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<p>In this paper we derive necessary optimality conditions for a Mayer problem involving a controlled sweeping process characterized by a moving set which is merely locally prox-regular (not necessarily smooth) and satisfies a constraint qualification condition formulated exclusively in terms of its normal vectors. We employ a penalization method based on the distance from the moving constraint, which allows convergence estimates that are uniform with respect to the control and, moreover, the strong <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$W^{1,2}$</EquationSource> </InlineEquation>-convergence of the approximating solutions. An example of nonsmooth mechanics with finite degrees of freedom is presented.</p>

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Penalization and Necessary Optimality Conditions for a Class of Nonsmooth Sweeping Processes

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  • 发表日期：

    2025年8月29日

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  • Piernicola Bettiol,
  • Giovanni Colombo,
  • Paolo Gidoni

摘要

In this paper we derive necessary optimality conditions for a Mayer problem involving a controlled sweeping process characterized by a moving set which is merely locally prox-regular (not necessarily smooth) and satisfies a constraint qualification condition formulated exclusively in terms of its normal vectors. We employ a penalization method based on the distance from the moving constraint, which allows convergence estimates that are uniform with respect to the control and, moreover, the strong W 1 , 2 $W^{1,2}$ -convergence of the approximating solutions. An example of nonsmooth mechanics with finite degrees of freedom is presented.