Constrained controllability of second order retarded nonlinear systems with nonlocal condition

Abstract

In this paper, the constrained controllability of the second order retarded nonlinear systems with nonlocal condition has been established by using the theory of cosine families and the generalized open mapping theorem. A new set of sufficient conditions for the constrained controllability of retarded nonlinear systems is established under the assumption that the associated linear system is controllable. By using the Banach fixed point theorem, the existence of mild solution for the considered system with nonlocal delay condition has been deduced. Finally, an example is provided to verify the effectiveness of the obtained theory.

Results on the approximate controllability of fractional hemivariational inequalities of order $1< r<2$

“…The notion of exact controllability has an essential role in mathematical control theories and technology. In recent years, many authors have done fruitful achievements on exact and approximate controllability of different nonlinear dynamical problems; one can refer to the research articles [16,30].…”

Relative controllability of a stochastic system using fractional delayed sine and cosine matrices

“…A solution representation for the linear inhomogeneous differential equation with constant coefficients and pure delay was established using the form of sine and cosine delayed matrices of polynomials of degree dependent on the value of delay in [11]. A set of sufficient conditions for the constrained controllability of retarded nonlinear systems is established using the Banach fixed point theorem, and the existence of a mild solution for the considered system with nonlocal delay condition was established in [23]. Using the delay Grammian matrix involving the delayed matrix sine, the authors presented sufficient and necessary conditions of controllability for linear problem governed by second-order delay differential equations in [25].…”

Approximate controllability of fractional neutral evolution systems of hyperbolic type

“…By using the Sadovskii fixed point theorem, the authors in [23] studied the controllability of fractional differential systems with nonlocal conditions . By using the theory of cosine families and the generalized open mapping theorem, the authors in [14] investigated the constrained controllability of the second order retarded nonlinear systems with nonlocal condition.…”