Singular Arcs on Average Optimal Control-Affine Problems

TL;DR

This work addresses average optimal control for control-affine systems with parameters described by a probability measure, formulating $(P)$ with $J[u]=\int_\Omega g(x(T,\omega),\omega)\,d\mu(\omega)$ and dynamics $\dot{x}=f_0(x,\omega)+f_1(x,\omega)u$. It derives a Pontryagin Maximum Principle for the ensemble setting, introducing a measurable adjoint $p(t,\omega)$ and a switching function $\Psi(t)=\int_\Omega p(t,\omega)f_1(\bar{x}(t,\omega),\omega)\,d\mu(\omega)$, and shows how bang-bang and singular arcs arise, with explicit formulas for singular controls via $\int_\Omega p[\cdot,\cdot]\,d\mu(\omega)$. For scalar controls under commutative dynamics, the singular-arc condition $\int_\Omega p[ f_0,[f_0,f_1] ]\,d\mu + \bar{u}\int_\Omega p[ f_1,[f_0,f_1] ]\,d\mu =0$ and the second-derivative relation $\ddot{\Psi}=0$ determine $\bar{u}$ when permissible. The paper validates the framework through a Sterile Insect Technique (SIT) model under uncertainty using a sample-average numerical scheme, showing convergence and strong agreement with a standard solver, thereby demonstrating practical applicability to uncertain OCPs.

Abstract

In this paper we address optimal control problems in which the system parameters follow a probability distribution, and the optimization is based on average performance. These problems, known as Riemann-Stieltjes optimal control or optimal ensemble control problems, involve uncertainties that influence system dynamics. Focusing on control-affine systems, we apply the Pontryagin Maximum Principle to characterize singular arcs in a feedback form. To demonstrate the practical relevance of our approach, we apply it to the sterile insect technique, a biological pest control method. Numerical simulations confirm the effectiveness of our framework in addressing control problems under uncertainty.

Figures (3)

• Figure 1: Relative distances of the cost functional between successive iterations.
• Figure 2: Relative distances of the control between successive iterations.
• Figure 3: In dashed red, the optimal control calculated with the formula \ref{['control_formula']}. In grey, the optimal control calculated with the GEKKO library. In blue the switching function $\Psi$ given by \ref{['switching_func']}.

Theorems & Definitions (9)

• Lemma III.1
• proof
• Theorem III.2: Existence of an Optimal Control
• proof
• Definition IV.1: $W^{1,1}$-local minimizer
• Theorem IV.1
• Lemma V.1
• Theorem V.2: Characterization of the optimal control
• proof