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Minimax problems for ensembles of control-affine systems

Alessandro Scagliotti

Abstract

In this paper, we consider ensembles of control-affine systems in $\mathbb{R}^d$, and we study simultaneous optimal control problems related to the worst-case minimization. After proving that such problems admit solutions, denoting with $(Θ^N)_N$ a sequence of compact sets that parametrize the ensembles of systems, we first show that the corresponding minimax optimal control problems are $Γ$-convergent whenever $(Θ^N)_N$ has a limit with respect to the Hausdorff distance. Besides its independent interest, the previous result plays a crucial role for establishing the Pontryagin Maximum Principle (PMP) when the ensemble is parametrized by a set $Θ$ consisting of infinitely many points. Namely, we first approximate $Θ$ by finite and increasing-in-size sets $(Θ^N)_N$ for which the PMP is known, and then we derive the PMP for the $Γ$-limiting problem. The same strategy can be pursued in applications, where we can reduce infinite ensembles to finite ones to compute the minimizers numerically. We bring as a numerical example the Schrödinger equation for a qubit with uncertain resonance frequency.

Minimax problems for ensembles of control-affine systems

Abstract

In this paper, we consider ensembles of control-affine systems in , and we study simultaneous optimal control problems related to the worst-case minimization. After proving that such problems admit solutions, denoting with a sequence of compact sets that parametrize the ensembles of systems, we first show that the corresponding minimax optimal control problems are -convergent whenever has a limit with respect to the Hausdorff distance. Besides its independent interest, the previous result plays a crucial role for establishing the Pontryagin Maximum Principle (PMP) when the ensemble is parametrized by a set consisting of infinitely many points. Namely, we first approximate by finite and increasing-in-size sets for which the PMP is known, and then we derive the PMP for the -limiting problem. The same strategy can be pursued in applications, where we can reduce infinite ensembles to finite ones to compute the minimizers numerically. We bring as a numerical example the Schrödinger equation for a qubit with uncertain resonance frequency.
Paper Structure (5 sections, 16 theorems, 68 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 5 sections, 16 theorems, 68 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Preliminaries
  2. Minimax optimal control problem: existence result
  3. $\Gamma$-convergence for minimax problems
  4. Pontryagin Maximum Principle via $\Gamma$-convergence
  5. Application to quantum control and numerical examples

Key Result

Lemma 1.1

For every $u\in\mathcal{U}$, let $X_u:[0,T]\times\Theta \to \mathbb{R}^d$ be the application defined in eq:def_evol_ens collecting the trajectories of the ensemble of control systems eq:ens_ctrl_sys. Then, for every $R>0$ there exists $\bar{R}>0$ such that, if $\|u\|_{L^p}\leq R$, we have for every $(t,\theta)\in[0,T]\times \Theta$.

Figures (1)

  • Figure 1: We compared the results obtained by minimizing numerically \ref{['eq:def_qubit_fun_av']} (red), \ref{['eq:def_qubit_fun_N']} (yellow), and by considering the control prescribed by \ref{['eq:control_qubit']} (blue). The left plots represent $|\langle \psi^{\mathrm{tar}} | \psi^{\alpha}_u(T) \rangle|$ (desirably, as close as possible to $1$) as $\alpha$ ranges in $[-0.5,0.5]$, while at the right-hand side we reported the graphs of the control used to drive the ensemble. In the pictures above, we set $T=20$, and $\varepsilon_1=0.5, \varepsilon_2=0.1$, while below we set $T=50$, $\varepsilon_1=0.25, \varepsilon_2=0.08$.

Theorems & Definitions (45)

  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • Remark 1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2
  • ...and 35 more