Controllability Of Fractional Differential Equations Research Articles

Approximate controllability of fractional differential equations driven by Markovian switching and Lévy noise with infinite delay

Motivated by applications of nonlinear dynamical systems and based on some existing works, we introduce a general class of Atangana-Baleanu nonlinear fractional neutral differential equations driven by Markovian switching and Lévy noise with infinite delay. In this paper, we aim to derive new results on the existence of solutions as well as approximate controllability for the corresponding Atangana-Baleanu equations in Banach spaces. Under suitable conditions with the help of semigroup theory, fixed point theorem as well as stopping time argument, we prove the main results. As an application, we present two examples with numerical simulations to support our theory. The results obtained in this paper improve and extend some related conclusions on this topic.

Relative controllability of linear state-delay fractional systems

AbstractIn this paper, our focus is on exploring the relative controllability of systems governed by linear fractional differential equations incorporating state delay. We introduce a novel counterpart to the Cayley-Hamilton theorem. Leveraging a delayed perturbation of the Mittag-Leffler function, along with a determining function and an analog of the Cayley-Hamilton theorem, we establish an algebraic Kalman-type rank criterion for assessing the relative controllability of fractional differential equations with state delay. Moreover, we articulate necessary and sufficient conditions for relative controllability criteria concerning linear fractional time-delay systems, expressed in terms of a new $$\alpha $$ α -Gramian matrix and define a control which transfer the system from any initial state to any final state within a given time. The theoretical findings are exemplified through the presentation of illustrative examples.

Relative controllability in fractional differential equations with pure delay

In this paper, we study relative controllability of fractional differential equations with pure delay. Delayed Gram‐type matrix criterion and rank criterion for relative controllability are established with the help of the explicit solution formula. An example is given to illustrate our theoretical results.

Controllability of Fractional Differential Equation of Order α∈(1,2] With Non‐Instantaneous Impulses

AbstractThe objective of this paper is to establish the sufficient condition for the exact controllability of a control system governed by the fractional differential equation of order α∈(1,2] with non‐instantaneous impulses in a Hilbert space X. The results are obtained using the α‐order cosine family and Banach fixed point theorem. Also, the exact controllability of an integrodifferential problem is studied. Finally, a few examples are given to illustrate the applications of these abstract results.

ILC method for solving approximate controllability of fractional differential equations with noninstantaneous impulses

In this paper, we utilize fractional calculus, theory of semigroup and fixed point approach to prove existence and approximate controllability results for a class of fractional differential equations (FDEs) with noninstantaneous impulses by constructing a suitable composite control function, imposing that the associated linear problem is approximately controllable on the terminal subinterval and dividing our global task into many subtasks on each subinterval. Next, we apply P-type iterative learning control (ILC) updating law to generate a sequence of control functions to find a desired control function to guarantee the error between the output and the desired reference trajectories tending to zero via a suitable norm in the sense of uniform convergence. As a result, the limit of the sequence of control functions is the required control to guarantee the above problem is approximately controllable. In addition, a numerical example is illustrated to demonstrate ILC scheme to solve approximate controllability of time fractional impulsive PDEs via tracking the given continuous and discontinuous trajectory.

Remarks on the controllability of fractional differential equations

In this paper, we study the approximate controllability of a linear fractional differential equation by using the method of regularization of Tikhonov. New concepts and results about controllability are established. Then, under the condition of the positivity of the controllability operator, we obtain that the linear system can be steered to an arbitrary small neighbourhood of the fractional integral of the state at final time.

Approximate controllability of fractional differential equations via resolvent operators

Of concern are the existence and approximate controllability of fractional differential equations governed by a linear closed operator which generates a resolvent. Using the analytic resolvent method and the continuity of a resolvent in the uniform operator topology, we derive the existence and approximate controllability results of a fractional control system.

Approximate Controllability of Fractional Differential Equations with State-Dependent Delay

The systems governed by delay differential equations come up in different fields of science and engineering but often demand the use of non-constant or state-dependent delays. The corresponding model equation is a delay differential equation with state-dependent delay as opposed to the standard models with constant delay. The concept of controllability plays an important role in physics and mathematics. In this paper, first we study the approximate controllability for a class of nonlinear fractional differential equations with state-dependent delays. Then, the result is extended to study the approximate controllability fractional systems with state-dependent delays and resolvent operators. A set of sufficient conditions are established to obtain the required result by employing semigroup theory, fixed point technique and fractional calculus. In particular, the approximate controllability of nonlinear fractional control systems is established under the assumption that the corresponding linear control system is approximately controllable. Also, an example is presented to illustrate the applicability of the obtained theory.