We consider the ensemble controllability problem for a linear time-invariant system <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\dot{x}(t,\theta )=A(\theta )x(t,\theta )+B(\theta )u(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where A and B are continuous matrices with respect to the parameter <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>, which belongs to some compact set <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Theta \subset \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Θ</mi> <mo>⊂</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. Given any continuous initial state datum <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\theta \mapsto x^0(\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>↦</mo> <msup> <mi>x</mi> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and any continuous target state <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\theta \mapsto x^1(\theta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>↦</mo> <msup> <mi>x</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we investigate the numerical computation of a <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-independent open loop control u such that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$x^0$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>x</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation> is steered, in a given time <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$T>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, at a distance <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$x^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>x</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> in the uniform norm (with respect to the parameter). We approach the problem both theoretically and numerically. Using the Fenchel–Rockafellar duality, we first prove the existence and uniqueness of the ensemble control of a minimal <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm. The numerical recovery of the optimal control is obtained by solving the dual problem, which consists in the unconstrained minimization of a non-differentiable functional in the space of Radon measures.

Numerical Aspect of Uniform Ensemble Controllability for Linear Systems

期刊论文
发表日期：

2026年8月1日

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Martin Lazar,
Jérôme Lohéac

摘要

We consider the ensemble controllability problem for a linear time-invariant system $\dot{x}(t,\theta )=A(\theta )x(t,\theta )+B(\theta )u(t)$ $\dot{x} (t, θ) = A (θ) x (t, θ) + B (θ) u (t)$ , where A and B are continuous matrices with respect to the parameter $\theta $ $θ$ , which belongs to some compact set $\Theta \subset \mathbb {R}$ $Θ \subset R$ . Given any continuous initial state datum $\theta \mapsto x^0(\theta )$ $θ \mapsto x^{0} (θ)$ and any continuous target state $\theta \mapsto x^1(\theta )$ $θ \mapsto x^{1} (θ)$ , we investigate the numerical computation of a $\theta $ $θ$ -independent open loop control u such that $x^0$ $x^{0}$ is steered, in a given time $T>0$ $T > 0$ , at a distance $\varepsilon >0$ $ε > 0$ of $x^1$ $x^{1}$ in the uniform norm (with respect to the parameter). We approach the problem both theoretically and numerically. Using the Fenchel–Rockafellar duality, we first prove the existence and uniqueness of the ensemble control of a minimal $L^2$ $L^{2}$ norm. The numerical recovery of the optimal control is obtained by solving the dual problem, which consists in the unconstrained minimization of a non-differentiable functional in the space of Radon measures.