This paper proposes novel fractional cosine and sine matrix functions derived from a determining matrix. It introduces a new controllability criterion based on the controllability Gramian for achieving exact controllability in oscillating fractional linear systems with damping term. We present Cayley-Hamilton type theorem for determining matrix. Finally, we prove a novel Kalman type rank criterion for controllability in oscillating fractional linear systems with damping, which is a novel contribution even in systems without damping. Numerical examples are provided to validate these findings.

A document by Javad Asadzade. Click on the document to view its contents.

In this paper, we study the optimal control of a discrete-time stochastic differential equation (SDE) of mean-field type, where the coefficients can depend on both a function of the law and the state of the process. We establish a new version of the maximum principle for discrete-time mean-field type stochastic optimal control problems. Moreover, the cost functional is also of the mean-field type. This maximum principle differs from the classical principle one since we introduce new discrete-time mean-field backward (matrix) stochastic equations. Based on the discrete-time mean-field backward stochastic equations where the adjoint equations turn out to be discrete backward SDEs with mean field, we obtain necessary first-order and sufficient optimality conditions for the stochastic discrete mean-field optimal control problem. To verify, we apply the result to production and consumption choice optimization problem.

The representation of an explicit solution to the Prabhakar fractional differential delayed system is studied employing the far-famed Laplace transform technique. Second, the existence uniqueness of the solution is debated together with the Ulam-Hyers stability of a semilinear Prabhakar fractional differential delayed system. Thirdly, the necessary and sufficient circumstances for the controllability of linear Prabhakar fractional differential delayed system are determined by describing the Gramian matrix. A sufficient circumstance for the relative controllability of a semilinear Prabhakar fractional differential delayed system is studied via the Krasnoselskii's fixed point theorem. Numerical examples are offered to verify the theoretical findings.

{In the present paper, a $\mu$-delayed Mittag-Leffler type function is introduced as a fundamental function. By means of $\mu$-delayed Mittag-Leffler type function, an exact analytical solution formula to non-homogeneous linear delayed Langevin equations involving two distinct $\mu$-Caputo type fractional derivatives of general orders is given. Also, a global solution of nonlinear version of delayed Langevin equations is inferred from the findings on hand and is verified with the aid of the functional(substitutional) operator. In terms of exponential function, we estimate $\mu$-delayed Mittag-Leffler type function. Existence uniqueness of solutions to nonlinear delayed Langevin fractional differential equations are obtained with regard to the weighted norm defined in accordance with exponential function. The notion of stability analysis in the sense of solutions to the described Langevin equations is discussed on the grounds of the fixed point approach. Numerical and simulated examples are shared to exemplify the theoretical findings. This paper provides novel results.

Sobolev type fractional functional evolution equations have many applications in the modeling of many physical processes. Therefore, we investigate fractional-order time-delay evolution equation of Sobolev type with multi-orders in a Banach space and introduce an analytical representation of a mild solution via a new delayed Mittag-Leffler type function which is generated by linear bounded operators. Furthermore, we derive an exact analytical representation of solutions for multi-dimensional fractional functional dynamical systems with nonpermutable and permutable matrices. We also study stability analysis of the given time-delay system in Ulam-Hyers sense with the help of Laplace transform.