Control problems for semilinear second order equations with cosine families

Jin-Mun Jeong,Seong-Ho Cho,Eun-Young Ju

doi:10.1186/s13662-016-0854-y

Abstract

This paper aims to obtain the approximate controllability for the second order nonlinear control systems with a strongly cosine family and the associated with sine family.

Highlights

1 Introduction The first part of this paper gives some basic results on the regularity of solutions of the abstract semilinear second order initial value problem
C(t) is called a strongly continuous cosine family if the following conditions are satisfied: c( ) C(s + t) + C(s – t) = C(s)C(t), for all s, t ∈ R, and C( ) = I; c( ) C(t)x is continuous in t on R for each fixed x ∈ X
The following basic results on cosine and sine families are from Propositions . and . of [ ]

Summary

Introduction

In [ ], when f : R → X is continuously differentiable, x ∈ D(A), y ∈ E, and k ∈ W , ( , T), the existence of a solution w ∈ L ( , T; D(A)) ∩ W , ( , T; E) of In Section we establish to the approximate controllability for the second order nonlinear system C(t) is called a strongly continuous cosine family if the following conditions are satisfied: c( ) C(s + t) + C(s – t) = C(s)C(t), for all s, t ∈ R, and C( ) = I; c( ) C(t)x is continuous in t on R for each fixed x ∈ X.

Objectives

Results

Conclusion