Abstract
Systems of ordinary differential equations such as x ′(t) = f(t,x(t)) are routinely used today to model a wide range of phenomena, in areas as diverse as aeronautics, power generation, robotics, economic growth, and natural resources. The great success of this paradigm is due in part to the fact that it suggests a natural mechanism through which the behavior of the system can be influenced by external factors. This is done by introducing an explicit control variable in the differential equation, a time-varying parameter that can be chosen (within certain limits) so as to attain a certain goal. This leads to the controlled differential equation
We present in this chapter the celebrated Pontryagin maximum principle, a set of necessary conditions for the optimal control of such a system.
The proof of the maximum principle, given in the book of Pontryagin, Boltyanskii, Gamkrelidze and Mischenko... represents, in a sense, the culmination of the efforts of mathematicians, for considerably more than a century, to rectify the Lagrange multiplier rule.
L. C. Young (Calculus of Variations and Optimal Control Theory)
As a child, I merely knew this; now I can explain it.
David Deutsch (The Fabric of Reality)
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Notes
- 1.
As in the calculus of variations, the anonymous norm ∥ x ∥ always refers to the relevant supremum norm, in this case sup t ∈ [ a,b ]| x(t)|.
- 2.
We bow here to the usage which has come to prevail. It would be more accurate to refer to H as the pre-Hamiltonian. The true Hamiltonian of the problem is in fact M. Ah well.
- 3.
While it is clear enough how this summarizes optimal behavior, the exact meaning of this feedback law as a dynamical system is somewhat unclear. The subject of “discontinuous feedback” addresses such issues.
- 4.
We should mention that minimal-time problems with linear dynamics, of which the soft landing problem is but one example, can be studied on a systematic basis using time reversal and a technique called “backing out of the origin.” See Lee and Markus [30].
- 5.
This is the compensating assumption used by Pontryagin and his collaborators.
- 6.
The nonsmooth maximum principle fails with ∂ L in the adjoint inclusion (see Exer. 22.29).
- 7.
See Clarke-Vinter [19] for details.
References
F. H. Clarke and R. B. Vinter. Applications of optimal multiprocesses. SIAM J. Control Optim., 27:1048–1071, 1989.
E. B. Lee and L. Markus. Foundations of Optimal Control Theory. Wiley, New York, 1967.
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Clarke, F. (2013). Necessary conditions. In: Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics, vol 264. Springer, London. https://doi.org/10.1007/978-1-4471-4820-3_22
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DOI: https://doi.org/10.1007/978-1-4471-4820-3_22
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