<p>In this manuscript, we explore the sufficient conditions for the existence and controllability of a new class of the nonlinear fractional Sobolev-type neutral integro-differential stochastic system with instantaneous impulses and finite delay in a separable Hilbert space. We investigate the existence of the mild solution for the proposed stochastic control problem while taking into account the instantaneous impulsive effects. For this purpose, the Riemann-Liouville fractional integral operator is used to convert the proposed stochastic control system into an equivalent fixed point problem. Then, we establish the controllability results by employing the theory of fractional calculus, semigroups of bounded linear operators, stochastic analysis, Mönch’s condition, and the measure of noncompactness. In our approach, we relax the compactness assumption on the semigroup operator. Instead of using strong Lipschitz-type conditions, we derive the sufficient conditions under non-Lipschitz assumptions. At the end, an example is given to demonstrate and support the theoretical results.</p>

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New results on existence and controllability of nonlinear fractional Sobolev-type delayed stochastic system with instantaneous impulses and noncompact semigroups

  • Om Prakash Kumar Sharma,
  • Ramesh Kumar Vats,
  • Ankit Kumar

摘要

In this manuscript, we explore the sufficient conditions for the existence and controllability of a new class of the nonlinear fractional Sobolev-type neutral integro-differential stochastic system with instantaneous impulses and finite delay in a separable Hilbert space. We investigate the existence of the mild solution for the proposed stochastic control problem while taking into account the instantaneous impulsive effects. For this purpose, the Riemann-Liouville fractional integral operator is used to convert the proposed stochastic control system into an equivalent fixed point problem. Then, we establish the controllability results by employing the theory of fractional calculus, semigroups of bounded linear operators, stochastic analysis, Mönch’s condition, and the measure of noncompactness. In our approach, we relax the compactness assumption on the semigroup operator. Instead of using strong Lipschitz-type conditions, we derive the sufficient conditions under non-Lipschitz assumptions. At the end, an example is given to demonstrate and support the theoretical results.