Elastomer-based soft-continuum robots with an extensible backbone exhibit high flexibility. These manipulators might show nonlinear kinematic behaviors due to, for example, the material hyperelasticity and means of actuation. Formulating a reliable kinematic model for an effective inverse kinematics control strategy is challenging, but is paramount for allowing effective manoeuvrability and controllability. In this article, we devise a kinematic modeling and control method for pneumatic-driven soft-continuum robots (up to 100% elongation ratio). The method is based on the Cosserat rod model including a pressure-dependent dynamic modulus. The kinematic model and control strategy are then expressed as nonlinear least-squares optimization problems. Hence, various inverse kinematics control modes can be achieved for a multisegment robot, e.g., tip position and orientation control of the overall robot or tip position control of each segment. Simulations and experiments are both conducted to validate the proposed method. The results highlight the high fidelity of the modeling technique and the effectiveness of the proposed inverse kinematics controller. In particular, the modeling and trajectory control errors for a two-segment robot are smaller than 4.5 mm, i.e., 5% of the robot's overall length.