Syed Abbas

Almost periodic solutions of neutral functional differential equations

In this paper we study a non-autonomous neutral functional differential equation in a Banach space. Applying the theory of semigroups of operators to evolution equations and Krasnoselskiis fixed point theorem we establish the existence and uniqueness of a mild almost periodic solution of the problem under consideration.

Dynamical analysis of fractional-order modified logistic model

In this paper, we study a fractional differential equation model of the single species multiplicative Allee effect. First we study the stability of equilibrium points. Further we give some sufficient conditions ensuring the existence and uniqueness of integral solution. In the last section we perform several numerical simulations to validate our analytical findings.

Existence and attractivity of k-almost automorphic sequence solution of a model of cellular neural networks with delay

Abstract In this paper we discuss the existence and global attractivity of k -almost automorphic sequence solution of a model of cellular neural networks. We consider the corresponding difference equation analogue of the model system using suitable discretization method and obtain certain conditions for the existence of solution. Almost automorphic function is a good generalization of almost periodic function. This is the first paper considering such solutions of the neural networks.

On Near-Optimal Mean-Field Stochastic Singular Controls: Necessary and Sufficient Conditions for Near-Optimality

Near-optimization is as sensible and important as optimization for both theory and applications. This paper deals with necessary and sufficient conditions for near-optimal singular stochastic controls for nonlinear controlled stochastic differential equations of mean-field type, which is also called McKean–Vlasov-type equations. The proof of our main result is based on Ekeland’s variational principle and some estimates of the state and adjoint processes. It is shown that optimal singular control may fail to exist even in simple cases, while near-optimal singular controls always exist. This justifies the use of near-optimal stochastic controls, which exist under minimal hypotheses and are sufficient in most practical cases. Moreover, since there are many near-optimal singular controls, it is possible to select among them appropriate ones that are easier for analysis and implementation. Under an additional assumptions, we prove that the near-maximum condition on the Hamiltonian function is a sufficient condition for near-optimality. This paper extends the results obtained in (Zhou, X.Y.: SIAM J. Control Optim. 36(3), 929–947, 1998) to a class of singular stochastic control problems involving stochastic differential equations of mean-field type. An example is given to illustrate the theoretical results.

Pseudo almost automorphic solutions of some nonlinear integro-differential equations

In this paper we discuss the existence and uniqueness of a pseudo almost automorphic solution of an integro-differential equation in a Banach space X. We achieve our results using the methods of fractional powers of operators and the Banach fixed point theorem. These results are new and complement the existing ones.

On necessary and sufficient conditions for near-optimal singular stochastic controls

In this paper we discuss the necessary and sufficient conditions for near-optimal singular stochastic controls for the systems driven by a nonlinear stochastic differential equations (SDEs in short). The proof of our result is based on Ekeland’s variational principle and some delicate estimates of the state and adjoint processes. It is well known that optimal singular controls may fail to exist even in simple cases. This justifies the use of near-optimal singular controls, which exist under minimal conditions and are sufficient in most practical cases. Moreover, since there are many near-optimal singular controls, it is possible to choose suitable ones, that are convenient for implementation. This result is a generalization of Zhou’s stochastic maximum principle for near-optimality to singular control problem.

Asymptotic almost automorphic solutions of impulsive neural network with almost automorphic coefficients

In this paper existence and asymptotic stability of asymptotically almost automorphic solution of impulsive neural networks with delay is discussed. The results are established by using various fixed point theorems and Lyapunov-like functional. As far as we know, this is the first paper to discuss such kind of solutions for impulsive neural networks. At the end, we give few examples to illustrate our theoretical findings. One can see that the numerical simulation results show asymptotically almost automorphic behaviour of the solution.

Analysis of Caputo Impulsive Fractional Order Differential Equations with Applications

We use Sadovskiis fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and compliment the existing one.

On Mean-Field Partial Information Maximum Principle of Optimal Control for Stochastic Systems with Levy Processes

In this paper, we study the mean-field-type partial information stochastic optimal control problem, where the system is governed by a controlled stochastic differential equation, driven by the Teugels martingales associated with some Lévy processes and an independent Brownian motion. We derive necessary and sufficient conditions of the optimal control for these mean-field models in the form of a maximum principle. The control domain is assumed to be convex. As an application, the partial information linear quadratic control problem of the mean-field type is discussed.

Stability analysis of two prey one predator model

In this paper we study a two prey one predator model with team approach. We analyze local stability behaviour of the system with the help of linearization and persistence behaviour of the system with the help of persistence of all three teams individually.