By Dingle R.

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Geophys. Res. R. Grasso, D. Sornette: J. Geophys. Res. 1 Introduction: The Brachistochrone Problem In this chapter we focus our attention on the open loop control of deterministic problems. We will see that the language of deterministic control theory is close to the language of classical mechanics. The deterministic control theory requires that the dynamics of the system under control is completely deﬁned by well-deﬁned equations of motion and accurate initial conditions. Although the theoretical description is not inﬂuenced by the degree of complexity of the system, the subsequently presented methods are useful if the system has only a few degrees of freedom.

This may be illustrated by a simple example. 10) and the given boundary conditions. Obviously, we will always succeed with this procedure, at least for a suﬃciently large class of functions u(t). In the future we deﬁne that such a pair {X(t), u(t)} is said to be admissible for the control problem. 16) for any suﬃciently small ε. In what follows we call the small diﬀerences δX(t) = X(t)−X ∗ (t) and δu(t) = u(t)−u∗ (t), respectively, variations around the (weak) minimum {X ∗ (t), u∗ (t)}. A weak minimum is not necessarily stable ˙ against arbitrary velocity variations δ X(t) or strong variations of the control function δu(t).

1 Introduction: The Brachistochrone Problem In this chapter we focus our attention on the open loop control of deterministic problems. We will see that the language of deterministic control theory is close to the language of classical mechanics. The deterministic control theory requires that the dynamics of the system under control is completely deﬁned by well-deﬁned equations of motion and accurate initial conditions. Although the theoretical description is not inﬂuenced by the degree of complexity of the system, the subsequently presented methods are useful if the system has only a few degrees of freedom.