The Calculus of Variations and Optimal Control
When the Tyrian princess Dido landed on the North African shore of the Mediterranean sea she was welcomed by a local chieftain. He offered her all the land that she could enclose between the shoreline and a rope of knotted cowhide. While the legend does not tell us, we may assume that Princess Dido arrived at the correct solution by stretching the rope into the shape of a circular arc and thereby maximized the area of the land upon which she was to found Carthage. This story of the founding of Carthage is apocryphal. Nonetheless it is probably the first account of a problem of the kind that inspired an entire mathematical discipline, the calculus of variations and its extensions such as the theory of optimal control. This book is intended to present an introductory treatment of the calculus of variations in Part I and of optimal control theory in Part II. The discussion in Part I is restricted to the simplest problem of the calculus of variations. The topic is entirely classical; all of the basic theory had been developed before the turn of the century. Consequently the material comes from many sources; however, those most useful to me have been the books of Oskar Bolza and of George M. Ewing. Part II is devoted to the elementary aspects of the modern extension of the calculus of variations, the theory of optimal control of dynamical systems.
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Integration of the EulerLagrange Equation
An Inverse Problem
The Weierstrass Necessary Condition
Jacobis Necessary Condition
Regular Optimal Trajectories
Examples of Extremal Control
adjoint equations admissible control bang-bang bang-bang control belongs calculus of variations class C1 conditions of Theorem consequence Consider the integral continuous function control constraint set corresponding cost function deduce an extremal defined denote derivative desired to transfer Dido's problem differential equation discontinuous Discuss extremal control discussed in Section end conditions end points Euler Euler-Lagrange equation example Exercise exists feasible functions of class furnishes Furthermore given initial global minimum hence illustrated in Figure implies instance interval Jacobi's equation Lagrange equation Lemma linear linearly independent mass flow rate maximize maximum principle minimizing function namely necessary conditions obtain optimal control optimal feedback control optimal trajectory Pareto-optimal piecewise continuous piecewise smooth functions prescribed problem of Section proof regular interior point respect rocket satisfied subinterval sufficiency theorem sufficient conditions suppose switching function switching sequence tangent plane target set terminal transversality condition terminal values time.optimal tions unique vector weak local minimum Weierstrass zero
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