To illustrate how actually to solve a control problem, in this last section we introduce some ad hoc calculus and geometry methods for the rocket car problem,  Example 5 in(EXAMPLES).

First of all, let us guess that to find an optimal solution we will need only to consider the cases a = 1 or a = −1. In other words, we will focus our attention only upon those controls for which at each moment of time either the left or the right rocket engine is fired at full power.

CASE 1: Suppose first that α ≡ 1 for some time interval, during which

And so

Let t₀ belong to the time interval where α ≡ 1 and integrate from t₀ to t:

In other words, so long as the control is set for α ≡ 1, the trajectory stays on the curve v² = 2q + b for some constant b.

CASE 2: Suppose now α ≡ −1 on some time interval. Then as above

And hence

Let t₁ belong to an interval where α ≡ −1 and integrate:

Consequently, as long as the control is set for α ≡ −1, the trajectory stays on the curve v² = −2q + c for some constant c.

GEOMETRIC INTERPRETATION. Formula (1.1) says if α ≡ 1, then (q(t), v(t))  lies on a parabola of the form

                                   v²= 2q + b.

Similarly, (1.2) says if α ≡ −1, then (q(t), v(t)) lies on a parabola

                                    v²= −2q + c.

References

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مواضيع ذات صلة

DYNAMIC PROGRAMMING-DYNAMIC PROGRAMMING AND THE PONTRYAGIN MAXIMUM PRINCIPLE

DYNAMIC PROGRAMMING-EXAMPLES

DERIVATION OF BELLMAN,S PDE-DYNAMIC PROGRAMMING

THE PONTRYAGIN MAXIMUM PRINCIPLE-MORE APPLICATIONS

THE PONTRYAGIN MAXIMUM PRINCIPLE-MAXIMUM PRINCIPLE WITH STATE CONSTRAINTS

THE PONTRYAGIN MAXIMUM PRINCIPLE-MORE APPLICATIONS

THE PONTRYAGIN MAXIMUM PRINCIPLE-MAXIMUMPRINCIPLEWITH TRANSVERSALITY CONDITIONS

THE PONTRYAGIN MAXIMUM PRINCIPLE-APPLICATIONS AND EXAMPLES

THE PONTRYAGIN MAXIMUM PRINCIPLE-STATEMENT OF PONTRYAGIN MAXIMUM PRINCIPLE

THE PONTRYAGIN MAXIMUM PRINCIPLE-REVIEW OF LAGRANGE MULTIPLIERS.

THE PONTRYAGIN MAXIMUM PRINCIPLE-CALCULUS OF VARIATIONS, HAMILTONIAN DYNAMICS

LINEAR TIME-OPTIMAL CONTROL-EXAMPLES