For a robot, say path planning in particular, what are the pros and cons of choosing classical control theory or optimal control (LQR for example) ?
Classical control theory requires Linear Time Invariant systems. Most robots of interest actually have non-linear dynamics. Of course many optimal control techniques also require linear systems. LQR for instance stands for Linear Quadratic Regulator meaning it takes a quadratic cost function and gives the optimal control for a system with linear dynamics. This is handled in both cases by linearizing the dynamics. In the case of classical control techniques linearizing the dynamics invalidates all the guarantees and the tools for analyzing stability and tuning gains generally don't work. In the case of optimal control methods then generated control will often still be good so long as the approximation error is low. Often it helps to simply try the different methods. Alternatively you may consider non-linear control methods.
Robotic path planning is a field unto itself. There are people doing research on integrating optimal nonlinear model based predictive control with path planning, but those tools aren't really ready for prime-time yet. A good place to get acquainted with this field would be to skim through Steven Lavalle's free path planning book:
After that you should have some idea what sort of algorithm you want to learn more about, hunt for existing code, implement your own code, etc...
Please see Lavalle's book for more details, in the meantime some basic terminology....
A path is a sequence of robot poses. A trajectory is sequence of time-pose pairs. Path planning creates a path, and control methods (classical, optimal, adaptive, etc) produce trajectories.
For example, when you plan your route from home to work, you come up with roads that you would take. That's your path, and you just did path planning. When you embark on the journey, you are at the particular place and the particular time. That's your trajectory.
A difference is in the toolset you used to produce path and trajectories. You planned the path based on the goal and the physical size of your car. You created a trajectory by following the path (reducing the error) and by pushing gas and brake pedals, and turning the steering wheel.
Hope this helps.
Besides the clear answer that helps distinguish planning from control, I would stress the point that classical approaches based on pole placement (e.g. PID) and optimal control techniques (e.g. LQR) respond to different design requirements:
The former methods are more oriented to vary the transient dynamics of the closed-loop system (faster/slower response) so as they allow for a principled way to reject disturbances; hence, key features are: frequency analysis, transient response, steady-state errors, robustness margins, ease of gain tuning.
The latter methods focus much more on meeting some given integral goals, that is to comply for example with objectives such as maximum level of control effort or final displacement attained in the minimum time that are all computed over the whole trajectory; thus, key features are: bounds on control output, state feedback, performance on the whole.
However, pole placement techniques are preferable for the difficulties encountered in LQR design of setting up proper cost functions, as described in wikipedia.