The state and output equation of a system are as under state

1. neither state controllable nor output controllable
2. state controllable but not output controllable
3. output controllable but not state controllable
4. both state controllable and output controllable

Given state equation:

x₁

(t)

x₁

(t)

u(t)

x₂

(t)

-1

-2

x₂

(t)

C(t) = [1 1]		x₁	(t)
		x₂	(t)

Here,

A =		0	1
		-1	-2

B =		0
		1

C = [1 1]
Check for controllability

AB =		0	1			0		=		1
		-1	-2			1				-2

∴ Q_C = [B: AB] = – 1, which is non-singular.
Hence, the state equation is controllable.
Check for observability :

A =		0	1
		-1	-2

then

A^T =		0	-1
		1	-2

C = [1 1]
then

C^T =		1
		1

Now,

A^TC^T =		0	-1			1
		1	-2			1

=		-1
		-1

θ₀ = [C^T : A^T C^T]

A^TC^T =		1	-1
		1	-1

= 0 i.e., singular.
Hence given system equation is not observable.
Therefore alternative (B) is the correct choice.