Data-driven modeling and feedback control play a vital role in several application areas ranging from robotics, control theory, manufacturing to management of assets, financial portfolios and supply chains. Many such problems in one way or another are related to variational problems in optimal control and machine learning.

The following work first presents, a generalized representer theorem approach to solving such variational problems when closed, densely defined operators, like the differential operators, are involved. Furthermore, loss functionals on infinite dimensional Hilbert spaces are considered to allow for greater freedom in problem formulations. The statement of the theorem presents a necessary and sufficient condition for the existence of linear representer for optimal solutions of such problems. Finally, examples, applying the theorem to neural networks, stochastic regression, and sparsity-inducing regularization problems are presented.

The second part of the thesis deals with applications of variational optimization in control problems. Examples from optimal control and model predictive control are presented for applications in the domain of autonomous vehicles and airborne wind energy systems. First, a combination of manifold learning and model predictive control is presented for obstacle avoidance in autonomous driving. Manifold learning is presented as a means to describe boundaries of star-shaped sets for which a single inequality constraint is sufficient to check containment of a point in the set's interior. The approach presented, learns the largest star-shaped set within a circular range such that all obstacle points remain outside the set. The inequality condition for checking containment in such sets is incorporated into a multi-phase, free-end-time optimal control problem to plan trajectories and control inputs moving the vehicle from one point to another while remaining within a given collection of star-shaped sets. The multi-phase, free-end-time problem is adapted to a moving horizon form to give a model predictive path following controller that avoids obstacles by virtue of the manifold learning scheme. A real-time, dynamically updated manifold is learned using point cloud data from a lidar-like sensor on the vehicle to avoid any apriori unknown or moving obstacles. Convergence and recursive feasibility guarantees for the MPC scheme are provided under mild assumptions on the behavior of the obstacles and dynamics of the vehicle. An automated parking scenario in the presence of static and dynamic obstacles is demonstrated in simulation for the complete process of optimal trajectory planning and path following.

Next, a continuous time, path following model predictive control scheme is shown for an Airborne Wind Energy (AWE) system. Here stability and convergence guarantees are provided by combining the model predictive controller with terminal constraints inspired from a convergent vector field design problem. A formal stability proof relying on Lyapunov stability arguments is presented to show that for such a design of vector field terminal constraints the path following controller converges to a zero tracking error on the desired path.

The last part of the thesis deals with uncertainty in AWE systems due to wind conditions and unknown aerodynamic characteristics. A Gaussian process data-driven optimisation technique and a direct adaptive nonlinear controller design are presented for the same.