Computation Of Optimal Trajectories Research Articles

Control Theory and Economic Policy Optimization: The Origin, Achievements and the Fading Optimism from a Historical Standpoint

Economists were interested in economic stabilization policies as early as the 1930’s but the formal applications of stability theory from the classical control theory to economic analysis appeared in the early 1950’s when a number of control engineers actively collaborated with economists on economic stability and feedback mechanisms. The theory of optimal control resulting from the contributions of mathematicians Lev Semenovich Pontryagin and Richard Ernest Bellman in the late 1950’s was first applied successfully to models of economic growth in the 1960’s by the economists who were interested in discovering the optimality properties of economic growth trajectories. It is shown that the collaborations of control engineers with econometricians in the 1970’s on the computation of optimal state and control trajectories in econometric models were the earliest attempts to demonstrate the possibility of applying deterministic, stochastic and adaptive optimal control to the numerical solution of optimal economic policies. We have explained why the collaborations of control engineers with econometricians on formulating and computing optimum system design in macroeconomic optimal planning models failed and why the economic applications of optimal control theory have proved to be more productive in the analysis of optimality conditions in mathematical economics and not in the computation of optimal trajectories in econometric models.

An algorithm to solve boundary value problems for differential iinclusions and applications in optimal control

Convergence results for simplicia1 fixed point algorithms applied to problems in Banach-spaces enable constructive proofs of the existence of fixed points for set valued operators [14]. Boundary value problems for differential inclusions will be interpreted in this context. The resulting new algorithm allows numerical treatment of boundary value problems with Peano-typedynamics. The necessary conditions of the Pontryagin Maximum Principle are discussed in this framework, leading to a new indirect method for the computation of optimal trajectories with its focus on global convergence conditions for compact control domains.

Regularization and the computation of optimal trajectories

A completely regular form for the differential equations governing the three-dimensional motion of a continuously thrusting space vehicle is obtained by using the Kustaanheimo-Stiefel regularization. The differential equations for the thrusting rocket are transformed using the K-S transformation and an optimal trajectory problem is posed in the transformed space. The canonical equations for the optimal motion in the transformed space are regularized by a suitable change of the independent variable. The transformed equations are regular in the sense that the differential equations do not possess terms with zero divisors when the motion encounters a gravitational force center. The resulting equations possess symmetry in form and the coefficients of the dependent variables are slowly varying quantities for a low-thrust space vehicle.