Non-Exponential Stabilization of Nonlinear Discrete-Time Systems

In this paper, we investigate the stabilization problem for a class of nonlinear discrete-time control systems. We consider semilinear systems with a nonlinear perturbation satisfying a Hölder-type condition. Based on a new discrete inequality of Gronwall type, we establish sufficient conditions for the global and local weak stabilizability of semilinear and nonlinear discrete-time systems, respectively.     

Exact Controllability of Semilinear Evolution Systems and Its Application

In this paper, we obtain several abstract results concerning the exact controllability of semilinear evolution systems. First, we prove the null local exact controllability of semilinear first-order systems by means of the contraction mapping principle; in this case, we do not assume any compactness. Next, we derive the global and/or local exact controllability of semilinear second-order systems by means of the Schauder fixed-point theorem; in this case, we assume only the embedding of the related spaces having some compactness, which is reasonable for many concrete problems. Our main result shows that the observability of the dual of the linearized system implies the exact controllability of the original semilinear system. Finally, we apply our abstract results to the exact controllability of the semilinear wave equation.     

Controllability of Some Nonlinear Systems in Hilbert Spaces

In this paper, several abstract results concerning the controllability of semilinear evolution systems are obtained. First, approximate controllability conditions for semilinear systems are obtained by means of a fixed-point theorem of the Rothe type; in this case, the compactness of the linear operator is assumed. Next, the exact controllability of semilinear systems with nonlinearities having small Lipschitz constants is derived by means of the Banach fixed-point theorem; in this case, the compactness of the operators is not assumed. In both cases, it is proven that the controllability of the linear system implies the controllability of the associated semilinear system. Finally, these abstract results are applied to the controllability of the semilinear wave and heat equations.     

Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces

In this paper, we establish the existence results for semilinear differential systems with nonlocal initial conditions in Banach spaces. The approaches used are fixed point theorems combined with convex-power condensing operators. The first result obtained will be applied to a class of semilinear parabolic equations.     

On the Existence of Positive Solutions for a Class of Semilinear Elliptic Systems

We study the existence of positive solutions for a class of semilinear ellipcic systems in general domains via the blow up argument and degree theory. The main idea can be used to establish the existence of positive solutions for the Navier problems of polyharmonic semilinear equations in general domains.     

Relaxation of nonlinear impulsive controlled systems on Banach spaces

Relaxation control for a class of semilinear impulsive controlled systems is investigated. Existence of mild solutions for semilinear impulsive controlled systems is proved. By introducing a regular countably additive measure, we convexify the original control systems and obtain the corresponding relaxed control systems. The existence of optimal relaxed controls and relaxation results is also proved.     

Strong solutions of semilinear matched microstruture models

The subject of this article is a matched microstructure model for Newtonian fluid flows in fractured porous media. This is a homogenized model which takes the form of two coupled parabolic differential equations with boundary conditions in a given (two-scale) domain in Euclidean space. The main objective is to establish the local well-posedness in the strong sense of the flow. Two main settings are investigated: semilinear systems with linear boundary conditions and semilinear systems with nonlinear boundary conditions. With the help of analytic semigroups, we establish local well-posedness and investigate the long-time behavior of the solutions in the first case: we establish global existence and show that solutions converge to zero at an exponential rate.     

Propagation of Singularities of Solutions to Hyperbolic‐Parabolic Coupled Systems

In this paper, the authors study the propagation of singlarities for a semilinear hyperbolic‐parabolic coupled system, which comes from the model of thermoelasticity. Both of the Cauchy problem and the problem inside of a domain are considered. We obtain that the microlocal singularities of solutions to the semilinear hyperbolic‐parabolic coupled system are propagated along null bicharacteristics of the hyperbolic operator by using the theory of paradifferential operators. Furthermore, for the Cauchy problem of the semilinear coupled system, if the initial data have singularities at the origin, we prove that the solutions have the same order regularity with respect to spatial variables as in hyperbolic problems in the forward characteristic cone issuing from the origin, which improves the previous results for semilinear systems in thermoelasticity.     

Approximate controllability of impulsive semilinear stochastic system with delay in state

Many practical systems in physical and biological sciences have impulsive dynamical behaviors during the evolution process that can be modeled by impulsive differential equations. This article studies the approximate controllability of impulsive semilinear stochastic system with delay in state in Hilbert spaces. Assuming the conditions for the approximate controllability of the corresponding deterministic linear system, we obtain the sufficient conditions for the approximate controllability of the impulsive semilinear stochastic system with delay in state. The results are obtained by using Banach fixed point theorem. Finally, two examples are given to illustrate the developed theory.     

Null controllability of semilinear integrodifferential systems in Banach space

Sufficient conditions for null controllability of semilinear integrodifferential systems with unbounded linear operators in Banach space are established. The results are obtained using semigroup of linear operators, fractional powers of operators, and the Schauder fixed point theorem. An application to partial integrodifferential equations is given.     

Bifurcation from trivial solution for elliptic systems with nonlinear boundary conditions

In this paper, we investigate semilinear elliptic systems having a parameter with nonlinear Neumann boundary conditions over a smooth bounded domain. The objective of our study is to analyse bifurcation component of positive solutions from trivial solution and their stability. The results are obtained via classical bifurcation theorem from a simple eigenvalue, by studying the eigenvalue problem of elliptic systems.     

Some existence results for a class of degenerate semilinear elliptic systems

Using a variational approach, we investigate a class of degenerate semilinear elliptic systems with measurable, unbounded nonnegative weights, where the domain is bounded or unbounded. Some existence results are obtained.     

Approximate controllability of the non-autonomous impulsive evolution equation with state-dependent delay in Banach spaces

In this paper, we consider the non-autonomous semilinear impulsive differential equations with state-dependent delay. The approximate controllability results of the first-order systems are obtained in a separable reflexive Banach space, which has a uniformly convex dual. In order to establish sufficient conditions of the approximate controllability of such a system, we have used the theory of linear evolution systems, properties of the resolvent operator and Schauder’s fixed point theorem. Finally, we provide two concrete examples to validate our results.     

非局部半线性积微分系统的可控制性

本文运用半群理论和Schauder不动点定理,在Banach空间研究了一类非局部半线性积微分系统的可控制性.最后,举例说明了所得结果.     

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.     

A prior1 bounds for a class of stationary diffusion systems

We establish a priori bounds for a class of semilinear elliptic systems which have semilinear boundary conditions. These bounds are obtained as a consequence of imposing a convex Lyapunov-like structure upon our systems.     

On infinite-dimensional control systems with state and control constraints

In this paper we examine infinite-dimensional control systems governed by semilinear evolution equations and having both state and control constraint. We introduce the relaxed system and show that the original trajectories are dense in an appropriate function space in the relaxed ones. We also determine the dependence of the solution set on the initial conditions. Then using those results we establish necessary and sufficient conditions for optimality for some optimization problems. Finally we prove some controllability results.     

Existence and nonexistence results for critical growth polyharmonic elliptic systems

In this work, we consider semilinear elliptic systems for the polyharmonic operator having a critical growth nonlinearity. We establish conditions for existence and nonexistence of nontrivial solutions to these systems.     

Generalized Quasilinearization Method for a Class of Semilinear Elliptic Systems

In this paper, the method of generalized quasilinearization is extended to a class of semilinear elliptic systems, and the sequences which are the solutions of linear differential equations that converge to the unique solution of the given semilinear elliptic system are obtained.     

Qualitative properties of blow-up solutions to some semilinear elliptic systems in non-convex domains

In this paper we study qualitative properties of boundary blow-up solutions to some semilinear elliptic cooperative systems in bounded non-convex domains. In particular, by a careful adaptation of the celebrated moving plane procedure of Alexandrov–Serrin, we deduce symmetry and monotonicity results for blow-up solutions for this class of systems.