1z)(ϱ),(E2z)(ϱ))+Bx(ϱ), ϱ in V, ϱ≠ϱj,Δz(ϱj)=mj, Δz′(ϱj)=˜mj, j=1,2,⋯,n,z(0)=z0, z′(0)=z1. | (1.1) |
In the above
(E1z)(ϱ)=∫ϱ0e1(ϱ,s,z(s))ds, (E2z)(ϱ)=∫σ0e2(ϱ,s,z(s))ds, |
with CDrϱ represents Caputo fractional derivative of order r in (1,2); A maps from D(A)⊂Q into Q denotes the sectorial operator of type (P,τ,r,ϕ) on the Banach space Q; the function f maps from [0,σ]×Q×Q×Q into Q and e1,e2:S×Q×Q→Q are appropriate functions, in which S={(ϱ,s):0≤s≤ϱ≤σ}. The bounded linear operator B:Y→Q, the control function x in L2(V,Y), in which Y is also a Banach space. The continuous functions mj, ˜mj:Q→Q and 0=ϱ0<ϱ1<ϱ2<⋯<ϱj<⋯<ϱn=σ; Δz(ϱj)=z(ϱ+j)−z(ϱ−j), where z(ϱ+j)=limϵ+→0z(ϱℓ+ϵ) and z(ϱ−j)=limϵ−→0z(ϱℓ+ϵ) denote the right and lift limits of z(ϱ) at ϱ=ϱj. Δz′(ϱj) has also a similar theories.
The following sections represent the remaining portions of this article: Section 2 starts with a description of some basic concepts and the results of the preparation. In Section 3, we look at the existence of mild solutions for the impulsive fractional Volterra-Fredholm type (1.1). Lastly, an application for establishing the theory of the primary results is shown.
We will implement some definitions, sectorial operator assumptions, R-L and Caputo fractional derivative definitions, and preliminaries in this section, which will be used throughout the article.
The Banach space C(V,Q) maps from V into Q is a continuous with ‖z‖C=supϱ∈V‖z(ϱ)‖.
PC(V,Q)={z:V→R:z∈C((ϱj,ϱj+1],R), j=0,⋯n and ∃ z(ϱ+j) and z(ϱ−j),j=1,⋯,n with z(ϱ−j)=z(ϱj)}, |
with ‖z‖PC=supϱ∈V‖z(ϱ)‖. Consider L(Q) represents the Banach space of every linear and bounded operators on Banach space Q.
Definition 2.1. [31] The integral fractional order β with such a lower limit of zero for f maps from [0,∞) into R+ is simply referred to as
Iβf(ϱ)=1Γ(β)∫ϱ0f(s)(ϱ−s)1−βds,ϱ>0, β∈R+. |
Definition 2.2. [31] The fractional order β of R-L derivative with the lower limit of zero for f is known as
LDβf(ϱ)=1Γ(j−β)djdϱj∫ϱ0f(s)(ϱ−s)j−β−1ds,ϱ>0, β∈(j−1,j), β∈R+. |
Definition 2.3. [31] The fractional derivative of order β in Caputo's approach with the lower limit zero for f is designated just for
CDβf(ϱ)=LDβ(f(ϱ)−j−1∑i=0f(i)(0)i!ϱi),ϱ>0, β∈(j−1,j), β∈R+. |
Definition 2.4. [35] The closed and linear operator A is called the sectorial operator of type (P,τ,r,ϕ) provided that there exists ϕ in R, τ in (0,π2), and there exists a positive constant P such that the r-resolvent of A exists outside the sector
ϕ+Sτ={τ+ρr:ρ in C(V,Q), |Arg(−ρr)|<τ}, | (2.1) |
and
‖(ρrI−A)−1‖≤P|ρr−ϕ|, ρr∉ϕ+Sτ. |
It is also simple to prove that A represents the infinitesimal generator of an r-resolvent family {Gr(ϱ)}ϱ≥0 in Banach space if one assumes A stands for a sectorial operator of type (P,τ,r,ϕ), where
Gr(ϱ)=12πi∫ceρrR(ρr,A)dρ. |
Definition 2.5. A function z in PC(V,Q) is called a mild solution of the system (1.1) provdied that
z(ϱ)={Nr(ϱ)z0+Mr(ϱ)z1+∫ϱ0Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))ds+∫ϱ0Gr(ϱ−s)Bx(s)ds, 0≤ϱ≤ϱ1,Nr(ϱ)z0+Mr(ϱ)z1+∑jq=1Nr(ϱ−ϱj)mj+∑jq=1Mr(ϱ−ϱj)˜mj+∫ϱ0Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))ds+∫ϱ0Gr(ϱ−s)Bx(s)ds, ϱj<ϱ≤ϱj+1, | (2.2) |
where
Nr(ϱ)=12πi∫ceρrρr−1R(ρr,A)dρ, Mr(ϱ)=12πi∫ceρrρr−2R(ρr,A)dρ,Gr(ϱ)=12πi∫ceρrR(ρr,A)dρ, |
with c being a suitable path such that ρr∉ϕ+Sτ for ρ belongs to c.
We consider now definition of exact controllability.
Definition 2.6. The system (1.1) is said to be controllable on V iff for all z0,z1,zw in Q, there exists x∈L2(V,Y) such that a mild solution z of (1.1) fulfills z(σ)=zw.
Let us recall some notations about the measure of noncompactness (see [2,3]).
Definition 2.7. The Hausdorff MNC δ discovered on for every bounded subset θ of Q by
δ(θ)=inf{κ>0:Afinitenumberofballswithradiismallerthan κ cancover θ}. |
Definition 2.8. [7] Suppose that Q+ is the positive cone of an ordered Banach space (Q,≤). The value of Q+ is called MNC on Q of N defined on for any bounded subsets of the Banach space Q iff N(¯co θ)=N(θ) for every θ⊆Q, in which ¯co θ denotes the closed convex hull of θ.
Definition 2.9. [2,8] For every bounded subsets θ,θ1,θ2 of Q.
(i) monotone iff for every bounded subsets θ, θ1, θ2 of Q we obtain: (θ1⊆θ2)⇒(N(θ1)≤N(θ2));
(ii) non singular iff N({b}∪θ)=N(θ) for every b in Q, θ⊂Q;
(iii) regular iff N(θ)=0 iff θ in Q, which is relatively compact;
(iv) δ(θ1+θ2)≤δ(θ1)+δ(θ2), where θ1+θ2={u+v:u in θ1,v in θ2};
(v) δ(θ1∪θ2)≤max{δ(θ1),δ(θ2)};
(vi) δ(βθ)≤|β|δ(θ), for every β∈R;
(vii) If the Lipschitz continuous function T maps from G(T)⊆Q into Banach space X along with l>0, then δX(Tθ)≤lδ(θ), for θ⊆G(T).
Lemma 2.10. [2] If P subset of C([b,σ],Q) is bounded and equicontinuous, in addition, δ(P(ϱ)) is continuous for all b≤ϱ≤σ and
δ(P)=sup{δ(P(ϱ)),σ∈[b,σ]},wherebyP(ϱ)={u(ϱ):z∈P}⊆Q. |
Lemma 2.11. [24] Suppose that the functions {yv}∞v=1 is a sequence of Bochner integrable from V→Q including the evaluation ‖yv(ϱ)‖≤δ(ϱ), for every ϱ in V and for all k≥1, where δ∈L1(V,R), then the function α(ϱ)=δ({yv(ϱ):v≥1}) in L1(V,R) and fulfills
δ({∫ϱ0yv(s)ds:v≥1})≤2∫ϱ0α(s)ds. |
Now, we consider the some conditions of sectorial operator of type (P,τ,r,ϕ).
Theorem 2.12. [35,36] Assume that A is a sectorial operator of type (P,τ,r,ϕ). In addition, the subsequent on ‖Nr(ϱ)‖ hold:
(i) For ζ∈(0,π), and suppose that ϕ≥0, we get
‖Nr(ϱ)‖≤M1(τ,ζ)Pe[M1(τ,ζ)(1+ϕϱr)]12[(1+sinζsin(ζ−τ))1r−1]πsin1+1rτ(1+τϱr)+Γ(r)Pπ(1+ϕϱr)|cosπ−ζr|rsinτsinζ, |
for ϱ>0, and M1(τ,ζ)=max{1,sinζsin(ζ−τ)}.
(ii) For ζ∈(0,π), and suppose that ϕ<0, we get
‖Nr(ϱ)‖≤(eP[(1+sinζ)1r−1]π|cosζ|1+1r+Γ(r)Pπ|cosζ||cosπ−ζr|r)11+|ϕ|ϱr, ϱ>0. |
Theorem 2.13. [35,36] Suppose that A is a sectorial operator of type (P,τ,r,ϕ). In addition, the subsequent on ‖Gr(ϱ)‖, and ‖Mr(ϱ)‖ hold:
(i) For ζ∈(0,π) and assume that ϕ≥0. we get
‖Gr(ϱ)‖≤P[(1+sinζsin(ζ−τ))1r−1]πsinτ(1+τϱr)1rϱr−1e[M1(τ,ζ)(1+ϕϱr)]1r+Pϱr−1π(1+ϕϱr)|cosπ−ζr|rsinτsinζ, |
‖Mr(ϱ)‖≤P[(1+sinζsin(ζ−τ))1r−1]M1(τ,ζ)πsinτr+2r(1+τϱr)r−1rϱr−1e[M1(τ,ζ)(1+ϕϱr)]1r+PrΓ(r)π(1+ϕϱr)|cosπ−ζr|rsinτsinζ, |
for ϱ>0, where M1(τ,ζ)=max{1,sinτsin(ζ−τ)}.
(ii) For 0<ζ<π, assume that ϕ<0, we get
‖Gr(ϱ)‖≤(eP[(1+sinζ)1r−1]π|cosζ|+Pπ|cosζ||cosπ−ζr|)ϱr−11+|ϕ|ϱr,‖Mr(ϱ)‖≤(eP[(1+sinζ)1r−1]ϱπ|cosζ|1+2r+rΓ(r)Pπ|cosζ||cosπ−ζr|)11+|ϕ|ϱr, |
for ϱ>0.
Lemma 2.14. [23] Suppose that T is closed convex subset of Q and zero in T, let the continuous function E maps from T into Q and that fulfills Mönch's condition, which is, (N⊆T is countable, N⊆¯co({0}∪E(N))⇒¯N is compact). Hence, E has a fixed point in T.
The existence of mild solutions for the Eq. (1.1) will be shown in this section. The following assumptions are required: It is straightforward to show that they are bounded because of the estimations on Nr(ϱ), Mr(ϱ) and Gr(ϱ) in Theorems 2.12 and 2.13.
(H1) The operators Nr(ϱ), Mr(ϱ), and Gr(ϱ). for every ϱ∈V, there exists a ˆP>0 such that
sup0≤ϱ≤σ‖Nr(ϱ)‖≤ˆP, sup0≤ϱ≤σ‖Mr(ϱ)‖≤ˆP, sup0≤ϱ≤σ‖Gr(ϱ)‖≤ˆP. |
(H2) f:V×Q×Q×Q→Q fulfills:
● (i) f(⋅,χ,u,z) is measurable for every (χ,u,z) in Q×Q and f(ϱ,⋅,⋅,⋅) is continuous for all ϱ∈V, z∈Q, f(⋅,χ,u,z) is strongly measurable;
● (ii) there exists p1∈(0,p) and ς1∈L1p1([0,σ],R+), and ω:R+→R+ is integrable function such that ‖f(ϱ,χ,u,z)‖≤ς1(ϱ)ω(‖χ‖Q+‖u‖+‖z‖), for all (ϱ,χ,u,z) in S×Q×Q×Q, where ω satisfies limv→∞infω(v)v=0;
● (iii) there exists 0<p2<p and ς2∈L1p2(V,R+) such that for every bounded subset S1⊂Y and W1⊂Q,
δ(f(ϱ,W1,S1,S2))≤ς2(ϱ)[δ(W1)+δ(S1)+δ(S2)], for a.e. ϱ∈V, |
W1(φ)={e(φ):e∈W1} and δ is the Hausdorff MNC.
(H3)e1:S×Q×Q→Q satisfies:
● (i) e1(⋅,s,z) is measurable for all (s,z)∈Q×Q, e1(ϱ,⋅,⋅) is continuous for all ϱ∈V;
● (ii) there exists F0>0 such that ‖e1(ϱ,s,z)‖≤F0(1+‖z‖Q), for every ϱ in V, z∈Q;
● (iii) there exists p3∈(0,p) and ς3∈L1p3(V,R+) such that for every bounded subset S3⊂Q,
δ(e1(ϱ,s,S3))≤ς3(ϱ,s)[δ(S3)] for a.e. ϱ∈V, |
with ς∗3=sups∈V∫s0ς3(ϱ,s)ds<∞.
(H4)e2:S×Q×Q→Q satisfies:
● (i) e2(⋅,s,z) is measurable for any (s,z) in Q×Q, e2(ϱ,⋅,⋅) is continuous for all ϱ∈V;
● (ii) there exists F1>0 such that ‖e2(ϱ,s,z)‖≤F1(1+‖z‖Q), for every ϱ∈V, z in Q;
● (iii) there exists p4∈(0,p) and ς4∈L1p4(V,R+) such that for every bounded subset S4⊂S,
δ(e2(ϱ,s,S4))≤ς4(ϱ,s)[δ(S4)] for a.e. σ∈V, |
with ς∗4=sups∈V∫s0ς4(ϱ,s)ds<∞;
(H5) the operator B maps from L2(V,Y) into L1(V,Q) is bounded and W:L2(V,Y)→L1(V,Q) is defined by
Wx=∫σ0G(σ−s)Bx(s)ds, |
fulfills:
(i) There exist a positive contants Pσ, Px such that ‖B‖≤Pσ and ‖W−1‖≤Px when W have an inverse W−1 acquires the value belongs to L2(V,Y)/KerW.
(ii) For p5 in (0,p) and for all bounded subset T∈Q, there exists ς5∈L1p5(V,R+) such that δ(W−1(T)(ϱ))≤ς5(ϱ)δ(T).
Consider the operator Π:PC(V,Q)→PC(V,Q) determined by
(Πz)(ϱ)={Nr(ϱ)z0+Mr(ϱ)z1+∫ϱ0Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))ds+∫ϱ0Gr(ϱ−s)Bx(s)ds, 0≤ϱ≤ϱ1,Nr(ϱ)z0+Mr(ϱ)z1+∑jq=1Nr(ϱ−ϱj)mj+∑jq=1Mr(ϱ−ϱj)˜mj+∫ϱ0Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))ds+∫ϱ0Gr(ϱ−s)Bx(s)ds, ϱj<ϱ≤ϱj+1. | (3.1) |
Theorem 3.1. If (H1)–(H5) are fuflilled. In addition, the system (1.1) is controllable if
2ˆP‖ς2‖[1+2ˆPPσ‖ς5‖](1+(ς∗3+ς∗4))<1. | (3.2) |
Proof. We introduce the control xz(⋅) for arbitrary function z∈PC(V,Q) and using (H5)(i), presented by
xz(ϱ)=W−1{zw−Nr(ϱ)z0−Mr(ϱ)z1−∫ϱ0Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))ds, 0≤ϱ≤ϱ1,zw−Nr(ϱ)z0−Mr(ϱ)z1−∑jq=1Nr(ϱ−ϱj)mj−∑jq=1Mr(ϱ−ϱj)˜mj−∫ϱ0Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))ds, ϱj<ϱ≤ϱj+1. |
We can see that the operator Π provided in (3.1) has a fixed point by using the control mentioned above. Moreover, if Π allows a fixed point, it is simple to deduce that (Πz)(σ)=zσ, that suggests that xz(ϱ) drives the mild solution of (1.1) from the initial state z0 and z1 to the final state zσ in time σ.
Step 1: There exists ℓ>0 such that Π(Gℓ)⊂Gℓ, where Gℓ={z∈PC(V,Q): ‖z‖≤ℓ}. Indeed, if the above assumption is fails, there is a function for every zℓ∈Gℓ and xzℓ∈L2(V,Y) according to Gℓ such that (Πzℓ)(ϱ) not in Gℓ, for every ϱ∈[0,ϱ1], we get
‖xz(ϱ)‖=‖W−1[zw−Nr(ϱ)z0−Mr(ϱ)z1−∫ϱ0Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))ds]‖≤Px[‖zw‖+ˆP‖z0‖+ˆP‖z1‖+ˆP∫ϱ0‖f(s,z(s),(E1z)(s),(E2z)(s))‖ds]≤Px[‖zw‖+ˆP‖z0‖+ˆP‖z1‖+ˆP‖ς1‖L1p1(V,R+)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))]. |
Likewise, for every ϱ∈(ϱj,ϱj+1], j=1,2,⋯,n, we can have
‖xz(ϱ)‖=‖W−1[zw−Nr(ϱ)z0−Mr(ϱ)z1−j∑q=1Nr(ϱ−ϱj)mj−j∑q=1Mr(ϱ−ϱj)˜mj−∫ϱ0Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))ds]‖≤Px[‖zw‖+ˆP‖z0‖+ˆP‖z1‖+ˆPj∑q=1‖mj‖+ˆPj∑q=1‖˜mj‖+ˆP∫ϱ0‖f(s,z(s),(E1z)(s),(E2z)(s))‖ds]≤Px[‖zw‖+ˆP‖z0‖+ˆP‖z1‖+ˆPj∑q=1‖mj‖+ˆPj∑q=1‖˜mj‖+ˆP‖ς1‖L1p1(V,R+)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))]. |
As a result, there exists ȷ1,ȷ2>0 such that
‖xz(ϱ)‖=Px{‖zw‖+ˆP‖z0‖+ˆP‖z1‖+ˆP‖ς1‖L1p1(V,R+)×ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))=ȷ1, 0≤ϱ≤ϱ1,‖zw‖+ˆP‖z0‖+ˆP‖z1‖+ˆP∑jq=1‖mj‖+ˆP∑jq=1‖˜mj‖+ˆP‖ς1‖L1p1(V,R+)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))=ȷ2, ϱj<ϱ≤ϱj+1. |
Using the assumptions (H1)–(H5), and for every ϱ∈[0,ϱ1], we get
ℓ<‖(Πz)(ϱ)‖≤‖Nr(ϱ)z0‖+‖Mr(ϱ)z1‖+∫ϱ0‖Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))‖ds+∫ϱ0‖Gr(ϱ−s)Bxz(s)‖ds≤ˆP‖z0‖+ˆP‖z1‖+ˆP∫ϱ0‖f(s,z(s),(E1z)(s),(E2z)(s))‖ds+ˆP∫ϱ0‖Bxz(s)‖ds≤ˆP‖z0‖+ˆP‖z1‖+ˆP‖ς1‖L1p1(V,R+)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))+ˆPPσPxσ[‖zw‖+ˆP‖z0‖+ˆP‖z1‖+ˆP‖ς1‖L1p1(V,R+)×ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))]. |
Similarly, for every ϱ∈(ϱj,ϱj+1], j=1,2,⋯,n, we can have
ℓ<(Πz)(ϱ)≤‖Nr(ϱ)z0‖+‖Mr(ϱ)z1‖+‖j∑q=1Nr(ϱ−ϱj)mj‖+‖j∑q=1Mr(ϱ−ϱj)˜mj‖+∫ϱ0‖Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))‖ds+∫ϱ0‖Gr(ϱ−s)Bxz(s)‖ds≤ˆP‖z0‖+ˆP‖z1‖+ˆPj∑q=1‖mj‖+ˆPj∑q=1‖˜mj‖+ˆP∫ϱ0‖f(s,z(s),(E1z)(s),(E2z)(s))‖ds+ˆP∫ϱ0‖Bxz(s)‖ds≤ˆP‖z0‖+ˆP‖z1‖+ˆPj∑q=1‖mj‖+ˆPj∑q=1‖˜mj‖+ˆP‖ς1‖L1p1(V,R+)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))+ˆPPσPxσ[‖zw‖+ˆP‖z0‖+ˆP‖z1‖+ˆPj∑q=1‖mj‖+ˆPj∑q=1‖˜mj‖+ˆP‖ς1‖L1p1(V,R+)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))]. |
Therefore
ℓ<{ˆP‖z0‖+ˆP‖z1‖+ˆP‖ς‖L1p1(V,R+)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))+ˆPPσPxσ[‖zw‖+ˆP‖z0‖+ˆP‖z1‖+ˆP‖ς1‖L1p1(V,R+)×ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))], 0≤ϱ≤ϱ1,ˆP‖z0‖+ˆP‖z1‖+ˆP∑jq=1‖mj‖+ˆP∑jq=1‖˜mj‖+ˆP‖ς1‖L1p1(V,R+)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))+ˆPPσPxσ[‖zw‖+ˆP‖z0‖+ˆP‖z1‖+ˆP∑jq=1‖mj‖+ˆP∑jq=1‖˜mj‖+ˆP‖ς1‖L1p1(V,R+)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))], ϱj<ϱ≤ϱj+1. | (3.3) |
If we divide (3.3) by ℓ, and assuming ℓ→∞. In addition, by (H2)(ii), one can get 1≤0. This is a contradiction. Therefore, ℓ>0, Π(Gℓ)⊆Gℓ.
Step 2: We verify that Π is continuous on PC(V,Q). For such a study, let z(v) tends to z∈PC(V,Q). there exists ℓ>0 such that ‖z(v)(ϱ)‖≤ℓ for any v and ϱ∈V, so z(v)∈PC(V,Q) and z∈PC(V,Q). By the hypotheses (H2)–(H5), we get
Fv(s)=f(s,z(s),(Fz(v))(s),(Hz(v))(s)),F(s)=f(s,z(s),(E1z)(s),(E2z)(s)). |
From Lebesgue's dominated convergence theorem, we get
∫ϱ0‖Fv(s)−F(s)‖ds→0 as v→∞, ϱj<ϱ≤ϱj+1. |
Then,
‖(Πz(v))(ϱ)−(Πz)(ϱ)‖=‖Nr(ϱ)z0+Mr(ϱ)z1+j∑q=1Nr(ϱ−ϱj)mj+j∑q=1Mr(ϱ−ϱj)˜mj+∫ϱ0Gr(ϱ−s)Fv(s)ds+∫ϱ0Gr(ϱ−s)Bxz(v)(s)ds−Nr(ϱ)z0−Mr(ϱ)z1−j∑q=1Nr(ϱ−ϱj)mj−j∑q=1Mr(ϱ−ϱj)˜mj−∫ϱ0‖Gr(ϱ−s)F(s)ds−∫ϱ0Gr(ϱ−s)Bxz(s)ds‖≤∫ϱ0‖Gr(ϱ−s)[Fv(s)−F(s)]‖ds+∫ϱ0‖Gr(ϱ−s)B[xz(v)(s)−xz(s)]‖ds≤ˆP∫ϱ0‖Fv(s)−F(s)‖ds+ˆPPσ∫ϱ0‖xz(v)(s)−xz(s)‖ds, | (3.4) |
where
‖xz(v)(s)−xz(s)‖≤PxˆP[∫ϱ0‖Fv(s)−F(s)‖ds]. | (3.5) |
Since the inequality (3.4) and (3.5), we get
‖(Πz(v))(ϱ)−(Πz)(ϱ)‖→0 as v→∞. |
Then, Π is continuous on PC(V,Q).
Step 3: Now, we show that {(Πz):z∈Gℓ} is equicontinuous family.
(Πz)(ϱ)={Nr(ϱ)z0+Mr(ϱ)z1+∫ϱ0Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))ds+∫ϱ0Gr(ϱ−s)Bx(s)ds, 0≤ϱ≤ϱ1,Nr(ϱ)z0+Mr(ϱ)z1+∑jq=1Nr(ϱ−ϱj)mj+∑jq=1Mr(ϱ−ϱj)˜mj+∫ϱ0Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))ds+∫ϱ0Gr(ϱ−s)Bx(s)ds, ϱj<ϱ≤ϱj+1. |
Suppose 0≤ℏ1<ℏ2≤ϱ1. In addition, for every ϱ∈[0,ϱ1], we get
‖(Πz)(ℏ2)−(Πz)(ℏ1)‖=‖Nr(ℏ2)z0+Mr(ℏ2)z1+∫ℏ20Gr(ℏ2−s)f(s,z(s),(E1z)(s),(E2z)(s))ds+∫ℏ20Gr(ℏ2−s)Bxz(s)ds−Nr(ℏ1)z0−Mr(ℏ1)z1−∫ℏ10Gr(ℏ1−s)f(s,z(s),(E1z)(s),(E2z)(s))ds−∫ℏ10Gr(ℏ1−s)Bxz(s)ds‖≤‖[Nr(ℏ2)−Nr(ℏ1)]z0‖+‖[Mr(ℏ2)−Mr(ℏ1)]z1‖+∫ℏ2ℏ1‖Gr(ℏ2−s)f(s,z(s),(E1z)(s),(E2z)(s))‖ds+∫ℏ10‖[Gr(ℏ2−s)−Gr(ℏ1−s)]f(s,z(s),(E1z)(s),(E2z)(s))‖ds+∫ℏ2ℏ1‖Gr(ℏ2−s)Bxz(s)‖ds+∫ℏ10‖[Gr(ℏ2−s)−Gr(ℏ1−s)]Bxz(s)‖ds≤‖Nr(ℏ2)−Nr(ℏ1)‖‖z0‖+‖Mr(ℏ2)−Mr(ℏ1)‖‖z1‖+ˆP∫ℏ2ℏ1ς1(s)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))ds+∫ℏ10‖Gr(ℏ2−s)−Gr(ℏ1−s)‖ς1(s)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))ds+ˆPPσ‖xz‖Lμ(V,Y)(ℏ2−ℏ1)+Pσ∫ℏ10‖Gr(ℏ2−s)−Gr(ℏ1−s)‖‖xz(s)‖ds. |
Similarly, for every ϱ∈(ϱj,ϱj+1], j=1,2,⋯,n, we can have
‖(Πz)(ℏ2)−(Πz)(ℏ1)‖=‖Nr(ℏ2)z0+Mr(ℏ2)z1+∑0<ϱj<ℏ2Nr(ℏ2−ϱj)mj+∑0<ϱj<ℏ2Mr(ℏ2−ϱj)˜mj+∫ℏ20Gr(ℏ2−s)f(s,z(s),(E1z)(s),(E2z)(s))ds+∫ℏ20Gr(ℏ2−s)Bx(s)ds−Nr(ℏ1)z0−Mr(ℏ1)z1−∑0<ϱj<ℏ1Nr(ℏ1−ϱj)mj−∑0<ϱj<ℏ1Mr(ℏ1−ϱj)˜mj−∫ℏ10Gr(ℏ1−s)f(s,z(s),(E1z)(s),(E2z)(s))ds−∫ℏ10Gr(ℏ1−s)Bx(s)ds‖≤‖[Nr(ℏ2)−Nr(ℏ1)]z0‖+‖[Mr(ℏ2)−Mr(ℏ1)]z1‖+∑ℏ1<ϱj<ℏ2‖Nr(ℏ2−ϱj)mj‖+∑0<ϱj<ℏ2‖[Nr(ℏ2−ϱj)−Nr(ℏ1−ϱj)]mj‖+∑ℏ1<ϱj<ℏ2‖Mr(ℏ2−ϱj)˜mj‖+∑0<ϱj<ℏ2‖[Mr(ℏ2−ϱj)−Mr(ℏ1−ϱj)]˜mj‖+∫ℏ2ℏ1‖Gr(ℏ2−s)f(s,z(s),(E1z)(s),(E2z)(s))‖ds+∫ℏ10‖[Gr(ℏ2−s)−Gr(ℏ1−s)]f(s,z(s),(E1z)(s),(E2z)(s))‖ds+∫ℏ2ℏ1‖Gr(ℏ2−s)Bx(s)‖ds+∫ℏ10‖[Gr(ℏ2−s)−Gr(ℏ1−s)]Bx(s)‖ds≤‖Nr(ℏ2)−Nr(ℏ1)‖‖z0‖+‖Mr(ℏ2)−Mr(ℏ1)‖‖z1‖+ˆP∑ℏ1<ϱj<ℏ2‖mj‖+∑0<ϱj<ℏ2‖Nr(ℏ2−ϱj)−Nr(ℏ1−ϱj)‖‖mj‖+ˆP∑ℏ1<ϱj<ℏ2‖˜mj‖+∑0<ϱj<ℏ2‖Mr(ℏ2−ϱj)−Mr(ℏ1−ϱj)‖‖˜mj‖+ˆP∫ℏ2ℏ1ς1(s)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))ds+∫ℏ10‖Gr(ℏ2−s)−Gr(ℏ1−s)‖ς1(s)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))ds+ˆPPσ‖x‖Lμ(V,Y)(ℏ2−ℏ1)+Pσ∫ℏ10‖Gr(ℏ2−s)−Gr(ℏ1−s)‖‖x(s)‖ds. |
Thus, we get
‖(Πz)(ℏ2)−(Πz)(ℏ1)‖={‖Nr(ℏ2)−Nr(ℏ1)‖‖z0‖+‖Mr(ℏ2)−Mr(ℏ1)‖‖z1‖+ˆP∫ℏ2ℏ1ς1(s)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))ds+∫ℏ10‖Gr(ℏ2−s)−Gr(ℏ1−s)‖ς1(s)×ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))ds+ˆPPσ‖xz‖Lμ(V,Y)(ℏ2−ℏ1)+Pσ∫ℏ10‖Gr(ℏ2−s)−Gr(ℏ1−s)‖‖xz(s)‖ds, ϱ∈[0,ϱ1],‖Nr(ℏ2)−Nr(ℏ1)‖‖z0‖+‖Mr(ℏ2)−Mr(ℏ1)‖‖z1‖+ˆP∑ℏ1<ϱj<ℏ2‖mj‖+∑0<ϱj<ℏ2‖Nr(ℏ2−ϱj)−Nr(ℏ1−ϱj)‖‖mj‖+ˆP∑ℏ1<ϱj<ℏ2‖˜mj‖+∑0<ϱj<ℏ2‖Mr(ℏ2−ϱj)−Mr(ℏ1−ϱj)‖‖˜mj‖+ˆP∫ℏ2ℏ1ς1(s)ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))ds+∫ℏ10‖Gr(ℏ2−s)−Gr(ℏ1−s)‖ς1(s)×ω(ℓ+σF0(1+ℓ)+σF1(1+ℓ))ds+ˆPPσ‖x‖Lμ(V,Y)(ℏ2−ℏ1)+Pσ∫ℏ10‖Gr(ℏ2−s)−Gr(ℏ1−s)‖‖x(s)‖ds, ϱ∈(ϱj,ϱj+1]. | (3.6) |
The aforementioned inequality's RHS of the system (3.6) tends to zero independently of z∈Gℓ as ℏ2→ℏ1 by using the continuity of functions ϱ→‖Nr(ϱ)‖, ϱ→‖Mr(ϱ)‖, and ϱ→‖Gr(ϱ)‖. Therefore, Π(Gℓ) is equicontinuous.
Step 4: Next, we prove that Mönch's condition holds.
Consider U⊆Gℓ is countable and U⊆conv({0}∪Π(U)), we show that λ(U)=0, where λ is the Hausdorff measure of noncompactness. Let U={zv}∞v=1. We check that Π(U)(σ) is relatively compact in PC(V,Q), for every ϱ∈(ϱj,ϱj+1]. From Theorem 2.11, and
δ({xz(v)(s)}∞v=1)=δ{W−1(zw−Nr(ϱ)z0−Mr(ϱ)z1−∑0<ϱj<ℏ2Nr(ϱ−ϱj)mj−∑0<ϱj<ℏ2Mr(ϱ−ϱj)˜mj−∫ϱ0Gr(ϱ−s)f(s,z(v)(s),(Fz(v))(s),(Hz(v))(s))ds)}∞v=1≤2ς5(s)ˆP(∫ϱ0ς2(s)[δ(U(s))+δ({Fz(v)(s)}∞v=1)+δ({Hz(v)(s)}∞v=1)]ds)≤2ς5(s)ˆP(∫ϱ0ς2(s)δ(U(s))ds+2∫ϱ0ς2(s)(ς∗3+ς∗4)δ(U(s))ds). |
From Lemma 2.11, and assumptions (H1)–(H5), we get
δ({Πz(v)(s)}∞v=1)=δ({Nr(ϱ)z0+Mr(ϱ)z1+j∑q=1Nr(ϱ−ϱj)mj+j∑q=1Mr(ϱ−ϱj)˜mj+∫ϱ0Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))ds+∫ϱ0Gr(ϱ−s)Bxz(s)ds}∞v=1)≤δ({∫ϱ0Gr(ϱ−s)f(s,z(s),(E1z)(s),(E2z)(s))ds}∞v=1)+δ({∫ϱ0Gr(ϱ−s)Bxz(s)ds}∞v=1)≤2ˆP(∫ϱ0ς2(s)[δ(U(s))+δ({Fz(v)(s)}∞v=1)+δ({Hz(v)(s)}∞v=1)]ds)+2ˆPPσ(∫ϱ0δ({xz(v)(s)}∞v=1)ds)≤2ˆP(∫ϱ0ς2(s)δ(U(s))ds+∫ϱ0ς2(s)(ς∗3+ς∗4)δ(U(s))ds)+4ˆP2Pσ×(∫ϱ0ς5(s)ds)(∫ϱ0ς2(s)δ(U(s))ds+∫ϱ0ς2(s)(ς∗3+ς∗4)δ(U(s))ds)≤2ˆP‖ς2‖[1+2ˆPPσ‖ς5‖](1+(ς∗3+ς∗4))δ(U(ϱ)). |
By Lemma 2.10, we get
δ({Πz(v)(s)}∞v=1)≤M∗δ(U(ϱ)). |
Therefore by using Mönch's condition, one can obtain
δ(Π)≤δ(conv({0}∪(Π(U)))=δ(Π(U))≤M∗δ(U), |
this implies δ(U)=0. Hence, Π has a fixed point in Gℓ. Thus, the fractional integrodifferential equations (1.1) has a fixed point fulfilling z(σ)=zσ. Thus, the fractional integrodifferential equations (1.1) is exact controllable on [0,σ].
Suppose the impulsive fractional mixed Volterra-Fredholm type integrodifferential systems of the form:
{∂r∂ϱrz(ϱ,ω)=∂2∂ϱ2z(ϱ,ω)+cos[z(ϱ,ω)+∫ϱ0(ϱ−ι)2sinz(ι,ω)dι]+∂∂ωz(ϱ,ω)+∫σ0cosz(ϱ,ω)dι+ξα(ϱ,ω),ϱ∈V=[0,1], ω∈[0,π], ϱ≠ϱj, j=1,2,⋯,n,z(ϱ,0)=z(ϱ,1)=0, ϱ∈V,z(ϱ+j,ω)−z(ϱ−j,ω)=mj, z′(ϱ+j,ω)−z′(ϱ−j,ω)=˜mj, j=1,2,⋯,n,z(0,ω)=z0(ω), z′(0,ω)=z1(ω), | (4.1) |
where ∂32∂ϱ32 denotes fractional partial derivative of r=32. 0=ϱ0<ϱ1<ϱ2<⋯<ϱj<⋯<ϱn=σ; z(ϱ+j)=lim(ϵ+,ω)→(0+,ω)z(ϱj+ϵ,ω) and z(ϱ−j)=lim(ϵ−,ω)→(0−,ω)z(ϱj+ϵ,ω).
Consider Q=Y=L2([0,π]), and let A maps from D(A)⊂Q into Q be presented as Az=z′′ along with domain D(A), which is
D(A)={z in Q:z, z′ are absolutely continuous, z′′ in Q,z(0)=z(π)=0}. |
Further, A stands for infinitesimal generator of an analytic semigroup {G(ϱ), ϱ≥0} determined by G(ϱ)z(s)=z(ϱ+s), for every z in Q. G(ϱ) is not compact semigroup on Q and δ(G(ϱ)U)≤δ(U), then δ stands for the Hausdorff MNC.
In addition, A has discrete spectrum with eigenvalues −μ2, μ∈N, and according normalized eigen functions given by yμ(z)=√(2/π)sin(μπz). Then, yμ stands for an orthonormal basic of Q. For more details refer to [35].
G(ϱ)=∞∑μ=1e−μ2ϱ⟨z,yμ⟩yμ, z∈Q, |
G(ϱ) is compact for any ϱ>0 and G(ϱ)≤e−ϱ for any ϱ≥0 [44].
A=∂2∂ϱ2 represents sectorial operator of type (P,τ,r,ϕ) and generates r-resolvent families Nr(ϱ), Mr(ϱ), and Gr(ϱ) for ϱ≥0. Since A=∂2∂ϱ2 is an m-accretive operator on Q with dense domain (H1) fulfilled.
Az=∞∑μ=1μ2⟨z,yμ⟩yμ, z∈D(A). |
Determine
f(ϱ,z(ϱ),(E1z)(ϱ),(E2z)(ϱ)(ω))=cos[z(ϱ,ω)+∫ϱ0(ϱ−ι)2sinz(ι,ω)dι]+∂∂ωz(ϱ,ω)+∫σ0cosz(ϱ,ω)dι, |
(E1z)(ϱ)=∫ϱ0sinz(ι,ω)dι,(E2z)(ϱ)=∫10cosz(ϱ,ω)dι. |
Assume that B:Q→Q is determined by
(Bx)(ϱ)(ω)=ξα(ϱ,ω), ω∈[0,π], |
For ω∈[0,π], the linear operator W specified by
(Wx)(ω)=∫10G(1−s)ξα(s,ω)ds, |
fulfilling (H2)–(H5). Thus, the system (4.1) can be rewritten as
{CDrϱz(ϱ)=Az(ϱ)+f(ϱ,z(ϱ),(E1z)(ϱ),(E2z)(ϱ))+Bx(ϱ), ϱ∈V, ϱ≠ϱj,Δz(ϱj)=mj, Δz′(ϱj)=˜mj, j=1,2,⋯,n,z(0)=z0, z′(0)=z1. | (4.2) |
As a result, Theorem 3.1's requirements are all fulfilled. The system (4.1) is therefore exact controllable on V according to Theorem 3.1.
In this study, we mainly concentrated on the exact controllability outcomes for fractional integrodifferential equations of mixed type via sectorial operators of type (P,τ,r,ϕ), employing fractional calculations, impulsive systems, sectorial operators, and fixed point technique. Lastly, an example for clarifying the theory of the important findings is constructed. The effectiveness of such research discoveries can be effectively increased to exact controllability using varied fractional differential structures (Hilfer system, A-B system, stochastic, etc.). Moreover, null controllability outcomes of impulsive fractional stochastic differential systems via sectorial operators will be the subject of future research.
This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444). Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R157), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
The authors declare no conflict of interest.
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