An analysis on the approximate controllability results for Caputo fractional hemivariational inequalities of order 1 < r < 2 using sectorial operators

In this paper, we investigate the effect of hemivariational inequalities on the approximate controllability of Caputo fractional differential systems. The main results of this study are tested by using multivalued maps, sectorial operators of type (P, η, r, γ ), fractional calculus, and the fixed point theorem. Initially, we introduce the idea of mild solution for fractional hemivariational inequalities. Next, the approximate controllability results of semilinear control problems were then established. Moreover, we will move on to the system involving nonlocal conditions. Finally, an example is provided in support of the main results we acquired

Approximate controllability results for fractional semilinear integro-differential inclusions in Hilbert spaces

In this paper, we consider a class of fractional integro-differential inclusions in Hilbert spaces. This paper deals with the approximate controllability for a class of fractional integro-differential control systems. First, we establishes a set of sufficient conditions for the approximate controllability for a class of fractional semilinear integro-differential inclusions in Hilbert spaces. We use Bohnenblust-Karlin's fixed point theorem to prove our main results. Further, we extend the result to study the approximate controllability concept with nonlocal conditions. An example is also given to illustrate our main results.Comment: arXiv admin note: substantial text overlap with arXiv:1502.0008

Approximate controllability and optimal control of impulsive fractional semilinear delay differential equations with non-local conditions

In this paper we study the approximate controllability and existence of optimal control of impulsive fractional semilinear delay differential equations with non-local conditions. We use Sadovskii's fixed point theorem, semigroup theory of linear operators and direct method for minimizing a functional to establish our results. At the end we give an example to illustrate our analytical findings.Comment: 15 page

Partial-Approximate Controllability of Nonlocal Fractional Evolution Equations via Approximating Method

In this paper we study partial-approximate controllability of semilinear nonlocal fractional evolution equations in Hilbert spaces. By using fractional calculus, variational approach and approximating technique, we give the approximate problem of the control system and get the compactness of approximate solution set. Then new sufficient conditions for the partial-approximate controllability of the control system are obtained when the compactness conditions or Lipschitz conditions for the nonlocal function are not required. Finally, we apply our abstract results to the parial-approximate controllability of the semilinear heat equation and delay equation

Controllability of the impulsive semi linear beam equation with memory and delay

The semilinear beam equation with impulses, memory and delay is considered. We obtain the approximate controllability. This is done by employing a technique that avoids fixed point theorems and pulling back the control solution to a fixed curve in a short time interval. Demonstrating, once again, that the controllability of a system is robust under the influence of impulses and delays.Comment: 10 page

A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515

Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces

We introduce a new notion called fractional stochastic nonlocal condition, and then we study approximate controllability of class of fractional stochastic nonlinear differential equations of Sobolev type in Hilbert spaces. We use Hölder's inequality, fixed point technique, fractional calculus, stochastic analysis and methods adopted directly from deterministic control problems for the main results. A new set of sufficient conditions is formulated and proved for the fractional stochastic control system to be approximately controllable. An example is given to illustrate the abstract results

Finite-approximate controllability of evolution systems via resolvent-like operators

In this work we extend a variational method to study the approximate controllability and finite dimensional exact controllability ( finite-approximate controllability) for the semilinear evolution equations in Hilbert spaces. We state a useful characterization of the finite-approximate controllability for linear evolution equation in terms of resolvent-like operators. We also find a control so that, in addition to the approximate controllability requirement, it ensures finite dimensional exact controllability. Assuming the approximate controllability of the corresponding linearized equation we obtain sufficient conditions for the finite-approximate controllability of the semilinear evolution equation under natural conditions. The obtained results are generalization and continuation of the recent results on this issue. Applications to heat equations are treated

Partial complete controllability of deterministic semilinear systems

In this paper the concept of partial complete controllability for deterministic semilinear control systems in separable Hilbert spaces is investigated. Some important systems can be expressed as a first order differential equation only by enlarging the state space. Therefore, the ordinary controllability concepts for them are too strong. This motivates the partial controllability concepts, which are directed to the original state space. Based on generalized contraction mapping theorem, a sufficient condition for the partial complete controllability of a semilinear deterministic control system is obtained in this paper. The result is demonstrated through appropriate examples.Publisher's Versio

Optimal Solutions to Relaxation in Multiple Control Problems of Sobolev Type with Nonlocal Nonlinear Fractional Differential Equations

We introduce the optimality question to the relaxation in multiple control problems described by Sobolev type nonlinear fractional differential equations with nonlocal control conditions in Banach spaces. Moreover, we consider the minimization problem of multi-integral functionals, with integrands that are not convex in the controls, of control systems with mixed nonconvex constraints on the controls. We prove, under appropriate conditions, that the relaxation problem admits optimal solutions. Furthermore, we show that those optimal solutions are in fact limits of minimizing sequences of systems with respect to the trajectory, multi-controls, and the functional in suitable topologies.Comment: This is a preprint of a paper whose final and definite form will be published in Journal of Optimization Theory and Applications, ISSN 0022-3239 (print), ISSN 1573-2878 (electronic). Submitted: 26-Dec-2014; Revised: 14-Apr-2015; Accepted: 19-Apr-201