In many control problems, the control is subject to satisfy a time varying algebraic equation. In this paper, we show that if a bounded measurable control <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u_0(.)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> exists such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$g(x_0(.),u_0(.))=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$x_0(.)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a trajectory of an analytic dynamical system corresponding to the control <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u_0(.)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and g is an analytic function, then there exists an analytic trajectory x(.) corresponding to an analytic control u(.) satisfying the same constraint. The proof of this result is based on stratification of subanalytic sets.

Existence of Analytic Solutions of Algebraic Equations in a Control Problem

期刊论文
发表日期：

2025年9月27日

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Hassan Hammouri

摘要

In many control problems, the control is subject to satisfy a time varying algebraic equation. In this paper, we show that if a bounded measurable control $u_0(.)$ $u_{0} (.)$ exists such that $g(x_0(.),u_0(.))=0$ $g (x_{0} (.), u_{0} (.)) = 0$ , where $x_0(.)$ $x_{0} (.)$ is a trajectory of an analytic dynamical system corresponding to the control $u_0(.)$ $u_{0} (.)$ , and g is an analytic function, then there exists an analytic trajectory x(.) corresponding to an analytic control u(.) satisfying the same constraint. The proof of this result is based on stratification of subanalytic sets.