Approximate controllability of fractional differential equations via resolvent operators

Abstract

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. MSC: 34K37; 47A10; 49J15

Approximate Controllability for Functional Equations with Riemann-Liouville Derivative by Iterative and Approximate Method

“…On the other hand, the notion of controllability plays a central role in the study of the theory of control and optimization. Therefore, there are a lot of works on the controllability, approximate controllability, and optimal control of linear and nonlinear differential and integral systems in various frameworks (see [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] and references therein).…”

Approximate Controllability of Fractional Nonlinear Hybrid Differential Systems via Resolvent Operators

“…Following the idea, as in [20], the suggested control function for system (1) can be written in the form.…”

“…We prove the approximate controllability of the fractional control system (1) by using the mild solution (20) and the control defined by (26). More precisely, we prove the existence of at least one state ∈ ( , ) satisfying (20) and (26) following the same arguments presented in [20], but using Dhage fixed point theorem. For this lets Θ ( ) = ( , ( )) ,…”

“…The fractional control systems involving a linear closed (unbounded) operator which generates resolvent operators were considered recently by many authors [15,[18][19][20]. The lack of the semigroup property of the generated resolvent operator was the most popular difficulty that has been faced by the interested researchers.…”

“…However, some authors used the idea of analytic sectorial operators to overcome this problem. For more details, we refer the reader to the papers [20][21][22][23] and references therein. To the best of our knowledge, there is not any investigation in the controllability problem via resolvent operators applied on hybrid systems such as the system that has been discussed in the article [24].…”