Introduction:
In control systems, the design of an optimal controller plays a crucial role in achieving desired system performance. For a linear time invariant (LTI) system, certain conditions must be satisfied in order to design an optimal controller. Let's explore the given options and understand why option 'B' is the correct answer.

Explanation:

a) The system is unstable but observable:
- An unstable system is inherently difficult to control, as it tends to diverge over time.
- Observability refers to the ability to determine the internal state of the system based on its output.
- It is challenging to design an optimal controller for an unstable system, even if it is observable, because the system itself is not stable. Therefore, option 'A' is not the correct answer.

b) The system is controllable and observable:
- Controllability refers to the ability to steer the system from any initial state to any desired state using control inputs.
- Observability refers to the ability to determine the internal state of the system based on its output.
- For a linear time invariant system, an optimal controller can be designed if the system is both controllable and observable.
- Controllability allows us to manipulate the system inputs to achieve the desired performance, while observability enables us to estimate the system's internal states for feedback control.
- Therefore, option 'B' is the correct answer.

c) The system is stable and unobservable:
- Stability indicates that the system's output remains bounded for any bounded input.
- Unobservability means that it is not possible to determine the internal state of the system based solely on its output.
- Although the system is stable, it is not possible to estimate the internal states, which hinders the design of an optimal controller. Hence, option 'C' is not the correct answer.

d) The system is uncontrollable but stable:
- Uncontrollability refers to the inability to steer the system from certain initial states to desired states using control inputs.
- Stability indicates that the system's output remains bounded for any bounded input.
- Even if the system is stable, the lack of controllability restricts the design of an optimal controller. Therefore, option 'D' is not the correct answer.

Conclusion:
For a linear time invariant system, the correct answer is option 'B' - the system must be both controllable and observable in order to design an optimal controller. Controllability allows for manipulation of the system inputs, while observability enables estimation of the system's internal states. Both factors are essential for the design of an optimal controller to achieve desired system performance.