5 . Consider the linear time - invariant dynamical system { ( 1 ) = [ _ 1 _ 2 ] * ( * ) + [ 1 ] u ( t ) . * ( + ) = [ 1 0 ] I ( t ) . with state I E F', input " EK, and output VER. The parameter " E * is constant . ( a ) For what values of a is the system ( zero - input ) asymptotically stable ?" ( `) For what values of a is the system bounded - input hounded _ output stable ?" ( c ) Which of the following statements , if any , are true ? i . If a_ I . there exists a bounded input for which the output will remain bounded . ii . If a _ O, there exists a bounded input for which the corresponding output will* grow unbounded .` lil . If a = O there exists a bounded input for which the corresponding output will grow unbounded .\\ it . If a_ O, there may exist a bounded input for which the output will remain bounded . ( I ) Consider the zero -input case , in which ult ) = O , and define the function V (I ) - FI PE , with ! =\\ [ at =\\\\. Is Vlaj & Lrapunar function ?" If so, under what conditions on It ?" Clearly justify your response . Hint : Recall that the trace of a matrix is equal to the product of it's Eigenvalues , and the determinant of a matrix is equal to the sum of its Eigenvalues