A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application

In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space R N . As an application, we then show the exact null-controllability for this semilinear heat equation in R N . The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under consideration. The latter is crucial when one uses a suitable approximation argument to derive the global null-controllability for the semilinear heat equation in R N . This allows us to overcome the well-known problem of the lack of compactness embedding arising in the study of null-controllability for nonlinear PDEs in generally unbounded domains.

State feedback containment control of multi-agents system with lipschitz nonlinearity

<p>This paper studies the containment control problem of the leader-follower configuration in a multi-agents system included with a type of nonlinearity such as Lipschitz with respect to continuous-time and directed spanning forest communication network topology. A state feedback containment controller is designed and proposed with control theory and the Laplacian network structure where it utilizes the relative information of each agent. The controller designed ensures that the followers are contained by the leaders that form the convex hull formation. For the containment to happen, a minimum of one leader is needed to have a direct communication trajectory to the followers. Lyapunov stability theory is used to provide the stability conditions after analyzing the network structure. Finally, it has been shown from simulation that the followers are contained successfully with the proposed controller.</p>

On Existence of Sequences of Weak Solutions of Fractional Systems with Lipschitz Nonlinearity

In this article, the variational method together with two control parameters is used for introducing the proof for the existence of infinitely many solutions for a new class of perturbed nonlinear system having p -Laplacian fractional-order differentiation.

Periodic solutions for a differential inclusion problem involving the p(t)-Laplacian

In the present paper, we consider the nonlinear periodic systems involving variable exponent driven by p(t)-Laplacian with a locally Lipschitz nonlinearity. Our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue-Sobolev space. Applying the non-smooth critical point theory, we establish some new existence results.

Infinite Existence Solutions of Fractional Systems with Lipschitz Nonlinearity

The paper deals with the existence of infinitely many solutions of a class of perturbed nonlinear fractional p -Laplacian differential systems using one control parameter combined with the variational method.

A Case Study on Designing a Sliding Mode Controller to Stabilize the Stochastic Effect of Noise on Mechanical Structures: Residential Buildings Equipped with ATMD

This study aims to stabilize the unwanted fluctuation of buildings as mechanical structures subjected to earth excitation as the noise. In this study, the ground motion is considered as a Wiener process, in which the governing stochastic differential equations have been presented in the form of Ito equation. To stabilize the vibration of the system, the ATMD system is considered and located on the upmost story of the building. A sliding mode controller has been utilized to control the ATMD system, which is a robust controller in the presence of uncertainty. For this purpose, the design of a sliding mode controller for the general dynamic system with Lipschitz nonlinearity and considering the Ito relations has been accomplished. The mentioned design has been implemented considering the presence of the Weiner process and existence of uncertainty in the structure and actuator. Then, the obtained general control law has been generalized to control the ATMD system. The results show that the designed controller is effective to reduce the effect of the unwanted impused vibrations on the building.