(Solved): [15 Marks] Consider the electrical circuit shown below. (a) Using KVL, KCL, the inductor law, the ...

[15 Marks] Consider the electrical circuit shown below. (a) Using KVL, KCL, the inductor law, the capacitor law and Ohm's law, write down a state space model for this circuit of the form \[ \dot{x}=A x+B u \] where the state vector is \( x=\left[\begin{array}{c}i_{L} \\ V_{C}\end{array}\right] \), and \( u \) is the applied voltage. [5 Marks] (b) For the parameter values \( R=1 \Omega L=0.1 H, C=0.1 F \), find the eigenvalues of the matrix \( A \) and determine if the system is asymptotically stable for \( u=0 \). [4 Marks] (c) For the parameter values in part (b), find a quadratic Lyapunov function for the system. [6 Marks]