This report documents synthesis of control law using Lyapunov functions for a range of nonlinear systems. Finding Lyapunov functions for a nonlinear system is a non-trivial task and to overcome this challenge, we have considered the Lyapunov function to be a Sum of Squares (SOS) polynomial. Using this approach, the problem of finding a suitable Lyapunov function is posed as an Semidefinite Program (SDP) which can be solved using a suitable solver. In this project, we have used results from Algebraic Geometry specifically Putinar's Positivstellensatz so as to restrict the search of Lyapunov function to a semialgebraic set.

A major drawback in using this approach is scalability to bigger problems. As the number of states increase, the size of the SDP increases as the computational time grows polynomial in number of states and exponentially based on the degree. Thus, practical implementation of this approach becomes difficult and we have used a sparse version of Putinar's Positivstellensatz so as to overcome the aforementioned challenge in practical implementation. We begin by finding Lyapunov function for Van der Pol's model of a nonlinear oscillator. Thereafter, we have considered complex systems such as a wind turbine and Orsted Satellite and found Lyapunov function using this approach. Finally, based on the obtained Lyapunov function we have attempted, methods for Nonlinear control design such as Sontag's formula and Lyapunov Redesign.

Keywords - Nonlinear systems, Lyapunov function, Sum of Squares, Semidefinite Programming, Semialgebraic sets, Putinar's Positivstellensatz, Sparse Putinar's Positivstellensatz, computational complexity