ABSTRACT
We considered before optimization control problems for systems described by linear and nonlinear equations. However, these equations were linear with respect to the control. Now we analyze systems such that the state operator is nonlinear with respect to the control. Therefore, we shall use an additional technique. This is the Implicit function theorem that is an extension of the Inverse function theorem. This result can guarantee the differentiability of the implicit operator that is control–?state mapping. However, this theorem uses the solvability of the linearized equation in the natural spaces that is an analog of the general assumption of the Inverse function theorem. If this property is false, then the dependence of state function from the control can be non-differentiable. However, we can determine its extended derivatives. We consider optimization control problems for the systems described by a nonlinear elliptic equation with a nonlinear control as an application.
