Nonlinear Impulsive Evolution Equations Research Articles

Optimal control and feedback control for nonlinear impulsive evolutionary equations

The aim of this paper is to study the existence, optimal control and feedback control for a class of nonlinear impulsive evolution equations involving weakly continuous operators. We first obtain the existence result of the evolution equation without impulse in any subinterval by Rothe method, and construct a solution of impulsive evolution equation by using this result. Moreover, we obtain the existence of optimal control pairs for the optimal control problem and feasible pairs for the feedback control system by establishing several sufficient assumptions. An application on a quasistatic frictional contact problem is provided.

Mild Solution of Second-Order Impulsive Integro-Differential Evolution Equations of Volterra Type in Banach Spaces

In this paper, by using the cosine family theory, measure of non-compactness and the generalization of Darbo-fixed point theorem, we establish the existence of mild solutions of nonlinear second-order impulsive integro-differential evolution equations of Volterra type in Banach spaces. The result obtained herein generalizes and improves some known results.

A Study on Impulsive Hilfer Fractional Evolution Equations with Nonlocal Conditions

Abstract In this paper, we concern with the existence of mild solution to nonlocal initial value problem for nonlinear Sobolev-type impulsive evolution equations with Hilfer fractional derivative which generalized the Riemann–Liouville fractional derivative. At first, we establish an equivalent integral equation for our main problem. Second, by means of the properties of Hilfer fractional calculus, combining measure of noncompactness with the fixed-point methods, we obtain the existence results of mild solutions with two new characteristic solution operators. The results we obtained are new and more general to known results. At last, an example is provided to illustrate the results.

Periodic boundary value problems for controlled nonlinear impulsive evolution equations on Banach spaces

This paper deals with the Periodic boundary value problems for Controlled nonlinear impulsive evolution equations. By using the theory of semigroup and fixed point methods, some conditions ensuring the existence and uniqueness. Finally, two examples are provided to demonstrate the effectiveness of the proposed results.

A general class of periodic boundary value problems for controlled nonlinear impulsive evolution equations on Banach spaces

This paper deals with the periodic boundary value problems for nonlinear impulsive evolution equations with controls. By using the theory of semigroup and fixed point methods, we address some conditions ensuring the existence and uniqueness. Finally, two examples are provided to prove the effectiveness of the proposed results.

Nonlinear impulsive evolution equations with nonlocal conditions and optimal controls

This paper is concerned with controlled nonlinear impulsive evolution equations with nonlocal conditions. The existence of PC-mild solutions is proved, but the uniqueness cannot be obtained. By constructing approximating minimizing sequences of functions, the existence of optimal controls of systems governed by nonlinear impulsive evolution equations is also presented.

Periodic boundary value problems for nonlinear impulsive evolution equations on Banach spaces

In this paper, we consider periodic boundary value problems for two new type nonlinear impulsive evolution equations on Banach spaces. By using the theory of semigroup, a mixed type Gronwall inequality and fixed point methods, we establish several sufficient conditions on the existence of mild solutions for such problems. Finally, examples are given to illustrate our main results.

Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations

In this paper, we study optimal relaxed controls and relaxation of nonlinear fractional impulsive evolution equations. Firstly, existence of piecewise continuous mild solutions for the original fractional impulsive control system is presented. Secondly, fractional impulsive relaxed control system is constructed by using a regular countably additive measure and making the original control system convexified. Thirdly, optimal relaxed controls and relaxation theorems are obtained. Finally, application to initial-boundary value problem of fractional impulsive parabolic control system is considered.

Second-order nonlinear impulsive evolution equations with time-varying generating operators and optimal controls

In this article, the second-order nonlinear impulsive evolution differential equations with time-varying generating operators is considered. Constructing evolution systems generated by time-varying operator matrix, we introduce suitable mild solution of the second-order nonlinear impulsive evolution differential equations. The existence and uniqueness of the mild solutions and the continuous dependence on initial value are proved. The existence of the optimal controls for a Lagrange problem of the systems governed by the second-order nonlinear impulsive evolution equations is also presented. An example is given for demonstration.

Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls

This paper is concerned with the second order nonlinear impulsiveevolution differential equations perturbed by unbounded operator onBanach space. Discussing the perturbation of time-varying operatormatrix and constructing the corresponding evolution system generatedby operator matrix we introduce the reasonable mild solution ofsecond order nonlinear impulsive evolution differential equationsand prove the existence of mild solutions. Existence of optimalcontrols for a Lagrange problem of systems governed by the secondorder nonlinear impulsive evolution equations is also presented. Anexample is given for demonstration.

Relaxed Control for a Class of Strongly Nonlinear Impulsive Evolution Equations

Relaxed control for a class of strongly nonlinear impulsive evolution equations is investigated. Existence of solutions of strongly nonlinear impulsive evolution equations is proved and properties of original and relaxed trajectories are discussed. The existence of optimal relaxed control and relaxation results are also presented. For illustration, one example is given.

Optimal Feedback Control for a Class of Strongly Nonlinear Impulsive Evolution Equations

In this paper, we consider the optimal feedback control problems of a system governed by strongly nonlinear impulsive differential equations which contains monotone operators and nonlinear nonmonotone perturbations. Based on the existence of feasible pairs, an existence result of optimal control pairs for the Lagrange problem is presented.

Optimal control for a class of strongly nonlinear impulsive equations in Banach spaces

In this paper, we consider the optimal control problem of system governed by a class of strongly nonlinear impulsive evolution equations. Based on the existence of strongly nonlinear impulsive evolution equations, which contain nonlinear monotone operators and nonmonotone perturbations, we prove the existence of optimal controls for the general Lagrange problem. The necessary conditions for optimality are also presented.

Strongly nonlinear impulsive evolution equations and optimal control

Strongly nonlinear impulsive evolution equations are investigated. Existence of solutions of strongly nonlinear impulsive equations is proved and some properties of the solutions are discussed. These results are applied to Lagrange problems of optimal control and we proved existence results. For illustration, an example of a quasi-linear impulsive parabolic differential equation and the corresponding optimal control is also presented.