Network systems are composed of dynamical entities with many interactions, where the overall behavior of the system depends on the interactions between these individual entities. Network controllability is essential to achieve the desired performance of a system. The tools from classical control theory pose challenges while addressing controllability issues of network systems due to their larger size, time varying topology in some cases and insufficient knowledge of edge weights in some other cases. This work aims to solve two optimization problems related to the network design for controllability: (i) Topology design- identify the optimum topology of a static network with higher-order node dynamics to maximize the energy-based measure, the average controllability, (ii) Input design- optimal selection of input nodes of a temporal network for structural controllability. The proposed optimization problems are combinatorial and NP-hard to solve. Using the graph-theoretical approach, we propose polynomial-time algorithms with approximation guarantees to solve the optimization problems. We validate the performance of our proposed algorithms on real-world data sets of functional brain networks for the optimal input selection problem.