Approximate Controllability of Hilfer Fractional Stochastic Evolution Inclusions of Order 1 < q < 2
Abstract
:1. Introduction
2. Preliminaries
- Take the real separable Hilbert spaces , .
- A Wiener process with linear bounded covariance operator such that is represented as .
- From to , the set of bounded linear operators is represented by .
- Let us assume that there is a complete orthonormal system in , a bounded sequence series of non-negative real numbers ∋, and a sequence of independent Brownian motions such that
- Let represent the space of all Hilbert–Schmidt operators with the inner product from to .
- Letting be a Banach space containing all strongly measurable, square-integrable, and -valued random variables, we may define the norm as follows: , where defines the expectation.
- Let
- .
- is the space of all -adapted, -valued measurable square integrable processes on .
- For any continuous map from into , let be the Banach space and its norm be
- ; with norm . Here, is a Banach space.
- (i)
- If , we have
- (ii)
- If , we have
- (ii)
- If, for every , the set is a nonempty, closed subset of , and if, ∀ open set of containing , ∃ an open neighbourhood of such that , then is termed upper semicontinuous (u.s.c.) on .
- (iii)
- When is a relatively compact bounded subset of every bounded subset , then is considered to be completely continuous.
- (iv)
- When a multivalued map has nonempty compact values and is completely continuous, it can only be said to be u.s.c. if it has a closed graph, meaning that , , and imply .
- (v)
- If there is an in ∋, then has a fixed point.
- (vi)
- The multivalued map is considered measurable if the function , defined by
- For each is measurable;
- For all is u.s.c.;
- For each there exists , such that
- for all
- ;
- is continuous in χ on ∀ fixed point .
- For all the operators are uniformly continuous;
- The inequality below is true for any and ,
- The controllability Grammian is defined by
- The resolvent operator
- :
- as in the strong topology.
3. Main Results
- (H1):
- The continuous function , and there exists a positive constant such that the function satisfies that
- (H2):
- The multi-valued map is an -Caratheodory function that satisfies the below conditions:
- For each the function is u.s.c; and for each the function is measurable. And, for each fixed the set
- There exists a positive function such that
4. Example
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Shukla, A.; Panda, S.K.; Vijayakumar, V.; Kumar, K.; Thilagavathi, K. Approximate Controllability of Hilfer Fractional Stochastic Evolution Inclusions of Order 1 < q < 2. Fractal Fract. 2024, 8, 499. https://doi.org/10.3390/fractalfract8090499
Shukla A, Panda SK, Vijayakumar V, Kumar K, Thilagavathi K. Approximate Controllability of Hilfer Fractional Stochastic Evolution Inclusions of Order 1 < q < 2. Fractal and Fractional. 2024; 8(9):499. https://doi.org/10.3390/fractalfract8090499
Chicago/Turabian StyleShukla, Anurag, Sumati Kumari Panda, Velusamy Vijayakumar, Kamalendra Kumar, and Kothandabani Thilagavathi. 2024. "Approximate Controllability of Hilfer Fractional Stochastic Evolution Inclusions of Order 1 < q < 2" Fractal and Fractional 8, no. 9: 499. https://doi.org/10.3390/fractalfract8090499