Optimal Control is the methodology by which a set of control laws are generated  for which a particular performance index (PI) is minimized or maximized. In an optimal control the PI may be a cost function, benefit function, or may be any other PI like Integral Square Error (ISE), Integral Absolute Error (IAE), Integral Time Square Error (ITSE) , Norm of control signals etc. 

Guaranteed Cost Control is more a robust control than optimal control. In  Guaranteed Cost Control a robust control law is developed such that the cost function associated with the closed-loop system with has an upper bound irrespective of parameter uncertainties and unmodelled dynamics. Further, the upper bound (of cost function) can be optimized by incorporating with a minimization problem. 

Yes, I agree with Prof. Raajeeb Dey. These must be highlighted. First of all Optimality and robustness are not mutually exclusive. A control system is should be both optimal and robust. Optimal controls are generally close loop. At times it may also be  open loop . For example when you design a LQR by Hamiltonian formulation, its open loop (Page no 142 D.C Naidu) but when you solve it by Riccati equation its close loop (Page no 143 D.C Naidu).  In a general optimal control problem the optimality gets priority and system may not be robust although robustness is desirable. But a Gauranteed cost control is a Robust  as well as optimal control and the robustness requirement must to satisfied along with optimality at the time of design. We can also design  Guaranteed cost control using the the PI like IAE, ITSE etc. It all depends on the objectives and requirement of the design. I exactly do not know whether guaranteed cost control problem can handle nonlinearity or not ,but  I have seen several papers on Guaranteed cost control for non linear plants.

@ Rajeeb da.. yes of course when you have so many performance criteria, like tracking ability, optimality, robustness etc. any design is a trade of among these. So based on the priority these names are given like  Optimal control and Guaranteed cost control etc. So I think with all our discussion we should conclude that Guaranteed cost control is a subclass of optimal control problem where robustness criteria is a necessary constraint to be satisfied

@Prasanta and Rajeeb: In your discussion i miss the concept of time dependence of the system under control. For optimal control you need a time independent system, either linear (PID controller for a second order system) or nonlinear. In general, an optimal controller minimizes costs for a wide range of starting conditions. But it is not necessarily stable or optimal for all conditions.

In contrast to an optimal controller, a robust control system does not minimize costs, but it "guarantees" maximum costs under all initial conditions. A simple example is a two point temperature controller. Another example are complex systems which are difficult to model - especially in the presence of unknown perturbations. In this case (in technical applications), system identification parameters are time dependent and an optimal control system can become unstable instead of optimal. That happened when we tried to control laser welding processes [1].

@Deepa Maths: In my optinion, opotimal and guaranteed cost systems have to be distinguishe by  the range of input parameters (or starting conditions):

1. In an optimal control system, cost function is minimized for a limited range of input parameters.

2. In a guaranteed cost system, a maximum cost is guarianteed for a infinite range of input parameters.

[1] Blug, A.; Abt, F.; Nicolosi, L.; Heider, A.; Weber, R.; Carl, D. et al. (2012): The full penetration hole as a stochastic process: controlling penetration depth in keyhole laser-welding processes. In: Appl. Phys. B 108 (1), S. 97–107.

Thank you for all our answer... Thank you so much... 

@Viswanath Devan Optimal Control Systems D.C. Naidu CRC press