Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces

In this paper, we investigate the controllability for a class of abstract impulsive neutral functional differential systems with infinite delay where the linear part is non-densely defined and satisfies the Hille–Yosida condition. The approach used is the Schauder fixed point theorem combined with the operator semigroups. Particularly, the compactness of the operator semigroups is not needed in this article.     

Existence and controllability results for fractional semilinear differential inclusions

In this paper, we prove the existence and controllability results for fractional semilinear differential inclusions involving the Caputo derivative in Banach spaces. The results are obtained by using fractional calculation, operator semigroups and Bohnenblust–Karlin’s fixed point theorem. At last, an example is given to illustrate the theory.     

Controllability of Semilinear Differential Systems with Nonlocal Initial Conditions in Banach Spaces

In this note, we establish a sufficient condition for the controllability of a first-order semilinear differential system with nonlocal initial conditions in Banach spaces. The approach used is the Sadovskii fixed-point theorem combined with operator semigroups. Particularly, the compactness of the operator semigroups is not needed in this article. Y.K. Chang was supported by Tianyuan Youth Fund of Mathematics in China (10826063), NNSF of China (10801065), the Scientific Research Fund of Gansu Provincial Education Department (20868), and Qing Lan Talent Engineering Funds (QL-05-16A) of Lanzhou Jiaotong University. J.J. Nieto was supported by Ministerio de Educacion y Ciencia-FEDER Project MTM2007-61724, and by Xunta de Galicia-FEDER Project PGIDIT05PXIC20702PN.     

Localized null controllability and corresponding minimal norm control blowup rates of thermoelastic systems

In this paper, we consider the problem of null controllability for an elastic operator under square root damping. Such partial differential equation models can be described by analytic semigroups on the basic space of finite energy. Thus by inherent smoothing coming from the parabolic-like behavior of the dynamics, the problem of null controllability is appropriate for consideration. In particular, we will show that the solution variables can be steered to the zero state by means of iterations of locally supported steering controls acting on appropriate finite dimensional systems. The hinged boundary conditions considered here admit of a diagonalization of the spatial operator. The control strategy implemented in [A. Benabdallah, M. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal. 7 (2002) 585-599] is used to construct a suboptimal control for the problem, but here we expand upon their results by providing a bound for the energy function Emin(T), T>0. Our results are valid for localized mechanical and thermal control. The strategy relies heavily on the availability of a Carleman's estimate for finite linear combinations of eigenfunctions of the Dirichlet Laplacian.     

Controllability for Semilinear Neutral Functional Differential Inclusions via Analytic Semigroups

In this paper, we prove controllability results for semilinear neutral functional differential inclusions with finite or infinite delay in Banach spaces. Our theory makes use of analytic semigroups and fractional powers of closed operators, integrated semigroups and cosine families.     

Controllability for Neutral Stochastic Functional Integrodifferential Infinite Delay Systems in Abstract Space

In this paper, sufficient conditions are given for the controllability of a class of neutral stochastic functional integrodifferential equations with infinite delay in abstract space. The Nussbaum fixed point theorem is adapted to obtain the controllability results, which greatly extends the previous results to the stochastic settings with the help of analytic semigroups. An example is provided to illustrate the theory.     

Polynomial stability of semigroups generated by operator matrices

In this paper, we study the stability properties of strongly continuous semigroups generated by block operator matrices. We consider triangular and full operator matrices whose diagonal operator blocks generate polynomially stable semigroups. As our main results, we present conditions under which also the semigroup generated by the operator matrix is polynomially stable. The theoretical results are used to derive conditions for the polynomial stability of a system consisting of a two-dimensional and a one-dimensional damped wave equation.     

广义算子半群的PERRON问题

借助于广义算子半群和广义积分算子半群的关系,讨论广义算子半群的Perron型指数稳定性,研究了广义积分算子半群的渐近行为.     

具非局部条件的半线性发展方程的可控性

本文使用Schaefer不动点定理和强连续算子半群理论,建立了抽象空间中具非局部条件的半线性发展方程解的可控性,得到了可控性的充分条件.文末用例子说明了所得结果.     

Controllability results for functional semilinear differential inclusions in Fréchet spaces

In this paper, a recent nonlinear alternative for multivalued admissible contractions in Fréchet spaces due to Frigon combined with semigroups theory is used to investigate the controllability of some classes of semilinear functional and neutral functional differential inclusions in Fréchet spaces.     

Bounded controlling sequences,lower stability and certain penalty procedures

We present a unified approach to certain controllability and optimization questions such as approximate controllability with bounded sequences, necessary and sufficient conditions for lower stability, a problem of moments, penalty techniques. The control processes concerned are given by strongly continuous semigroups in Banach spaces, but the argument is led in the generality of continuous operators.     

Semigroups of Operators on Hardy Spaces and Cocycles of Holomorphic Flows

In this paper the strongly continuous semigroups of bounded cocycle-weighted composition operators for holomorphic flows on Hardy spaces are characterized. In the process, an easy-to-check criterion for an operator to be a weighted composition operator is established.     

Triangularizability of Polynomially Compact Operators

An operator on a complex Banach space is polynomially compact if a non-zero polynomial of the operator is compact, and power compact if a power of the operator is compact. Theorems on triangularizability of algebras (resp. semigroups) of compact operators are shown to be valid also for algebras (resp. semigroups) of polynomially (resp. power) compact operators, provided that pairs of operators have compact commutators.     

Hypercyclic semigroups and somewhere dense orbitsSemigroupes hypercycliques et orbites quelque part denses

We study hypercyclicity of linear strongly continuous semigroups. In the case of iterations of a single operator Bourdon and Feldman have recently proved that the existence of somewhere dense orbits implies hypercyclicity. We show the corresponding result for semigroups. As a consequence, a conjecture of Herrero concerning iterations of a single operator also holds for strongly continuous semigroups. To cite this article: G. Costakis, A. Peris, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 895–898.     

Existence Results on Semi-Infinite Intervals for Nonlocal Evolutions Equations and Inclusions via Semigroup Theory

In this paper, we prove existence and controllability results on semi-infinite intervals for first- and second-order evolutions equations and inclusions in Banach spaces with nonlocal conditions. Our theory makes use of semigroups, cosine families, nonlinear alternative of Leray–Schauder, and a diagonalization argument.     

Maximal dominated operator semigroups

Maximal operator semigroups, bounded in a certain sense, on real or complex vector spaces are studied. For any maximal semigroup \MM dominated by a certain pair of homogeneous functions there is an operator quasinorm for which \MM is exactly the semigroup of contractions in this quasinorm. Applications to Riesz spaces are given. In particular, maximal semigroups of matrices dominated by a given positive matrix are characterized. We thus answer the question implicitly posed in [2]. November 20, 1998     

On shape preserving semigroups

Motivated by positivity-, monotonicity-, and convexity preserving differential equations, we introduce a definition of shape preserving operator semigroups and analyze their fundamental properties. In particular, we prove that the class of shape preserving semigroups is preserved by perturbations and taking limits. These results are applied to partial delay differential equations.     

Boundary controllability for the time-fractional nonlinear Korteweg-de Vries (KdV) equation

In this paper, we study the time-fractional nonlinear Korteweg-de Vries (KdV) equation. By using the theory of semigroups, we prove the well-posedness of the time-fractional nonlinear KdV equation. Moreover, we present the boundary controllability result for the problem.     

CONTROLLABILITY OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this article, we give sufficient conditions for controllability of some partial neutral functional differential equations with infinite delay. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. The results are obtained using the integrated semigroups theory. An application is given to illustrate our abstract result.     

Robust viability of hybrid systems

Robust viability of hybrid systems is examined based on the controllability operator subject to transition dynamics uncertainty. Two modifications to the controllability operator are introduced, these being the uncertain controllability operator and the uncertainty operator. It is shown how existing fixed-point approximation algorithms can be generalized and applied to compute robustly viable sets for hybrid systems. The three-tank control problem is considered. Robust viable sets are computed for this system subject to transition dynamics uncertainties.