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Two-interval single and multiple shooting for x-prime equals u

Example

For $\dot{x} = u$ , $x (0) = 0$ , and mesh $0, 1 / 2, 1$ , take the compact control set $U_{c} = [- 1, 1]$ and let the two constant controls satisfy $v_{0}, v_{1} \in U_{c}$ . Since the dynamics are constant on each interval, every consistent explicit RK tableau with $\sum_{i} b_{i} = 1$ and stage abscissae $c_{i} \in [0, 1]$ gives the exact interval updates on this mesh. Single shooting yields $x_{1} = v_{0} / 2$ and $x_{2} = (v_{0} + v_{1}) / 2$ . Multiple shooting retains $x_{1}, x_{2}$ and imposes $x_{1} - v_{0} / 2 = 0$ , $x_{2} - x_{1} - v_{1} / 2 = 0$ . The shared objective $\frac{1}{4} (v_{0}^{2} + v_{1}^{2}) + \frac{1}{2} (x_{2} - 1)^{2}$ therefore has the same reduced value in both formulations.

Facts & Assumptions

Given: The two-cell mesh, compact control set $U_{c} = [- 1, 1]$ , constant controls $v_{0}, v_{1} \in U_{c}$ , exact real arithmetic, and a consistent explicit tableau with $\sum_{i} b_{i} = 1$ and $c_{i} \in [0, 1]$ .

[F1]

Piecewise-constant controls assign one control value per mesh cell (Time meshes and piecewise-constant control parameterization).

[F2]

The single and multiple shooting recursions use the same deterministic interval maps (Finite-horizon shooting and Runge–Kutta collocation transcription).

[F3]

Their feasible control sets and reduced objectives agree by projection (Exact single and multiple shooting have the same reduced feasible controls).

[A1]

The explicit tableau is consistent, meaning $\sum_{i} b_{i} = 1$ ; all arithmetic shown is exact.

Verification

1.1F1A1givenalgebra

On the first cell every stage slope equals $v_{0}$ , so its endpoint increment is $\frac{1}{2} v_{0} \sum_{i} b_{i} = \frac{1}{2} v_{0}$ . The second cell adds $\frac{1}{2} v_{1} \sum_{i} b_{i} = \frac{1}{2} v_{1}$ . Thus single shooting gives the stated $x_{1}$ and $x_{2}$ .

2.1F2F3step 1.1algebra∎

The multiple-shooting continuity equations uniquely impose $x_{1} = \frac{1}{2} v_{0}$ and $x_{2} = x_{1} + \frac{1}{2} v_{1} = \frac{1}{2} (v_{0} + v_{1})$ . Substitution into its objective gives exactly the single-shooting objective. This is the concrete bijection and reduced-objective identity stated in [F3].

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Sources

S. Gros and M. Diehl, Numerical Optimal Control, draft course manuscript, Chapter 13 §§13.1–13.2 (standard reference, not scraped)
J. Monnier, Optimal Control of ODEs: The Linear-Quadratic (LQ) Case, course notes, Chapter 2 §2.4 (standard reference, not scraped)