On solvability and optimal controls for impulsive stochastic integrodifferential varying-coefficient model

This article concentrates in analyzing optimal controls for stochastic integrodifferential equation (SIDE) in Hilbert space. Necessary parameters are imposed to demonstrate the system that follows a unique variation of parameter formula using Leray Schauder Alternative. Subsequently, the existence of optimal control is investigated for the considered Lagrange control problem. The theoretical example with the mechanical example of ethanol fuelled engine are discussed to validate the results obtained.


Introduction
Several writers study a family of regression and extended regression models that allow coefficient fluctuations as smooth functions of other variables.This class of models combines generalized additive models and dynamic generalized linear models into one cohesive system.When it comes to the proportional hazards model for survival data, this technique offers a unique way of modelling departures from the proportionate risks assumption.Over the past few decades, efforts have been made to increase the flexibility of linear regression models.Generalized additive models, which are smooth, non-parametric functions that can partially or completely replace the linear and parametric functions of regressors, have been the subject of research.Here, we have models with linear regressors that appear to be separate generalizations, but really have coefficients that can smoothly change in response to the values of other variables, or what we would call "effect modifiers".The random variable Y is dependent on a parameter η for its distribution.Additionally, there are predictors X l , X 2 , . . ., X p and R 1 , R 2 , . . ., R p .A model with variable coefficients has the form The model (1) states that the (unspecified) functions β l ( ), β 2 ( ), . . ., β p ( ) cause R 1 , R 2 , . . ., R p to alter the coefficients of the X l , X 2 , . . ., X p .A unique kind of interaction between each R j and X j is implied by the reliance of β j ( ) on R j .At times, R j can be easily confused with the variables X j ; in other scenarios, it might be a unique variable like "time".
As the linear predictor in the generalized linear model, η is connected to the mean μ = EY by the link function η = g(μ).Model (1) takes the form of the Gaussian model in its simplest example, where g(μ) = μ and Y is normally distributed with mean μ.
where var( ) = σ 2 and E( ) = 0. Several well-liked models include the log-linear models, in which η = log μ and Y has a Poisson distribution, and the linear logistic model, in which g(μ) = log{ μ 1−μ } and Y a binomial variate.Generalized additive models are an extension of generalized linear models, where the linear predictor is replaced with an additive sum of smooth functions.As we will see, the varying-coefficient model has specific instances such as the generalized additive model and the dynamic generalized linear model.

Examples of Varying-coefficient Model
(1) That term is linear in X if β j (R j ) = β j (the constant function).Model (1) is the standard linear model, often known as the extended linear model if every term is linear.In the case of X j = c (let's assume c = 1), the jth term is just β j (R j ), an ambiguous function R j .If every term in the model has the same form as (1) or is linear, then (1) represents a generalized additive model.(2) A linear function β j (R j ) = β j R j yields a product interaction of the type β j X j (R j ).(3) For the purposes of simplicity, let us assume that the model is a single-term normal linear model and that X j is the modifying variable R j .Thus, we obtain This model has been studied extensively by researchers and is often used for smoothing or nonparametric regression of Y against X. (4) We can have vector or scalar values for each R j .We will assume for most of the study that the R j are scalar; expansions to the vector-valued situation will also be examined.
Balasubramaniam and Tamilalagan [1] considered the following impulsive fractional stochastic integrodifferential system and investigated its solvability and optimal control properties.
In [2], the authors have established the existence results of Hilfer fractional integrodifferential equation of the form: where is the infinitesimal generator of an analytic semigroup of bounded linear operators S(t), t ≥ 0, on a separable Hilbert space H with inner product •, • and norm • .
Recently, Hamdy M. Ahmed [3] considered semilinear neutral fractional stochastic integrodifferential system with non-local condition of the form: Motivated by above works, we are concerned in considering impulsive SIDEs of the form: Here A : Eventually, (PC(J , L 2 ( , H )), .PC ) is a Banach space.l : J × H m+1 → H and σ : J × H n+1 → L(K, H ) are the suitable maps used in this article.Randomness has to be incorporated into mathematical models of real-world phenomena because random effects and noise have caused many real-world phenomena, such expanding population, heat conduction in materials with memory, stock prices and so forth, to fluctuate in recent years.Stochastic Differential Equations (SDEs) are differential equations that assume unpredictability.Because SDEs allow for the abstract representation of many issues, they are employed in a wide range of fields, including as engineering, finance and economics.Books [4,5] and publications [6,7] provide additional fundamental information on SDEs.Further information on the qualitative characteristics of mild solutions to different SIDEs and the fixed point approach may be found in [8][9][10] and the references therein.
When stochastic differential equations describe the system dynamics and performance index, an optimum control issue becomes a stochastic optimal control problem.Sathiyaraj et al. [11] recently shown optimum control and controllability for fractional SDEs with Poisson jumps.However, there aren't many papers in the literature that discuss optimum control problems.[12][13][14][15][16][17].Using the Lagrange multiplier method and the fractional variational principle, Agrawal [13] provided comprehensive information for fractional optimum control problems, accounting for fractional derivatives in the Riemann-Liouville sense.Using resolvent operators, Tamaligan et al. [18] examined the solvability and best controls for FSDE driven by Poisson jumps.Tang and Liu [19] discovered recently that the robustness of the feedback optimum control is not ensured by the regularity of the solution to the backward stochastic Riccati equations.They prove the equivalence between the solvability of the associated backward stochastic Riccati equations and the existence of the resilient optimum feedback control strategy operators, under suitable regularity requirements.In order to construct the online H ∞ optimization problems for a class of nonlinear systems without taking the system dynamics into account, Shuping He et al. [20] created a novel policy iterative technique.Additionally, using a unique policy iteration (PI) approach, Shuping He et al. [21] investigated the online adaptive optimum controller design for a class of nonlinear systems.Without utilizing the system internal parameters, the optimal law for controller design is solved through the appropriate algebraic Riccati equation (ARE) by employing the neural network linear differential inclusion (LDI) approach to linearize the nonlinear components in each iteration.This paper's model is more sophisticated than [21] since it incorporates a stochastic term with a timevarying coefficient.Using a successive approximation technique, Ramkumar et al. [22] examined the optimum management of a neutral FSDE with a Caputo fractional derivative.See [23][24][25][26][27][28] for a list of more articles that discuss the solvability and optimum control for fractional SDEs.
Novelty of the work: (1) Thus far, the literature has not addressed the optimal controllability for an impulsive stochastic time-varying-coefficient model.(2) The Leray Schauder Alternative confirms the existence and solvability of the mild solution of ( 6).(3) A comprehensive analysis of the 88 observations on the exhaust from an ethanol-fuelled engine in mechanical engineering is conducted to demonstrate the practical implementation of the stated hypothesis.
This paper's outline is: Section 2 establishes the concepts and preliminary steps needed to solve the aforementioned system.Section 3 proves the existence results of the expressed system (6).The system's existence results are established in Section 4. The illustrations are included in Section 5 to validate our findings.

Preliminaries and notations
Let ( , F, P) denote a complete probability space with increasing sub σ -algebra Definition 2.1: A one parameter family {R(t) : t ≥ 0} of bounded linear operators is called resolvent operator for For more background on the resolvent operator, we refer to [29][30][31].

Definition 2.2 ([32]):
Let E be a Banach space, ∈ E a closed convex subset, U ⊂ an open set (with respect to the topology of ) and such that θ ∈ U. Assume that F : Ū → is weakly sequence compact.If F Ū is relatively weakly compact then, either (i) F has a fixed point, or (ii) there is a point u ∈ ∂ U and λ ∈ (0, 1) with u = λFu, where θ be the zero vector of E. Ū and ∂ U denote the closure and the boundary of U in , respectively.

Remark 2.1:
We know that a strongly continuous operator is weakly sequential compact (WSC).The converse is not true in general (even if E is reflexive).Leray-Schauder alternative is useful to derive WSC operators.In our study, we assume that the infinitesimal generator A does not generate the compact semigroup.We use MNC to prove the existence of the mild solution.Leray Schauder alternative fits well in this situation.

Definition 2.3:
A stochastic process ϑ(t) ∈ PC(J , L 2 ( , H )) follows the variation of constant formula for the system (6) whenever Let us have the following hypotheses: fies the requirements: (i) For each t ∈ J the function σ (t, .): and lim r→∞ inf 1 Set the admissible set

Existence of mild solution
Theorem 3.1: Assume that hypotheses (A1)-(A5) are satisfied then the system (6) has at least one mild solution on J , given that Proof: Define the map : It is adequate to demonstrate seems to have a fixed point in PC(J , L 2 ( , H )).
We assume that there exists a positive number r such that B r ⊆ B r .If it is not true, then for each positive number r, there is a function ϑ r (•) ∈ B r but B r = B r , but ϑ r (t) > r for some t(r) ∈ J , where t(r) denotes that t is dependent on r.
Dividing r throughout and let r → ∞, which contradicts our assumption (9).
Step 2: To prove is continuous, let {ϑ n} be a sequence ϑ n → ϑ in PC(J , L 2 ( , H )) as n → ∞ then for t ∈ (t k , t k+1 ], we get
Step 4: To show ( ϑ)(t) is compact for 0 ≤ t ≤ B, firstly we need to prove ( ϑ)(0) is relatively compact in B r .For 0 < < B, ϑ ∈ B r , This implies that there are relatively compact sets to the set {( ϑ)(t) : ϑ ∈ B r }.Hence (t) is also relatively compact in B r .Hence has a fixed point ϑ(.) on B r .Thus all conditions of Leray Schauder alternative are satisfied, as a result, the system (6) possesses a mild solution.

Existence of optimal control
The Lagrange Problem (P) can be considered as follows: Considering, (ϑ 0 , u 0 where and ϑ u represents the mild solution of ( 6) which corresponds to the control u ∈ U ad .Regarding the existence of solutions to (P), let us assume the following: and almost all t ∈ J .(iv) There exist constants d ≥ 0, f > 0, μ ≥ 0 and μ ∈ L (J , R) Theorem 4.1: The Lagrange problem (P) permits at least one optimum pair if assumptions (A1)-(A6) are true and B is a strongly continuous operator, (i .e.) ∃ an admissible control pair × U ad } = +∞, there is nothing to demonstrate.Without losing generality, we suppose inf{I(ϑ u , u) | (ϑ u , u) A minimizing sequence of feasible pair {(ϑ m, u m)} ⊂ P ad exists according to the definition of infimum, where P ad = {(ϑ, u) : ϑ is a mild solution of system (1) corresponding to u ∈ U ad } I(ϑ m, u m) → ρ as m → +∞.As {u m} ⊆ U ad , m = 1, 2, . . ., u m is a bounded subset of the separable reflexive Banach space L p (J , U), ∃ a subsequence, relabelled as u m and u 0 ∈ L p (J , U) u m → u 0 weakly in L p (J , U).Since U ad is closed and convex.Through Marzur lemma [33], u 0 ∈ U ad .Let {u m} denotes the sequence of solutions of the system corresponding to u m, ϑ 0 is the mild solution that accords to u 0 .ϑ m, ϑ 0 fulfil the integral equations: and From the boundedness of {u m}, {u 0 }, and Theorem 3.1, where Then we have, E ϑ m − ϑ 0 2 w − → 0 as m → ∞ yields ϑ m w − → ϑ 0 as m → ∞. (H6) implies Balder's hypotheses [34].Henceforth, (ϑ, u) → E( B 0 L(t, ϑ(t), u(t))) satisfies the assumptions in the weak topology of L p (J , U) ⊂ L 1 (J , U) and strong topology of L 1 (J , U).As a result, on L p (J , U) I is weakly lower semicontinuous and by (A6)(iv), I > −∞, I reaches its infimum at u 0 ∈ U ad , (i.e.) Hence the proof of optimal controllability.
τ , e j e j , the normalized eigenfunctions are e j (s) = 2 π sin(js), j = 1, 2, . ... R(t)τ = ∞ j=1 e −j 2 t τ , e j e j ∀τ ∈ H . Thus {R(t)} t≥0 becomes uniformly bounded compact semigroup.Also, Let us consider the control function to be u : ϑ To exhibit 16 in the abstract form of (6), we incorporate Furthermore, we assume the functions (iv) For the map σ : J × R → R, the succeeding circumstances exist: (a) σ (t, .) is continuous for t Take into consideration, the cost function It is clear that the assumptions of Theorem 3.1 are met, implying there is at least one optimal pair.Hence justified.

Example 2: ethanol fuelled engine
An analysis of 88 observations on the exhaust from an engine running on ethanol was conducted for this mechanical engineering example.The concentration of nitric oxide and nitrogen dioxide, normalized by the engine's workload, is the response variable, represented by NO x .The engine's compression ratio C and the equivalency ratio E, which measures the fuel-air mixture, are the two predictors.The information is displayed in Figure 1.Plotting NO x vs E and C is displayed in Figure 1(a,b).
The basic model NO x ≈ E 2 is suggested by the strong quadratic-like influence of E and the seemingly little effect of C. NO x vs E is depicted in Figure 1(c), where C levels are classified as low, medium and high.This implies that C may be interacting with E. The structure of this interaction is seen in Figure 2. The fitted linear regressions of NO x on C in four nonoverlapping ranges of E are displayed by the broken lines.A linear model in C appears to fit well inside each range of E. However, the intercept and slope of the line both change as E does.This prompts us to think of a model of the kind Despite the plots' suggestion that β 0 (E) ≈ E 2 , we will fit β 0 (E) and β 1 (E) flexibly and leave them both undetermined.Cleveland et al. [35] examined this model, which is an illustration of a varying-coefficient model.The term l(t, ϑ(t), ϑ(A 1 (t)), ϑ(A 2 (t)), . . .ϑ(A m (t))) of ( 1) is comparable to the term in Equation (17).By integrating an additional engine with the suggested engine, we arrive at model (6), from which the best control of exhaust ethanol engines may be studied.

Conclusions
This article is devoted to studying the optimal controls for stochastic integrodifferential equation (SIDE) in Hilbert space.Necessary parameters are imposed to demonstrate the system that follows a unique variation of parameter formula using Leray Schauder alternative.Subsequently, the existence of optimal control is investigated for the considered Lagrange control problem.We  explored a class of regression and generalized regression models in which the coefficients are allowed to vary as smooth functions of other variables.General algorithms are presented for estimating the models flexibly.This class of models ties together generalized additive models and dynamic generalized linear models into one common framework.When applied to the proportional hazards model for survival data, this approach provides a new way of modelling departures from the proportional hazards assumption.There are some directions in which this work could be extended.
(1) The effect modifier R might be vector valued, in which case a multidimensional smoother would be used in the estimation procedure for the function β(R).The conditionally parametric models of Cleveland et al. [35] automatically allow for this case when all the terms are modelled conditionally on the same R. (5) Second order system in the frame work of ( 6) can be studied using sine and cosine operators.Numerical simulation will be interesting as well to justify the theory [36].

Figure 1 .
Figure 1.Using certain values of C coded as low (l), medium (m), or high (h) (intermediate values are coded with * ), (a) NO x versus E; (b) NO x versus C and (c) NO x versus E.

Figure 2 .
Figure 2. NO x vs C for (a) low, (b) medium, (c) high and (d) very high values of E: --, fitted linear regression; --fitted lines from the varying-coefficient model, chosen at the median value of E for the panel's data.

( 2 )
One would look for directions in the effectmodifier space that result in large changes in the coefficients.(3)Model (6) can be generalized to Hilfer fractional order model with the integral boundary conditions.(4) Proposed model (6) is w.r.t.time varying-coefficient, one can extend the same model with state varying-coefficient with the proper arguments and with the corresponding real-life applications.