  This paper presents a general description of a parameter estimation inverse
problem for systems governed by nonlinear differential equations. The inverse
problem is presented using optimal control tools with state constraints, where
the minimization process is based on a first-order optimization technique such
as adaptive monotony-backtracking steepest descent technique and nonlinear
conjugate gradient methods satisfying strong Wolfe conditions. Global
convergence theory of both methods is rigorously established where new linear
convergence rates have been reported. Indeed, for the nonlinear non-convex
optimization we show that under the Lipschitz-continuous condition of the
gradient of the objective function we have a linear convergence rate toward a
stationary point. Furthermore, nonlinear conjugate gradient method has also
been shown to be linearly convergent toward stationary points where the second
derivative of the objective function is bounded. The convergence analysis in
this work has been established in a general nonlinear non-convex optimization
under constraints framework where the considered time-dependent model could
whether be a system of coupled ordinary differential equations or partial
differential equations. Numerical evidence on a selection of popular nonlinear
models is presented to support the theoretical results. Nonlinear Conjugate
gradient methods, Nonlinear Optimal control and Convergence analysis and
Dynamical systems and Parameter estimation and Inverse problem
