Constraint Splitting and Projection Methods for Optimal Control of Double Integrator

Abstract

We consider the minimum-energy control of a car, which is modelled as a point mass sliding on the ground in a fixed direction, and so it can be mathematically described as the double integrator. The control variable, representing the acceleration or the deceleration, is constrained by simple bounds from above and below. Despite the simplicity of the problem, it is not possible to find an analytical solution to it because of the constrained control variable. To find a numerical solution to this problem we apply three different projection-type methods: (i) Dykstra’s algorithm, (ii) the Douglas–Rachford (DR) method and (iii) the Aragón Artacho–Campoy (AAC) algorithm. To the knowledge of the authors, these kinds of (projection) methods have not previously been applied to continuous-time optimal control problems, which are infinite-dimensional optimization problems. The problem we study in this article is posed in infinite-dimensional Hilbert spaces. Behaviour of the DR and AAC algorithms are explored via numerical experiments with respect to their parameters. An error analysis is also carried out numerically for a particular instance of the problem for each of the algorithms.

We dedicate our contribution to the memory of our friend and mentor Jonathan Borwein

Appendix

Algorithm 1b (Dykstra-b)

Steps 1–4:

(Initialization) Do as in Steps 1–4 of Algorithm 1.

Step 5:

(Stopping criterion) If $\|u^{k+1} - u^k\|{ }_{L^\infty } \le \varepsilon$, then return u ^k+1 and stop. Otherwise, set k := k + 1 and go to Step 2.

Algorithm 2b (DR-b)

Step 1:

(Initialization) Choose a parameter $\lambda \in \left ]0,1\right [$ and the initial iterate u ⁰ arbitrarily. Choose a small parameter ε > 0, and set k = 0.

Step 2:

(Projection onto $\mathcal {A}$) Set u ⁻ = λu ^k. Compute $\widetilde {u} = P_{\mathcal {A}}(u^-)$ by using (2.9).

Step 3:

(Projection onto $\mathcal {B}$) Set $u^- := 2\widetilde {u}-u^k$. Compute $\widehat {u} = P_{\mathcal {B}}(u^-)$ by using (2.22).

Step 4:

(Update) Set $u^{k+1} := u^k + \widehat {u} - \widetilde {u}$.

Step 5:

(Stopping criterion) If $\|u^{k+1} - u^k\|{ }_{L^\infty } \le \varepsilon$, then return $\widetilde {u}$ and stop. Otherwise, set k := k + 1 and go to Step 2.

Algorithm 3b (AAC-b)

Step 1:

(Initialization) Choose the initial iterate u ⁰ arbitrarily. Choose a small parameter ε > 0, two parameters α and β in $\left ]0,1\right [$, and set k = 0.

Step 2:

(Projection onto $\mathcal {A}$) Set u ⁻ = u ^k. Compute $\widetilde {u} = P_{\mathcal {A}}(u^-)$ by using (2.9).

Step 3:

(Projection onto $\mathcal {B}$) Set $u^- = 2\beta \widetilde {u}-u^k$. Compute $\widehat {u} = P_{\mathcal {B}}(u^-)$ by using (2.22).

Step 4:

(Update) Set $u^{k+1} := u^k +2\alpha \beta (\widehat {u}-\widetilde {u})$.

Step 5:

(Stopping criterion) If $\|u^{k+1} - u^k\|{ }_{L^\infty } \le \varepsilon$, then return $\widetilde {u}$ and stop. Otherwise, set k := k + 1 and go to Step 2.