M. Muslim

Existence and approximation of solutions to fractional differential equations

In this paper we shall study a fractional order semilinear differential equation in an arbitrary Banach space X. We used the analytic semigroup theory of linear operators and fixed point method to establish the existence and uniqueness of solutions of the given problem. We also prove the existence of a global solution. Existence and convergence of an approximate solution to the given problem is also proved in a separable Hilbert space. Finally, we give an example to illustrate the applications of the abstract results.

Existence and approximations of solutions to some fractional order functional integral equations

In this paper we shall study a fractional order functional integral equation. In the first part of the paper, we proved the existence and uniqueness of mile and global solutions in a Banach space. In the second part of the paper, we used the analytic semigroups theory oflinear operators and the fixed point method to establish the existence, uniqueness and convergence of approximate solutions of the given problem in a separable Hilbert space. We also proved the existence and convergence of Faedo-Galerkin approximate solution to the given problem. Finally, we give an example.

Approximation of solutions to history-valued neutral functional differential equations

In this paper, we consider a class of abstract neutral functional differential equations in a separable Hilbert space and study the approximation of solutions. An example is also given to illustrate the applications of the abstract results.

Approximation of solutions to retarded differential equations with applications to population dynamics

We consider a retarded differential equation with applications to population dynamics. We establish the convergence of a finite-dimensional approximations of a unique solution, the existence and uniqueness of which are also proved in the process.

Sampled-data reliable stabilization of T-S fuzzy systems and its application

In this article, based on sampled-data approach, a new robust state feedback reliable controller design for a class of Takagi–Sugeno fuzzy systems is presented. Different from the existing fault models for reliable controller, a novel generalized actuator fault model is proposed. In particular, the implemented fault model consists of both linear and nonlinear components. Consequently, by employing input-delay approach, the sampled-data system is equivalently transformed into a continuous-time system with a variable time delay. The main objective is to design a suitable reliable sampled-data state feedback controller guaranteeing the asymptotic stability of the resulting closed-loop fuzzy system. For this purpose, using Lyapunov stability theory together with Wirtinger-based double integral inequality, some new delay-dependent stabilization conditions in terms of linear matrix inequalities are established to determine the underlying systems stability and to achieve the desired control performance. Finally, to show the advantages and effectiveness of the developed control method, numerical simulations are carried out on two practical models.

Observer-based dissipative control for Markovian jump systems via delta operators

ABSTRACT This paper addresses the issue of observer-based dissipative control problem for a class of Markovian jump systems with random delay via delta operator approach. First, based on the construction of a novel Lyapunov functional together with the use of free-weighting matrix approach, a new set of sufficient conditions is established which ensures the stochastic asymptotic stability and dissipativity of the closed-loop augmented Markovian jump delta operator system. Next, the result is extended to design an observer-based state feedback dissipative control law such that the resulting closed-loop system is stochastically asymptotically stable with the desired dissipative performance index. Further, the existence of control laws is formulated in the form of linear matrix inequalities (LMIs) which can be easily solved by using some standard numerical packages. Also, the observer and control gains can be calculated by using the solutions of an obtained set of LMIs. It is worth pointing out that the dissipative control problem considered here includes the H∞ and passivity-based control problems as special cases. Finally, two numerical examples with simulation are presented to demonstrate the effectiveness of the obtained design technique.

Approximation of solutions to non-local history-valued retarded differential equations

In the present work we consider a retarded differential equation and prove the existence, uniqueness and convergence of approximate solutions. Also we consider the Faedo-Galerkin approximations of the solution and the convergence.

Faedo–Galerkin approximation of second order nonlinear differential equation with deviated argument

In this manuscript, we consider a second order nonlinear differential equation with deviated argument in a separable Hilbert space X. We used the strongly continuous cosine family of linear operators and fixed point method to study the existence of an approximate solution of the second order differential equation. We define the fractional power of the closed linear operator and used it to prove the convergence of the approximate solution. Also, we prove the existence and convergence of the Faedo–Galerkin approximate solution. Finally, we give an example to illustrate the application of these abstract results.