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In this dissertation, we study the coupled system of q$$ q $$‐fractional Langevin differential equations involving q$$ q $$‐Caputo derivative having q$$ q $$‐fractional integral conditions. With the help of some adequate conditions, we investigate the uniqueness and existence of mild solution of the aforementioned system. We also analyze various kinds of Ulam's stability. Banach fixed point theorem and Leray–Schauder of cone type are used to illustrate the existence and uniqueness results. We also used non‐linear functional analysis methods to explore variety of stability types. An example is provided to clearly demonstrate our theoretical outcomes. [ABSTRACT FROM AUTHOR]

This paper is devoted to the problem of indefinite stochastic linear quadratic (LQ) optimal control by piecewise constant controls in an infinite horizon case. By restricting the set of admissible controls to the class of piecewise constant stochastic processes, we reformulated the above control problem under the setting of systems modeled by Itô differential equations controlled by impulses. We show that the solution in a state feedback form of the indefinite stochastic LQ control problem is equivalent to the existence of a global stabilizing solution associated to a class of backward matrix linear differential equations with a Riccati type jumping operators. [ABSTRACT FROM AUTHOR]

In this study, we investigate the nonuniform exponential stability and nonuniform exponential instability concepts for a linear random one‐sided system. We derive necessary and sufficient conditions for these concepts. As an extension for this study, we establish Lyapunov‐type characterizations using Lyapunov functions and Lyapunov norms. Additionally, we illustrate the essential properties that are derived in these directions for the adjoint and inverse systems associated for the original linear random one‐sided system. [ABSTRACT FROM AUTHOR]