Local controllability of nonlinear systems

Abstract

We establish the local controllability about a fixed solution x0(t)of the vector system x(t)=F(t, x(t), x(t- w(t)), u(t), u(t- h(t))) on a bounded interval J by assuming the corresponding variational linear system is completely controllable on J. With an appropriate bound on the control functions u we also obtain the stability of solutions around x0(t). For the neutral system x(t)=G(x(t), x(t- 1), x(t- 1), u(t), u(t- h)) a similar result is obtained about the origin, where G is zero at the origin. We apply local controllability results for nonlinear systems in the areas of control design using bang-bang control functions, the capture problem in a conservative force field, and discrete economic systems where the control function represents a tax rate vector.