Conventional guidance laws are designed based on Lyapunov theorems on asymptotic stability or exponential stability. They will guide the line-of-sight angular rate to converge to zero or its small neighborhood, however, only as time approaches infinity. In this paper, new guidance laws with finite convergent time are proposed. The guidance laws are obtained based on new sufficient conditions derived in this paper for the finite time convergence of the line-of-sight angular rate. It is proved that, with the guidance laws, the line-of-sight angular rate will converge to zero or a small neighborhood of zero before the final time of the guidance process. Furthermore, such guidance laws will ensure finite time convergence and finite time stability in both the planar and three-dimensional environments. Simulation results show that the guidance laws are highly effective. Nomenclature a Mr , a M , a M = missile acceleration along the line-of-sight axes a r , a , a = relative acceleration along line-of-sight axes a Tr , a T , a T = target acceleration along line-of-sight axes N = navigation ratio q = line-of-sight angle _ q = derivative of q with respect to time q = second-order derivative of q with respect to time r = relative range _ r = derivative of r with respect to time r = second-order derivative of r with respect to time t = time u = missile acceleration normal to line of sight u r = missile acceleration along line of sight V M = missile velocity V T = target velocity w = target acceleration normal to line of sight w r = target acceleration along line of sight x M , y M , z M = position coordinates of missile in inertial frame x T , y T , z T = position coordinates of target in inertial frame = azimuth = elevation ' M = flight-path angle of missile ' T = flight-path angle of target M = heading angle of missile T = heading angle of target

The control parameterization method is a popular numerical technique for solving optimal control problems. The main idea of control parameterization is to discretize the control space by approximating the control function by a linear combination of basis functions. Under this approximation scheme, the optimal control problem is reduced to an approximate nonlinear optimization problem with a finite number of decision variables. This approximate problem can then be solved using nonlinear programming techniques. The aim of this paper is to introduce the fundamentals of the control parameterization method and survey its various applications to non-standard optimal control problems. Topics discussed include gradient computation, numerical convergence, variable switching times, and methods for handling state constraints. We conclude the paper with some suggestions for future research.

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In this paper, we develop a computational method for a class of optimal control problems where the objective and constraint functionals depend on two or more discrete time points. These time points can be either fixed or variable. Using the control parametrization technique and a time scaling transformation, this type of optimal control problem is approximated by a sequence of approximate optimal parameter selection problems. Each of these approximate problems can be viewed as a finite dimensional optimization problem. New gradient formulae for the cost and constraint functions are derived. With these gradient formulae, standard gradient-based optimization methods can be applied to solve each approximate optimal parameter selection problem. For illustration, two numerical examples are solved.